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John Baez has been shared in 324 public circles

You can see here the 50 latest shared circles.
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AuthorFollowersDateUsers in CircleCommentsReshares+1Links
Becky Collins16,609Science Circle :Circle of very #social #engagerspeople and companiesTo be included in my shares (#sharedcircle), be so kind to:1 - Do +1 t the post2 - Comment the post and specify your "category" (job or interest) Ex: Fashion, SEO, Companies, Social Media Marketing, Sailing, Photography, Bloggers/Writers, Web graphics and design, Italy, Artists, Sport, Finance/Economy ...3 - include the circle among your circles4 - share the circle (include yourself)Improve your popularity, be social be cool !Keep yourself updated, enjoy the Shared Circles Hellenic Alliance, you can share your shared circles inside the upcoming Community:https://plus.google.com/communities/112552559573595396104  #socialmedia  #media  #circles   #circleshare   #circlesharing  #circlecircle   #beckyscircle   #sharedcircles   #sharedpubliccircles  #sharedcircleoftheday  +Becky Collins ?2014-09-16 05:24:00459102
Richard Green81,215Engagers Showcase Circle, September 14, 2014If I sent you a notification, it means that you are included in my Engagers Showcase Circle. “Showcase” means that you are invited to leave a comment (on the original post) with a link to one of your own posts, which ideally should be one of your best recent posts.This circle consists of people who have engaged with one of my recent posts in the form of +1s, comments and reshares. I skipped over one post because it received too much engagement, but I'm including a link to it for completeness.Everyone mentioned below is also included in the circle.Do nuclear physicists have half life crises?https://plus.google.com/101584889282878921052/posts/ayw6WPGGaFESt Peter's Church, Heyshamhttps://plus.google.com/101584889282878921052/posts/9DEtmbdz15zSmiling cow?https://plus.google.com/101584889282878921052/posts/9NuqPpsgtBkThe look-and-say sequence and Conway's Cosmological Theoremhttps://plus.google.com/101584889282878921052/posts/jEQ7zxFpJt4Cordyline australis, the “cabbage tree”https://plus.google.com/101584889282878921052/posts/hktDAgyo6mA“Maurits, stop picking at it. You'll only make it worse.” by David Swarthttps://plus.google.com/101584889282878921052/posts/gj327Ywh33T“Phyllotactic Portrait of Fibonacci” by Robert Boschhttps://plus.google.com/101584889282878921052/posts/8LykdvHpRvPFountain in Williamson Parkhttps://plus.google.com/101584889282878921052/posts/HzZTLQaQ9RT“Youth” by Silvia Cordeddahttps://plus.google.com/101584889282878921052/posts/T2Lo3c2zLxvThe arithmetic derivative, the Goldbach conjecture, and the twin prime conjecturehttps://plus.google.com/101584889282878921052/posts/9nY35Ma1pbUGlobe Thistlehttps://plus.google.com/101584889282878921052/posts/i8mtiyVikWhTallinn (reshared from +Paul Harper)https://plus.google.com/101584889282878921052/posts/U4DAQxK5fkxCubes passing in the night (reshared from +Sean Walker)https://plus.google.com/101584889282878921052/posts/32JKvAFqP9SThe graph of arctanhttps://plus.google.com/101584889282878921052/posts/FLvyDupud1z“Hopf Knott” by Peter Sittner2014-09-14 15:25:57463212113212
Cableicous2,882Thanks for all teh cablezThanks muchly to all those who have contributed their #Cableicous  imagery for  this 14th circle of 99 people who have contributed their cableicous grandeur - your continued support of my cable fetish is much enjoyed.And a new circle of 99 begins...#photography #cables #cableicous #circleshare #2014 #cableriacirculus2014-09-10 10:47:199912442
Becky Collins15,950Dance Related Circle :Circle of very #social #engagerspeople and companiesTo be included in my shares (#sharedcircle), be so kind to:1 - Do +1 t the post2 - Comment the post and specify your "category" (job or interest) Ex: Fashion, SEO, Companies, Social Media Marketing, Sailing, Photography, Bloggers/Writers, Web graphics and design, Italy, Artists, Sport, Finance/Economy ...3 - include the circle among your circles4 - share the circle (include yourself)Improve your popularity, be social be cool !Keep yourself updated, enjoy the Shared Circles Hellenic Alliance, you can share your shared circles inside the upcoming Community:https://plus.google.com/communities/112552559573595396104  #socialmedia  #media  #circles   #circleshare   #circlesharing  #circlecircle   #beckyscircle   #sharedcircles   #sharedpubliccircles  #sharedcircleoftheday  +Becky Collins ?2014-09-02 05:00:4347130935
EDZUL FREDY KRISNAWAN0Meet the People who WILL...Take This Circle To The TopLet 'er Rip! You guys and gals are SUPERSTARS!!!Want to be added to the #hyperadd?1) Add me to your circles.2) Share, +1, and Comment This Share3) Reshare anything that interests you on my stream (profile) from today or the rest of this week.#circlesharing   #sharedcircles   #circles   #circlemaster  2014-08-29 11:28:1048511413
Wendy Thanh Hồng43GOOGLE FRIENDS! -  RESHARE if you want to be included *'"*:•:••:*:•-:¦:*  *SHARE AND BE  SHARED*  *:¦:-•:*:••-:•:''''*  This is a super Circle and in it I put together a group of really interesting and active people on Google Plus to add in your circles.I'm talking about the top   Google + users that share unique and original contents.Follow   this advice and grow your G+ community with people that share amazing content that will surprise you:boost   visibility on Google+ - Share the circle!If you want to be added to the next Circle you have to do these simple steps:1 - Include me in your circles 3 - Share the circle (Publicly) 4 - Add +1 to the post 5 - Follow  your dreams and smile to life.More you share More you get! :)I will thankful if you plus and share this circle!#circles #shared #share #add #friends #circle #share #sharecircle #circleshare2014-08-25 05:58:2348611315
Kieu Trinh0GOOGLE FRIENDS! -  RESHARE if you want to be included *'"*:•:••:*:•-:¦:*  *SHARE AND BE  SHARED*  *:¦:-•:*:••-:•:''''*  This is a super Circle and in it I put together a group of really interesting and active people on Google Plus to add in your circles.I'm talking about the top   Google + users that share unique and original contents.Follow   this advice and grow your G+ community with people that share amazing content that will surprise you:boost   visibility on Google+ - Share the circle!If you want to be added to the next Circle you have to do these simple steps:1 - Include me in your circles 3 - Share the circle (Publicly) 4 - Add +1 to the post 5 - Follow  your dreams and smile to life.More you share More you get! :)I will thankful if you plus and share this circle!#circles #shared #share #add #friends #circle #share #sharecircle #circleshare2014-08-25 05:34:0348613619
Cableicous2,902Thanks for all teh cablezThanks muchly to all those who have contributed their #Cableicous  imagery for another grand week of beautaliciousness.And a new week begins...#photography #cables #cableicous #circleshare #2014 #cableriacirculus2014-08-09 22:10:58406119
Richard Green77,832Engagers Showcase Circle, August 7, 2014If I sent you a notification, it means that you are included in my Engagers Showcase Circle. “Showcase” means that you are invited to leave a comment (on the original post) with a link to one of your own posts, which ideally should be one of your best recent posts.This circle consists of people who have engaged with one of my recent posts in the form of +1s, comments and reshares.Everyone mentioned below is also included in the circle.Our cat, Chesterhttps://plus.google.com/101584889282878921052/posts/ToxRHsMHytsFibonacci numbers and corridors of width 4https://plus.google.com/101584889282878921052/posts/gCTyaSV4ugzWalk in the rainhttps://plus.google.com/101584889282878921052/posts/gVPzuv7aKHALenticular cloud (reshared from +Sean R. Heavey)https://plus.google.com/101584889282878921052/posts/giTgt4PUd1GGlass Paperweight by Paul Stankardhttps://plus.google.com/101584889282878921052/posts/fLfKWxFj3f2“Mathematistan” by Martin Kuppehttps://plus.google.com/101584889282878921052/posts/AcUBb8Y9uBjCat's back on the menu, boys!https://plus.google.com/101584889282878921052/posts/aP3cZEnaqquWaterfall (reshared from +Keith Boone)https://plus.google.com/101584889282878921052/posts/S2pmsTTyiZzOak tree at “The Pig”https://plus.google.com/101584889282878921052/posts/cv8pi2ffX1NThe Bargate, Southamptonhttps://plus.google.com/101584889282878921052/posts/LsYSEpUS1bLCosmos flowerhttps://plus.google.com/101584889282878921052/posts/KqvLW32KyXfThe Ashton Memorialhttps://plus.google.com/101584889282878921052/posts/aK1E3XqWWSSThrough the castle windowhttps://plus.google.com/101584889282878921052/posts/MT7uBM2SUt7Friedman numbershttps://plus.google.com/101584889282878921052/posts/32tzjfB8NnMThe Norfolk Knifehttps://plus.google.com/101584889282878921052/posts/DHf4jfSkUKKThe lake at Wyresdale Parkhttps://plus.google.com/101584889282878921052/posts/3xFACaympiNCastle of the Clouds2014-08-07 21:46:51451213123238
Nick Warner645Check out these awesome Crowdfunders2014-08-07 16:05:31470001
Refurio Anachro4,492This july engagers circle comes packed with curiosity, brought to life by questions and answers, and people sharing their enthusiasm. Take your chance, get acquainted to this party of very nice people and deep thinkers, add it now!+John Baez had asked for an inside view of a mirror ellipsoid i was happy to provide. The result's actually a spheroid because it has a symmetry axis. Since then, i did quite some staring at ellipsoids, see below... https://plus.google.com/b/115434895453136495635/115434895453136495635/posts/5zcrptKx3C3This reshare of +Xah Lee's "math is programing" rant got me into ugly. Gracefully handling distractions would be nice to have more of. If you must know, i heard one can retrieve deleted comments using the search...https://plus.google.com/b/115434895453136495635/115434895453136495635/posts/TZpAUuQfzSn“Mathematistan” by Martin Kuppe offers innocuous glances aside from the popular maths mainstream. Thanks +Richard Green for sharing and noting similarities to the all time favorite "Hitchikers guide to the galaxy"!https://plus.google.com/b/115434895453136495635/115434895453136495635/posts/UnXikyENJo1A raytraced perspective down a mirror cylinder showing quite a lot about billiard trajectories on a bunch of elliptic tables. Learn what Birkhoff, Poincare, and Poncelet had to do with it...https://plus.google.com/b/115434895453136495635/115434895453136495635/posts/Vmrx7GMRe2iReshare of 6d toroid animations, thanks +Owen Maresh and +Cornus Ammonis for sharing!https://plus.google.com/b/115434895453136495635/115434895453136495635/posts/gSma1ksKFeiThis ellipsoid glossary and coordinate system came with puzzles! Special thanks to +Bruce Elliott for taking part in the fun. Apparently i didn't add a clean solution putting everything together at the end... More to come, stay tuned!https://plus.google.com/b/115434895453136495635/115434895453136495635/posts/L6C4Kob2bNeLast june's engagers circlehtt2014-08-03 23:22:2311314310
Marius Kiupelis279 If you received this notification you are in this circle♚♚  If you want to be shared in this circle ♚♚♚♚♚♚♚ Then just keep sharing! ♚♚♚♚♚2014-08-03 19:05:53201036
Lynda Chervil890Add this circle of excellent engagers, thinkers, innovators, and future tech leaders. My ++Solar Power++ Circle will provide you with the latest and greatest from the world of tech innovations, especially in the renewable energy sector. Simply add this circle and then share it!If you'd like to be added to this circle, please comment below, share and add the circle. Only those who qualify with content and expertise will be added. Thanks! #Tesla   #nikolatesla   #elonmusk   #solarenergy   #innovation   #technology   #solarpower   #hydroelectric   #science   #STEM   #research   #education   #futuretech   #futuretechnology  2014-07-31 14:24:32343125
Peter Edenist31,0152014 Super Sci-FI Circle : No, the Sky is not Falling!!! Also the Gravity is not sucking you in... please +1 this post to support it or you may have to take a trip in Snowpiercer, no need to thank me. Please reshare if you think this is a worthwhile circle. If you have been notified, you are in the circle!All the people in this circle are linked to our community (see link further down). As usual, please tag and recommend anyone who you think should be in this circle. Live long and prosper!Sci-FI Community here: http://goo.gl/s1NVd  Science Fiction Pics: http://goo.gl/sOSPK5Mighty Shiny Browncoats : http://goo.gl/9osg1tDoctor Who : http://goo.gl/z3uWX3Ultimate Star Wars : http://goo.gl/Wu8bv6Ultimate Star Trek : http://goo.gl/JJPql9Science on G+ community here: http://goo.gl/46uFH #sciencefiction #sf #scifi  2014-07-24 12:58:0746211960145
Becky Collins13,434Mobile Operator Circle:Circle of very #social #engagerspeople and companiesTo be included in my shares (#sharedcircle), be so kind to:1 - Do +1 t the post2 - Comment the post and specify your "category" (job or interest) Ex: Fashion, SEO, Companies, Social Media Marketing, Sailing, Photography, Bloggers/Writers, Web graphics and design, Italy, Artists, Sport, Finance/Economy ...3 - include the circle among your circles4 - share the circle (include yourself)Improve your popularity, be social be cool !Keep yourself updated, enjoy the Shared Circles Hellenic Alliance, you can share your shared circles inside the upcoming Community:https://plus.google.com/communities/112552559573595396104  #socialmedia  #media  #circles   #circleshare   #circlesharing  #circlecircle   #beckyscircle   #sharedcircles   #sharedpubliccircles  #sharedcircleoftheday  +Becky Collins ?2014-07-24 05:16:124763112
Able Lawrence100,166100K Engagers Celebration Circle Thank you all those who have followed me and engaged with my posts and taking me to the milestone of 100,000 followers. Goolge+ has been an exhilarating journey of 3 years and I would like to thank all those who have followed me and engaged on my posts whether they were on Science or Birds or Technology. The circle has been created using  * +Circloscope * which is the work of +Ehsan Ahmadi Gharacheh All of you are free to share your favorite posts in the comments and also reshare this circle. If you are included in the circle, you will get a notification.  2014-07-17 17:22:18340491787
Doug Hyatt5,196THIS IS MEGA CIRCLE 5    #DOUGHYATTCIRCLESHARING IT ADDED 1,000+ FOLLOWERS FOR ME IN A WEEK.1) ADD THE CIRCLE2) RESHARE THE CIRCLE AND BE SURE TO REMEMBER TO CHECK THE BOX WHICH INCLUDES YOURSELF.3)RESHARE TO PUBLIC EVERY DAY!BE SURE TO COPY THE HASHTAG LIST AND ADD IN YOUR COMMENTS LIKE I HAVE!  IT WILL GIVE YOU MAXIMUM RESULTS#doughyattcirclesharing#circles   #circlemania   #circletoday   #circleplus   #circleshare   #circlesharing   #CircleSharing   #sharecircle   #sharecirclethis   #worldsharecircle   #pleasecircleaddme   #circleoftheday   #newgenerationcircle  #monstercircle   #usacircle   #bestofthecircle   #shareofthecircle   #pleasecircle   #newbestcircl  add circle add circle * best share circle * share please pleaseplease add circle * best share circle * share please please#add #circle   #addcircle   #pleasecircle   #bestcircle   #mostcircle  #bestsharecircle   #newcircle   #top100circle   #best500circle   #top500circle   2014-07-16 15:49:10477151421
Richard Green74,896Engagers Showcase Circle, July 12, 2014If I sent you a notification, it means that you are included in my Engagers Showcase Circle. “Showcase” means that you are invited to leave a comment (on the original post) with a link to one of your own posts, which ideally should be one of your best recent posts.This circle consists of people who have engaged with one of my recent posts in the form of +1s, comments and reshares.Everyone mentioned below is also included in the circle.Thistlehttps://plus.google.com/101584889282878921052/posts/CKB7ks1VWZ3“Star Gate” by +Gary Matthews https://plus.google.com/101584889282878921052/posts/ean83YFix7sBubble (reshared from +Ann-Marie Jurek)https://plus.google.com/101584889282878921052/posts/PkybKJfXv14Gabriel's Hornhttps://plus.google.com/101584889282878921052/posts/MXNTuq8ZEBPHibiscus flowerhttps://plus.google.com/101584889282878921052/posts/Xsmw62v98CHStackable 12-sided dicehttps://plus.google.com/101584889282878921052/posts/5mLoukdZAA5Astronomical book priceshttps://plus.google.com/101584889282878921052/posts/S15LTNxJA9GPyritohedral symmetryhttps://plus.google.com/101584889282878921052/posts/XQUYMrKGiAQSt Stephen's Basilica (reshared from +Laura Orange)https://plus.google.com/101584889282878921052/posts/ApnSMBsUWFeCircular Pascal Arrays and Fibonacci numbershttps://plus.google.com/101584889282878921052/posts/NVqYKoRVnHg“Fungi tree” by +Tom Beddardhttps://plus.google.com/101584889282878921052/posts/J6S4bDc6QwQAs always, reshares of this circle are appreciated, and I look forward to seeing everyone's links. Thanks for reading my posts!2014-07-12 05:31:36476209126225
Refurio Anachro4,312June engagers circle: Say hello to almost 200 nice people who joined in to participate in math topics. If you're intelligent, curious, like well written posts and engaged discussion - these people will not dissappoint. Go on and add them now!Last month's posts:Spin story continues (part 3): Angular velocity and tilings in spin space. While i locked myself in a cycle, struggling to follow, +wendy krieger blew the bubble and lifted the party up to dimension 4! I couldn't resist but to guest post this gem. Next time i'll find a picture first, promised!:https://plus.google.com/115434895453136495635/posts/6F9sxQxiqEaClifford Algebra (part 1): Into Clifford algebra. Because with those one can easily describe spin in any dimension.https://plus.google.com/115434895453136495635/posts/1CtiDXou6RqUnicode math cheat sheet: ⁵√7+x² – ℵ₀ – ℝⁿ – x̅ ≟ x*⊕x⁎ – αβ ≺ ∇ϕhttps://plus.google.com/115434895453136495635/posts/GNoCEmBxSWSClifford Algebra (part 2): Multiplication in geometric algebra.https://plus.google.com/115434895453136495635/posts/NKqs5BwsCqqCardioid gears, a reshare of +Rob Kook's very unexpected video find:https://plus.google.com/115434895453136495635/posts/UMBXJihPgcaCan we get anything faster than light that way? While arguing below +John Baez' cool post i suddenly started hearing vikings...https://plus.google.com/115434895453136495635/posts/jjYu4KhHaCTClifford Algebra (part 3): Cayley graphs and a strange clock. My first attempt to quickly put my thoughts down for you to see. I guess i'll have to practice a bit to make it more appealing on shorter glance, so more people can benefit.https://plus.google.com/115434895453136495635/posts/bkHQAZLzpt8Last may's engagers circle:https://plus.google.com/115434895453136495635/posts/hGPebLWRcU2You're in this circle because you reshared, plussed or commented on one of my posts (possibly via +Spherical Reflections), or got into a discussion with me. Thank you!#engagers #engagerscircle #engagersshowcasecircle  #circleshare #circlesharing #circl2014-07-02 13:23:15194659
Peter Edenist30,747TESLA Super Science Circle 2014 - June Edition!!!! : A quality circle curated carefully and tended since 2011. Who is in this circle?  people who are either involved in science, love science or are active on the Science on G+ community. Community link is here http://goo.gl/46uFH and the Page is +Science on G+!Have you wondered where one can meet with like-minded people who love science and like to read about the real stuff? This is the ultimate SUL on science.... add and share to keep it alive.--------------------------------------------------------------------------------------------------Want to nominate someone? Please do so in comments. If you want to recommend something let us know. People who are notified are part of the circle.#science #tesla #scienceongplus #scienceeveryday 2014-06-19 13:39:49342755693
David Wees4,496This is a circle of 346 math educators. If you are interested in following the work of innovative mathematics educators, this circle is for you.If you are a mathematics educator, and I have not included you in this circle, let me know.2014-06-11 16:24:03346010
Dina Tika0Here is a group of Active Engagers, Circle Sharers, Awesome Plus Oners, and Cool People on Google Plus!   Circle Sharing is an awesome way to increase your followers and active engagers on your profile. Some of my favorite people that I've met here on Google + through Circle Sharing.    Want to be in the next Circle of Awesomeness? Follow the Steps Below!  ☛ Add the circle ☛ Share in the Public ☛ Plus 1 the Post. ☛ Comment. 2014-06-10 05:53:52479001
Refurio Anachro4,141May engagers circle: Meet the people who made the fabulous cheer of this week's spin party. Many curious people, all of them worth to add!Diagram 18: rotations and spinorshttps://plus.google.com/b/115434895453136495635/115434895453136495635/posts/dpPMfoH6zc5This is actually part two, but got posted earlier (see below). You just made my first +100, thank you! Stay tuned for more on spinning and spinors.https://plus.google.com/b/115434895453136495635/115434895453136495635/posts/TLkA6tCNwEeIf you want to know what an earlier stage of Diagram 18 looked like, see here. I posted 2/3rd of the final post with a question marking my progess. Also note the discussion below, thanks to everybody who helped me out!https://plus.google.com/b/115434895453136495635/115434895453136495635/posts/THePbpcPnqDYou're in this circle because you reshared, plussed or commented on one of my posts (possibly via +Spherical Reflections), or got into a discussion with me. Thank you!Last april's engagers circlehttps://plus.google.com/b/115434895453136495635/115434895453136495635/posts/RFMY2aYscJJ#engagers #engagerscircle #engagersshowcasecircle  #circleshare #circlesharing #circle #circleoftheday  #addcircle #findcircles #publicsharedcircles  #addpeople #plussers #awesomepeople #sciencecircle  #commenterscircle #mathematicians #friends 2014-06-02 11:57:02158447
John Nuntiatio37,49330k+ Circle - A circle worth adding and followingThe people from this circle have more than 30,000 followers.2014-06-02 08:54:46140496
Becky Collins10,282Mobile Circle :Circle of very #social #engagerspeople and companiesTo be included in my shares (#sharedcircle), be so kind to:1 - Do +1 t the post2 - Comment the post and specify your "category" (job or interest) Ex: Fashion, SEO, Companies, Social Media Marketing, Sailing, Photography, Bloggers/Writers, Web graphics and design, Italy, Artists, Sport, Finance/Economy ...3 - include the circle among your circles4 - share the circle (include yourself)Improve your popularity, be social be cool !Keep yourself updated, enjoy the Shared Circles Hellenic Alliance, you can share your shared circles inside the upcoming Community:https://plus.google.com/communities/112552559573595396104  #socialmedia   #media   #circles   #circleshare   #circlesharing   #circlecircle   #beckyscircle   #sharedcircles   #sharedpubliccircles   #sharedcircleoftheday  +Becky Collins ?2014-05-28 05:03:174777219
Jasmina Brozović53,426Notification CircleThis is a circle of people I don't wanna miss any new posts from!Great Circle idea by +Zvonimir FrasRead more :https://plus.google.com/u/0/+ZvonimirFras/posts/VsZjS7H7vJf#circles #circlethis #circleyoushare #sharedcircles #addcircle #sharedpubliccircles #circleshare #publicsharedcircles #circleoftheweek #circlesharing #publiccircle #sharedcircle #circleshare   #circle     #motivateme 2014-05-20 16:26:14179271031
Daniel Zawadzki7,897This #circle is #great, #amazing, #wonderful,Boost your visibility on Google+ - Share the circle!#1 - Click add people and create your circle#2 - share the circle (include yourself)#3 - add +1 to the post#circle   #circleshare   #circlesharing   #circleoftheday   #circleoftheweek   #share   #sharedcircles   #sharedpubliccircles   #sharedcircleoftheday   #engagers   #engagerscircle   #engagerspeople   2014-05-19 20:33:554709446123
Refurio Anachro4,070April engagers circle, a ball of intelligent, nice, and responsive people. Have a look!You're in this circle because you reshared, plussed or commented on one of my posts (possibly via +Spherical Reflections), or got into a discussion with me. Thank you!The following has been classic april fool's, of course. Works best on kids %-]https://plus.google.com/115434895453136495635/posts/SA2aNCZPrQkNot being a big fan of synchronous communication, i will try and challenge you in case you do want to hang out. That's a rather new ploy of my character, and it seems i still have to learn a lot on how to do this nicely...https://plus.google.com/115434895453136495635/posts/fM8DtSPKL2MI have been seen spreading rumours with +David Roberts, speculating that this year's Fields medal might go to Manjul Bhargava. Here for the fullest explanation, thanks again David:https://plus.google.com/115434895453136495635/posts/KUbagU2C7C7A raytraced landscape featuring two spherical mirrors. It's about dynamical billiards (chaos theory).https://plus.google.com/115434895453136495635/posts/BCduuGFS75LWhen i saw +Richard Green's post about Coxeter-Conway friezes i couldn't resist to divert my readers to have fun counting paths with +DrJamesTanton. Fibonacci inside:https://plus.google.com/115434895453136495635/posts/gKBcK1HNdMvEverybody should have played at least once with an interactive IFS demo! If you haven't, or want to try again - here's a nice one by +Felix Woitzel:https://plus.google.com/115434895453136495635/posts/N9HeysDgTzi #engagers   #engagerscircle   #engagersshowcasecircle   #circleshare   #circlesharing   #circle   #circleoftheday   #addcircle   2014-05-16 11:46:1281867
Richard Green68,138Engagers Showcase Circle, May 12, 2014If you received a notification, it means that you are included in my Engagers Showcase Circle. “Showcase” means that you are invited to leave a comment (on the original post) with a link to one of your own posts, which ideally should be one of your best recent posts.This circle consists of people who have engaged with one of my recent posts in the form of +1s, comments and reshares.Everyone mentioned below is also included in the circle.Peruvian glacier (reshared from +Irina Tcherednichenko)https://plus.google.com/101584889282878921052/posts/Z6NR7zaZa5qGrape hyacinthshttps://plus.google.com/101584889282878921052/posts/GZoHH2bnRUiSymmetries of a regular octagonhttps://plus.google.com/101584889282878921052/posts/bFcGcsurF5HHappy Easter!https://plus.google.com/101584889282878921052/posts/YkWFA9WmxVAPerfect Squared Squareshttps://plus.google.com/101584889282878921052/posts/HDU4mWD3wptOur dog Rexhttps://plus.google.com/101584889282878921052/posts/5JtEcnt5MpPPerrott's Follyhttps://plus.google.com/101584889282878921052/posts/8DzXHLn6fe4The Lonely Runner Conjecturehttps://plus.google.com/101584889282878921052/posts/YHJkEdKg3jz“Owl” by Andrea Mininihttps://plus.google.com/101584889282878921052/posts/12uh9J9MjMXPenrose nanotileshttps://plus.google.com/101584889282878921052/posts/7PZPSLaFkAx“Interlaced Stars” by Magnenauthttps://plus.google.com/101584889282878921052/posts/5vNjv1yjVMqBlack Forest View (reshared from +Ralph Reichert)https://plus.google.com/101584889282878921052/posts/YKPsvCFDyUv“Hippie Bus Apartments” by +Richie Montgomeryhttps://plus.google.com/101584889282878921052/posts/iCMXCBVzjd2Not crazy enoughhttps://plus.google.com/101584889282878921052/posts/d5EeDSoHe5u“Impossible Sphere” by Alexey Ermushevhttps://plus.google.com/101584889282878921052/posts/J2yUMw6YASqBeethoven, high on 2014-05-12 05:07:06489214129231
Cindy Wilson02014-05-10 14:34:25371700
Pamela Baskett02014-05-10 13:35:433711505
DRESS3,901If you want to be added to the next Circle you have to do these simple steps:1 - Include me in your circles 2 - Share this post3 - Add +1 to the post 4 - Leave a comment if you have done the 3 steps above#circle   #circleshare  #circle #circleshare #circle #circleshare #circle #circleshare #circle #circleshare #circle #circleshare #circle #circleshare #circle #circleshare #circle #circleshare #circle #circleshare #circle #circleshare #circle #circleshare #circle #circleshare #circle #circleshare #circle #circleshare #circle #circleshare #circle #circleshare #circle #circleshare #circle #circleshare 2014-05-05 12:20:413723411
Graeme McRae24,994Graeme's Circle of Active GooglePlussers! . . . This is a curated circle of active people on Google+* If you add this circle, you will see hundreds of new posts per day in your stream from this circle alone. It's a circle of people I started following a few months ago, and who have consistently posted something at least once a week during all of weeks since then. If you want an interesting stream, you should add this circle. If you're interested in how I curated this circle, and why I think it makes for a really great stream, read on. . .How did these people get in the circle?. . . They must have posted something interesting, at least once, because I don't follow people unless they post something interesting. By now, you're probably not surprised by the number of people on G+ whose pages are full of repeated posts, videos without comments, "what's up" and other uninteresting things. I don't follow those people.How did they stay in the circle since they were added?. . . After I add someone, I keep them in the circle of people I added around the same time. Then, several times a week, I go through all my circles and delete people who haven't posted (something visible to me) in the last 7 days. So the people who are left in the circle are consistent posters. That says something important about them, because consistency is a proxy for quality.Wait. What? Why is consistency is a proxy for quality?. . . I'll admit it: I don't have the science to back up that wild statement, but I'll tell you why I believe it to be true. People who post crap don't get a lot of followers, and the few followers they have don't bother to engage. After a while, the crappy-posting people start to think G+ is a ghost town, and they lose interest. On the other hand, the people who share (or re-share) interesting things get followers and engagement, slowly at first, but ever increasing, and this outpouring of love is what keeps them going. That's what I think anyway, and I'm sticking to it.People who share interesting things get lots of followers. . . This is another half-baked theory of mine, but hear me out. You might think this makes sense: Google+ offers suggestions of people to follow, especially to new users. Google+'s only source of names to suggest are other G+ers like yourself. The G+ers who have the most engagement are the ones Google+ knows will be good to follow, so these people get put out as suggestions more often, and some percentage of those suggestions are taken up. So the more engagement you have, the more Google+ will suggest you as a person to follow, and so the more followers you'll get. Completing the theory: the more interesting the stuff you share (or reshare), the more engagement you'll get, and hence the more followers you'll get.How can I get in your circle?. . . Don't bother trying to "get into" this circle. This circle isn't for you to get into. Think of it the other way around. This circle is for you to add, so you can follow a bunch of interesting and consistent people. The stream from this one circle will be chock-full of fascinating content for you to reshare. People will start to notice your reshares, and they will engage with you, and then Google+ will "notice" (it's a computer, but let's anthropomorphise, shall we?) you and suggest you to others, who will then follow you.No, really. How can I get in your circle?. . . First, unfollow me, then wait, oh, about a week. Spend that week posting a shitload of interesting things. Then follow me. Since you're so interesting, Google+ will notify me that you followed me (G+ doesn't notify me about every follower). I'll look at your posts, find one interestin2014-05-04 21:03:52371111319
nicholas syahputra821. Plus The Post2. Comment3. Add People To Circles4. Share The Circle!#circlesharing #circleshare #circles #circle #googleplustips #googleplus #indonesia #artists #artist #artistphotographeramateurorprofessional2014-04-30 07:03:1548531842
Ryan Johnson3,346My new some cool, fun & interesting people CirclePlease reshare it#circles           #circlescirclescircles           #circleshare         #circleshare2014-04-25 06:14:362357820
Becky Collins8,289Engagers Showcase Circle :Circle of very #social #engagerspeople and companiesTo be included in my shares (#sharedcircle), be so kind to:1 - Do +1 t the post2 - Comment the post and specify your "category" (job or interest) Ex: Fashion, SEO, Companies, Social Media Marketing, Sailing, Photography, Bloggers/Writers, Web graphics and design, Italy, Artists, Sport, Finance/Economy ...3 - include the circle among your circles4 - share the circle (include yourself)Improve your popularity, be social be cool !Keep yourself updated, enjoy the Shared Circles Hellenic Alliance, you can share your shared circles inside the upcoming Community:https://plus.google.com/communities/112552559573595396104  #socialmedia  #media  #circles   #circleshare   #circlesharing  #circlecircle   #beckyscircle   #sharedcircles   #sharedpubliccircles  #sharedcircleoftheday  +Becky Collins ?2014-04-23 05:59:02400201526
Circlemania !!!2,173in this circle you can find people keen to publish, share, track and unr to good communities. see and discover new friends and circles that may interest you. Do not forget to follow me as: +Andres Aguilar  #circle   #circleshare   #circlemaina   #circlemeup  2014-04-19 15:48:55498261934
Richard Green64,997Engagers Showcase Circle, April 17, 2014If you received a notification, it means that you are included in my Engagers Showcase Circle. “Showcase” means that you are invited to leave a comment (on the original post) with a link to one of your own posts, which ideally should be one of your best recent posts.This circle consists of people who have engaged with one of my recent posts in the form of +1s, comments and reshares. Two of the posts below attracted too much engagement to be usable for the circle, but I am including links to them for completeness.Everyone mentioned below is also included in the circle.Adding cubeshttps://plus.google.com/101584889282878921052/posts/WqimoVTZWL3Flower and snail shell (reshared from +Ralph Reichert)https://plus.google.com/101584889282878921052/posts/5st1n4VBMh5April Fools' Dayhttps://plus.google.com/101584889282878921052/posts/Mxa1TNjYo4r“Trinity” by Jen Starkhttps://plus.google.com/101584889282878921052/posts/gfRheBovSZUHyperbolic honeycombs (reshared from +Roice Nelson)https://plus.google.com/101584889282878921052/posts/59yPx8yoFbuCoxeter—Conway friezeshttps://plus.google.com/101584889282878921052/posts/4XvYb9biaLpTriple helix staircase (reshared from +Romain Brasselet)https://plus.google.com/101584889282878921052/posts/SRJ7CqGNrtLWhich airport is closest to you?https://plus.google.com/101584889282878921052/posts/FxxdDb6orPy“Spinning Cosmos” by Paul Friedlanderhttps://plus.google.com/101584889282878921052/posts/RjnVfRhBPmZStrč prst skrz krkhttps://plus.google.com/101584889282878921052/posts/h87PZWs2MSbMersenne primes and perfect numbershttps://plus.google.com/101584889282878921052/posts/cLWBTddiSyy“Urge to fly” by Hermin Abramovitchhttps://plus.google.com/101584889282878921052/posts/j2K97AvLvAoKeeper of Time (reshared from +Michael Quinn)https://plus.google.com/101584889282878921052/posts/QnaHrgKjsUUSeven mutual2014-04-17 04:49:32487185132233
Ryan Johnson2,352sharedcircle   #publiccircleshare     #circleshare     #circleoftheday  #circleshares     #circlesharing     #circlesshare    #publicsharedcircle   #publicsharedcircles   #sharedcircleoftheday   #sharedcircleoftheweek     #sharedcircles     #sharedpubliccircle   #todayspublicsharedcircle  #todayssharedcircle     #publiccircle   #circle     #circles    #awesome   #awesomeness   #awesomepeople   #shareyourcircle     #bestengagers   #followers   #followback     #awesomecircles  #topsharedcircle   #topsharedcircle     #myseoissocial     #besocial     #socializethesocial     #trust   #circles   #sharedcircle     #followers      #social     #socialnetworking     2014-04-16 05:12:352629212
Adam Black6,732Transhumans CircleBy popular Demand,  I am sharing some of my Circles. This is +Mark Bruce  Transhumans Circle  Plus Others Ive been adding.If you are tagged,  you are in it.Web Link for Mobile https://plus.google.com/u/0/+AdamBlack/posts/6iQmLpEj1Zm #sharedcircles   #GeekCulture   #ScienceSunday #Transhumans #Transhumanism #HumanPlus  2014-04-13 06:25:0424522733
John Nuntiatio33,101#circles #sharedcircles #circlesharing #circle #circleoftheday #bestofcircles #bestcircleshare #share #tech #business #sport #health #tennis #europe   #asia   #america   #africa   #australia   #football   #golf   #basketball   #news  2014-04-09 06:25:5632961213
Refurio Anachro3,603You're in this march engagers circle because you reshared, plussed or commented on one of my posts (possibly via +Spherical Reflections), or got into a discussion with me. It's been fun being with you and i'm looking forward to more of it soon! Thank you!Mar 4, 2014 - links a couple of my favorite videos featuring John Conway on the occasion of +Numberphile posting a new one:https://plus.google.com/115434895453136495635/posts/AHpMJ79biC3Mar 12, 2014 - reshared an excellent post by +Gerald Stuhrberg about Richard Couranthttps://plus.google.com/115434895453136495635/posts/CzMomxKK2HzMar 18, 2014 - Diagram 17: Bratteli diagram about C★ algebrashttps://plus.google.com/115434895453136495635/posts/6EiftoFjkxmMar 19, 2014 - reshared a four dimensional platonic covered with gears by Greg Egan, thanks +John Baezhttps://plus.google.com/115434895453136495635/posts/NeaNS4XDBXNMar 22, 2014 - linked a blog post by keegan, an animated quiascrystal with a simple recipe to get one. Don't miss the link to Roger Penroses latest public lecture video in the commentshttps://plus.google.com/115434895453136495635/posts/8RUh6oQzMdpMar 31 - a link to +Vladimir Voevodsky's latest public lecture, and to an implementation of the addictive 2048 game in Coq!https://plus.google.com/115434895453136495635/posts/9bnhXDeH1UXIf you're looking for even more, please have a look at my trusty +Spherical Reflections's page. I reshare interesting circles there, and occasionally post mandelbrot beach pictures. #engagers   #circle , #circles   #circleshare   #circlesharing   #sharedcircles   #publiccircle   2014-04-03 11:25:3978165
John Nuntiatio32,205#circles #sharedcircles #circlesharing #circle #circleoftheday #bestofcircles #bestcircleshare #share #tech #business #sport #health2014-04-03 06:40:30128164
Jean-Marc Luna36,938My buddies (432/1)of the G+ villageI am happy to share (finally) my first big circles (432/1 and 432/2) with you. It is also a way to thank you and to inform you of my consideration.You will find in these two circles all the people I follow regularly since my first steps on Google+ and which accompany me for soon two years in this beautiful adventure, bringing every day, by their sympathy, their humor, their knowledge, their generosity and their open-mindedness, a bit extra to my everyday life.I do not know everybody, I have crossed the path with some, I communicate more with some and many people became friends, thing which I did not expect by landing here.When I began, I was lucky enough to discover new interesting people, thanks to the system of circles sharing, and I hope that these both will be for you of the same utility.If you add a “plus”, maybe it will have an impact on climate change,if you share this circle, maybe Putin is going to get married with an ukrainian women in 2014,if you include me in your circles, maybe it will solve the problem of hunger in the world,Maybe ... but in fact, nothing is sure ;)A good day to all :)With a special thought for #dirktalamasca   #sharedcircles   #circles   #circlesharing  2014-03-31 15:25:36432622658
Daniel Zawadzki6,461Very soccial Circle#specialcircle      #topengagers   #circleoftheday     #saturdaycircle #weekendcircleHi friends! This is very imoprtant circle, most of top g+ pplif you want to add this circle:1 - include all circle in your circles 3 - shared the circle (include yourself) 4 - add +1 to the post More you share more you are great.2014-03-22 09:41:38472571572
Enrico Altavilla5,522I'm sharing with you my Science circle. It's focused mainly on physics, astrophysics and math but it also contains subjects (both people and pages) who share news about other scientific fields. Enjoy! 2014-03-21 06:45:031031015
Richard Green61,119Engagers Showcase Circle, March 20, 2014If you received a notification, it means that you are included in my Engagers Showcase Circle. “Showcase” means that you are invited to leave a comment (on the original post) with a link to one of your own posts, which ideally should be one of your best recent posts.This circle consists of people who have engaged with one of my recent posts in the form of +1s, comments and reshares. I ran out of room, so my most recent posts will be included next time instead.Everyone mentioned below is also included in the circle.Michelangelo and the Jordan curve theoremhttps://plus.google.com/101584889282878921052/posts/UN8TetdX48VThe Kelpieshttps://plus.google.com/101584889282878921052/posts/M6aWXwu6zESAntoine's necklacehttps://plus.google.com/101584889282878921052/posts/4anPfBNTyjXThe Farey-Ford tessellationhttps://plus.google.com/101584889282878921052/posts/GrZP3ajnRGUAlien Garden (reshared from +Kevin Clift)https://plus.google.com/101584889282878921052/posts/j55RpNfocNABetween a rock and a hard placehttps://plus.google.com/101584889282878921052/posts/WGNdNFneW1uAlan Turing: a universal Turing machine?https://plus.google.com/101584889282878921052/posts/MUjAnVtL7hbVisual artist Aliza Razellhttps://plus.google.com/101584889282878921052/posts/hbd3e6U1fhSBeet it (reshared from +Laura Ockel)https://plus.google.com/101584889282878921052/posts/U7RjV31mM7WChess played on a sphere... and other animalshttps://plus.google.com/101584889282878921052/posts/LGXUE73Z3MoPass 'Em Onhttps://plus.google.com/101584889282878921052/posts/iFCQcXfhNdDArizona sunset (reshared from +Debashish Samaddar)https://plus.google.com/101584889282878921052/posts/CR1yJ48g8CWThe inside of my fridge, through Glasshttps://plus.google.com/101584889282878921052/posts/SQK6KoFL2rmBatmobile 1?https://plus.google.com/101584889282878921052/posts/aV43AZenAd9Mary O'Malley ceramics (reshared from 2014-03-20 19:09:36469199157230
John Nuntiatio30,154#circles #sharedcircles #circlesharing #circle #circleoftheday #bestofcircles #bestcircleshare #share #tech #business #sport2014-03-18 08:53:47471025
John Nuntiatio30,154#circles #sharedcircles #circlesharing #circle #circleoftheday #bestofcircles #bestcircleshare #share #tech #business #sport2014-03-18 08:53:18471001
John Nuntiatio30,154#circles #sharedcircles #circlesharing #circle #circleoftheday #bestofcircles #bestcircleshare #share #tech #business #sport2014-03-18 08:52:50471125

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Top posts in the last 50 posts

Most comments: 219

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2014-08-03 09:59:26 (219 comments, 70 reshares, 244 +1s) 

My last post on the NASA "quantum vacuum plasma thruster" was mainly about the shoddy theory behind it - like how there's no such thing as a "quantum vacuum plasma".

But you could argue: hey, if the gizmo actually works, isn't that good enough?   

Unfortunately, the experiment has problems too.  In brief:

1.  They tested a device that was designed to work and one that was designed not to work.  They both worked.

2.  They tested the devices in a "vacuum chamber", but they didn't take the air out.

3.  They didn't carefully study all possible causes of experimental error... like their devices heating the air.

In a bit more detail:

1.  Their device, called the Cannae drive, was invented by a guy named Guido Fetta.  You can see a picture below.  It's not complicated! It's a... more »

Most reshares: 682

posted image

2014-08-10 02:51:19 (169 comments, 682 reshares, 554 +1s) 

Both rectangles are moving at constant speed

At least that's what the creator of this illusion says!  It looks like the yellow and blue rectangles are taking turns going forward - one step at a time.

This is an illusion that's so good it's hard to believe it's an illusion. When the black and white lines disappear, it's easy to see the rectangles are moving at constant speed.  But before that they seem to be taking turns, and pausing when they reach each new line.

Could the creator of this illusion be cheating - fooling you into thinking there's an illusion?   How can you tell, except by making your own version of this animated gif?

Hide one rectangle with your hand.  Then look closely at the other.  Try not to look at the black and white lines.  I think you'll see the rectangle is moving at constant speed.

Butif you... more »

Most plusones: 599

posted image

2014-08-13 03:16:36 (57 comments, 250 reshares, 599 +1s) 

Maryam Mirzakhani won the Fields medal yesterday.

As a child in Tehran, she didn't intend to become a mathematician - she just wanted to read every book she could find!  She also watched television biographies of famous women like Marie Curie and Helen Keller.  She started wanting to do something great... maybe become a writer.

She finished elementary school while the Iran-Iraq war was ending, and took a test that got her into a special middle school for girls.  She did poorly in math her first year, and it undermined her confidence.  “I lost my interest in math," she said.

But the next year she had a better teacher, and she fell in love with the subject.  She and a friend became the first women on Iranian math Olympiad team.  She won a gold medal the first year, and got a perfect score the next year.

After getting finishing her undergraduate workat Shar... more »

Latest 50 posts

posted image

2014-09-18 04:11:22 (36 comments, 30 reshares, 133 +1s) 

Packing tetrahedra

What's the densest way to pack regular tetrahedra?  Aristotle, after staying up too late grading Alexander the Great's homework, once claimed they could fill space completely.  But that's clearly false.

Here's the story.  To save space, I'll use tet to mean 'regular tetrahedron'.

In 1976, a guy named Hoylman showed that if you have tets centered at points in a lattice, all pointing the same way, the best density you can get is

18/49  ≈  36.73%

That's lousy - spheres can do 74.05%.  But Hoylman's work was good, because he corrected an earlier false claim by Minkowski, who was a genius when it came to lattices.

In 2006, Conway and Torquato made a big breakthrough.  First they packed 20 tets into an icosahedron - there's a beautiful easy way to do this, since anicosahed... more »

Packing tetrahedra

What's the densest way to pack regular tetrahedra?  Aristotle, after staying up too late grading Alexander the Great's homework, once claimed they could fill space completely.  But that's clearly false.

Here's the story.  To save space, I'll use tet to mean 'regular tetrahedron'.

In 1976, a guy named Hoylman showed that if you have tets centered at points in a lattice, all pointing the same way, the best density you can get is

18/49  ≈  36.73%

That's lousy - spheres can do 74.05%.  But Hoylman's work was good, because he corrected an earlier false claim by Minkowski, who was a genius when it came to lattices.

In 2006, Conway and Torquato made a big breakthrough.  First they packed 20 tets into an icosahedron - there's a beautiful easy way to do this, since an icosahedron has 20 triangles as faces.   You're left with a hole in the middle, but it's not very big.  Then they packed icosahedra as densely as they could.  This is the hard part.  But using this combination of tricks, they packed tets with a density of

71.65%

This is still a bit worse than spheres.  Much earlier, the brilliant mathematician Stanislaw Ulam had conjectured that the maximum density for packing equal-sized spheres was worse than for any other convex shape in 3 dimensions.  This conjecture is still open!  But if he's right, tets must be able to beat spheres.

In 2007, a bunch of people showed experimentally that you could get tets to beat spheres - they got densities of around 75%.  And in 2008, Elizabeth Chen figured out how to make a cluster of 18 tets, and then pack these clusters, to get a density of

77.86%

The race has picked up since then!  I won't tell the whole story, since it's quite long.  But this picture shows the current record, held by Elizabeth Chen, Michael Engel and Sharon Glotzer. 

These folks used Monte Carlo simulations to help them pack N tets into a cluster and then pack these clusters as densely as possible. 
I'm only showing three cases here, but their paper shows what they get up to N = 16. 

With N = 10 you get two wagon wheels.  With N = 11 you get something complicated.  With N = 12 you get 6 dimers arranged in a certain way.  And so on.  So far the winners in the density contest are N = 4, 8, 12 and 16.  These all give a density of

4000/4671  ≈   85.63%

So, this may be the best we can do!  But nobody has proved that.

All this stuff is actually related to physics, since now people can make 'fluids' of tiny hard tetrahedra.  And in 2009, some people showed that at high enough densities such a fluid will spontaneously transforms to a dodecagonal quasicrystal, which can be compressed to a density of 83.24%.  They did it using Monte Carlos simulations.

If you ask what are the practical application, I'll tell you: math like this isn't mainly about practical applications!  It's mainly about having fun while developing our ability to solve hard problems.

However, the same kinds of 'Monte Carlo optimization methods' used to tackle this problem are also important for keeping our economy humming ever faster as we burn more and more carbon, chop down rainforests, overfish the oceans and generally wreck the environment.  So don't complain - we mathematicians are playing our part!   :-)

Puzzle: I mentioned the numbers N = 4, 8, 12 and 16.   What does that suggest?

You can see more pictures of the best known packing here:

http://blog.wolfram.com/2010/08/30/tetrahedra-packing/

For more, try this:

• Jeffrey C. Lagarias and Chuanming Zong, Mysteries in packing regular tetrahedra,  AMS Notices, December 2012, 1540-1549, http://www.ams.org/notices/201211/rtx121101540p.pdf.

and this, where I got the picture:

• Elizabeth R. Chen, Michael Engel and Sharon C. Glotzer, Dense crystalline dimer packings of regular tetrahedra, Discrete and Computational Geometry 44 (2010), 253–280, http://arxiv.org/abs/1001.0586.

and this:

https://en.wikipedia.org/wiki/Tetrahedron_packing

#spnetwork arxiv:1001.0586 #geometry #packing  ___

posted image

2014-09-17 02:58:44 (42 comments, 41 reshares, 130 +1s) 

Packing spheres

What's the densest way to pack spheres?  Here are two equally good ways.

In fact there are infinitely many equally good ways!  We start by laying spheres on the plane in a hexagonal arrangement, as tightly as we can.  Then we put a second layer like this on top, with the new spheres resting in the gaps between the old ones.  Then we put on a third layer.  But now there are 2 really different ways to do it! 

The spheres in the third layer can be directly above the spheres in the first layer - that's the picture at right.  Or they can be not directly above - shown at left. 

As we continue, we keep getting more choices. 

One systematic choice is to make the layers alternate like ABABAB.....  That's called the hexagonal close packing, and that's how crystals of magnesium work.

Anothersystemati... more »

Packing spheres

What's the densest way to pack spheres?  Here are two equally good ways.

In fact there are infinitely many equally good ways!  We start by laying spheres on the plane in a hexagonal arrangement, as tightly as we can.  Then we put a second layer like this on top, with the new spheres resting in the gaps between the old ones.  Then we put on a third layer.  But now there are 2 really different ways to do it! 

The spheres in the third layer can be directly above the spheres in the first layer - that's the picture at right.  Or they can be not directly above - shown at left. 

As we continue, we keep getting more choices. 

One systematic choice is to make the layers alternate like ABABAB.....  That's called the hexagonal close packing, and that's how crystals of magnesium work.

Another systematic choice makes every third layer be the same, like ABCABC...  That's called the cubic close packing or face-centered cubic, and that's how crystals of lead work.

(Why "cubic?"  Because - even though it's not obvious! - you can also get this pattern by putting a sphere at each corner and each face of a cubical lattice. Trying to visualize this in your head is a great way to build your brain power.)

There are also uncountably many unsystematic ways to choose how to put down the layers of spheres, like ABACBCAC....  You just can't use the same letter twice in a row.

 In 1611, the famous astronomer Kepler conjectured that sphere packings of this sort were the densest possible.  They fill up

π / 3 √2  =  0.740480489...

of the space, and he claimed you can't do better. 

Proving this turned out to be very, very hard.  Wu-Yi Hsiang claimed to have a proof in 1993.   It was 92 pages long.  Experts said it had gaps (pardon the pun).  Hsiang has never admitted there's a problem.

Thomas Hales claimed to have a proof in 1998.   His proof took 250 pages... together with 3 gigabytes of computer programs, data and results!  

The famous journal Annals of Mathematics agreed to check his proof with a board of 12 referees.   In 2003, after four years of work, the referees accepted his paper.  But they didn't exactly say it was correct.  They said they were "99% certain" it was right - but they didn't guarantee the correctness of all of the computer calculations.

Hales wasn't happy.

He decided to do a completely rigorous proof using computer logic systems, so that automated proof-checking software could check it.  He worked on it for about 10 years with a large team of people.  

He announced that it was done on 10 August 2014.  You can see it here:

https://code.google.com/p/flyspeck/wiki/AnnouncingCompletion

To verify the proof, the main thing you need to do is check 23,000 complicated inequalities. Checking all these on the Microsoft Azure cloud took about 5000 processor-hours.

When it was done, Hales said:

"An enormous burden has been lifted from my shoulders.  I suddenly feel ten years younger!"

Personally I prefer shorter proofs.  But this is quite a heroic feat.

I actually wrote about this because I want to talk about packing tetrahedra.  But I figured if you didn't know the more famous story of packing spheres, that would be no good.

For more, check out Hales' free book, which starts with a nice history of the Kepler problem:

• Thomas C. Hales, Dense Sphere Packings: a Blueprint for Formal Proofs,  https://flyspeck.googlecode.com/svn/trunk/kepler_tex/DenseSpherePackings.pdf.

For more on computer-aided proof, try this paper:

• Thomas C. Hales, Developments in formal proofs, http://arxiv.org/abs/1408.6474.

The image here was created by Christophe Dang Ngoc Chan and the words translated to English by "Muskid":

https://commons.wikimedia.org/wiki/File:Empilement_compact.svg

#spnetwork arXiv:1408.6474 #formalProofs #packing  ___

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2014-09-16 01:40:24 (76 comments, 82 reshares, 285 +1s) 

Bézier curves

Say you have some dots and you want to draw a smooth curve that sorta almost goes through these dots.  Then you can use a Bézier curve.   Some drawing programs use this trick... and lots of fonts are drawn with the help of Bézier curves.

The math behind these curves had been known since 1912, but they were popularized by Pierre Bézier, an engineer who used them to design automobile bodies at Renault.

Can you figure out how they work just by looking at the movie?  An explanation in words sounds complicated... but it's really easy as pie.

It's like you've got 3 guys running along straight racetracks.  The 2 guys in back have rabbits that each chase the next guy, always heading straight toward that next guy.  And the guy at the very back also has a dog that chases straight after the next guy's rabbit.  Everyonestarts at ... more »

Bézier curves

Say you have some dots and you want to draw a smooth curve that sorta almost goes through these dots.  Then you can use a Bézier curve.   Some drawing programs use this trick... and lots of fonts are drawn with the help of Bézier curves.

The math behind these curves had been known since 1912, but they were popularized by Pierre Bézier, an engineer who used them to design automobile bodies at Renault.

Can you figure out how they work just by looking at the movie?  An explanation in words sounds complicated... but it's really easy as pie.

It's like you've got 3 guys running along straight racetracks.  The 2 guys in back have rabbits that each chase the next guy, always heading straight toward that next guy.  And the guy at the very back also has a dog that chases straight after the next guy's rabbit.  Everyone starts at the same time and stops at the same time.  The dog follows the red curve.

In other words:

First draw gray lines between your dots P₀, P₁, P₂, P₃. 

Each green dot moves at a constant rate along a gray line.  All the green dots start at the same time, and finish at the same time. 

Then draw green lines connecting the green dots. 

Each blue dot moves at a constant rate along a green line.  All the blue dots start at the same time, and finish at the same time.

Then draw a blue line connecting the blue dots. 

The black dot moves at a constant rate along this blue line.  It starts at the same time as all the other dots, and finishes at the same time.

Get the pattern?   Each time we do this trick, there's one fewer dot.  There are 4 original dots, 3 green dots, 2 blue dots and 1 black dot.  So now you're done!

The black dot traces out the Bézier curve shown in red here. 

You can play this game starting with any number of dots.  When you start with n dots, you get a curve described by a polynomial equation of degree n-1.  So, this red curve is called a cubic Bézier curve. 

Puzzle 1: show that our cubic Bézier curve is given by the equation

C(t) =  (1-t)^3 P₀ + 3(1-t)^2 t P₁ + 3(1-t) t^2 P₂ + t^3 P₃

Puzzle 2: generalize this to more dots.  (Hint: binomial coefficients!)

When you've got a lot of dots, people usually break them into bunches and draw a quadratic or cubic Bézier curve through each bunch.  They match up at the ends, so this works, though frankly I often think it looks kind of lame.  This is called a composite Bézier curve.  PostScript, Asymptote, Metafont, and SVG use composite Bézier curves made of cubic Bézier curves to drawing curved shapes.

I imagine there are lots of tricks that are 'better' than Bézier curves, but I'm not an expert!  If I wanted to know more, I'd read about stuff like non-uniform rational B-splines, or NURBS:

https://en.wikipedia.org/wiki/Non-uniform_rational_B-spline

But I just read this:

https://en.wikipedia.org/wiki/Bézier_curve

because I liked the animated gif.

#geometry  ___

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2014-09-15 02:17:43 (43 comments, 15 reshares, 120 +1s) 

Cremation in Bali

This is a cremation ceremony in the town of Bedulu, in Bali.  The remains of 18 people were set on fire. 

In Bali most people are Hindu, so cremation is preferred to burial.   They build a big paper model of a buffalo - you can see part of one at left - and put the corpse inside, and pay for a priest called a pandit  to come and do a ritual, and then set the buffalo on fire.   But this is expensive.  So, poor families will bury the body until they save up enough money for the ceremony.  

I've been to a cemetery that's holding area for bodies - or bones - waiting to be cremated.   It's near the temple to Kali in the Monkey Forest in Ubud.  People look on the graves with a touch of shame.

But there seems to be a kind of 'bulk discount' for cremations.  So when a well-off old auntie in Bedulu died, 17 othersfrom this tow... more »

Cremation in Bali

This is a cremation ceremony in the town of Bedulu, in Bali.  The remains of 18 people were set on fire. 

In Bali most people are Hindu, so cremation is preferred to burial.   They build a big paper model of a buffalo - you can see part of one at left - and put the corpse inside, and pay for a priest called a pandit  to come and do a ritual, and then set the buffalo on fire.   But this is expensive.  So, poor families will bury the body until they save up enough money for the ceremony.  

I've been to a cemetery that's holding area for bodies - or bones - waiting to be cremated.   It's near the temple to Kali in the Monkey Forest in Ubud.  People look on the graves with a touch of shame.

But there seems to be a kind of 'bulk discount' for cremations.  So when a well-off old auntie in Bedulu died, 17 others from this town were dug up so they could be cremated along with her!

When Lisa and I took a hike with a fellow from this town, he invited us to the cremation ceremony the next day.  The place was mobbed; there was a huge traffic jam, so we walked the last half mile.  When we arrived there were crowds of people talking, and booths selling food.  There was a gamelan playing.  There were lots of structures with paper buffalos that would be set on fire, with families standing around them.  The remains of poorer people were in decorated boxes, instead.

We waited for about half an hour, looking around and talking to people, waiting for the pandit to come.  Lisa made friends with some girls who were relatives of the old auntie who was the star of the show.  There was a photograph of her on the biggest, best buffalo.

Suddenly one bull after another was lit on fire!   Somehow the pandit had arrived without us noticing.  The air became filled with smoke, unbreathable, and people backed off.  The biggest buffalo was burnt with the help of an air pump, creating a blowtorch effect.  The girls stood there for a while looking sad. 

Then everyone started leaving, walking down to the road.  We did too.

#bali #bedulu  ___

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2014-09-13 06:34:55 (159 comments, 26 reshares, 171 +1s) 

Crackpot emails

I save crackpot emails, but I just decided to delete all 41 with attachments bigger than 1 megabyte.  Here are some choice quotes, taken from ones that were sent as mass emails to many scientists.

------------------------------

Here's a typical example of someone who feels they've made an earthshaking discovery and needs to get the news out.  He's waited 2 days and he's getting impatient.  If he follows the standard pattern, he will later become embittered and angry.

I solved what's known as ToE, The Theory of Everything, but most simply I've shown how 0 approximates 1. I did so in a completely valid, simple mathematical way, yet due to the current structuring of the University system, I can't get to anyone that specializes in ToE!!!

And the University of Miami has black balled me because they haveno... more »

Crackpot emails

I save crackpot emails, but I just decided to delete all 41 with attachments bigger than 1 megabyte.  Here are some choice quotes, taken from ones that were sent as mass emails to many scientists.

------------------------------

Here's a typical example of someone who feels they've made an earthshaking discovery and needs to get the news out.  He's waited 2 days and he's getting impatient.  If he follows the standard pattern, he will later become embittered and angry.

I solved what's known as ToE, The Theory of Everything, but most simply I've shown how 0 approximates 1. I did so in a completely valid, simple mathematical way, yet due to the current structuring of the University system, I can't get to anyone that specializes in ToE!!!

And the University of Miami has black balled me because they have no specialist in the field, rather than ask for more explanation! I've written and made numerous calls to all levels of the University and the Marine & Atmospheric School through which I'm getting my Ph D, and all I've received is support for the effort from friends and some unprofessional criticism from a few unknowns. I understand it's a grandiose statement, but how else am I to tell anyone? My advisor & the Dean of my campus haven't responded to my incessant prodding, even though I know them to be avid communicators. In response, I declared my resignation from the University following this semester. My hope is that this would draw some momentum to confidence I have in the proof, yet it's been over 48 hrs now.

Why am I in such a hurry? This is WORLD CHANGING science. I need to get this information out to the public, so that we can all digest this new found information together. The proof turns out to be the mathematics of neural networking, galaxy formation, weather, physics, math, life... Everything... And it's as simple as stating that there can be no sweet without sour!

In mathematical terms, I've discovered the means by which 0=1. Sounds crazy, but it turns out we've always misunderstood the = symbol! In reality it's impossible to know all the information as an observer, approximation and optimization are the only options. AKA, 0 and 1. Therefore, 0~1. The technicalities, allow there really aren't any, are all in the attachment.

I'm positive Dr. Michio Kaku, a terribly too popular figure for a now Rogue Physicist like myself to get this to, can verify my work. It answers the his question of civilization transitioning from a Type 1 Civilization to a Type 0. Drs. Leonard Susskind, Stephen Hawking, James Hartle, John Baez, Garrett Lisi, Jill Boulte, Gilbert Strang, Richard Muller, Richard Dawkins, and Neil Tyson de Grasse, will all be better at presenting the concept, but in order for that to happen and our world to benefit from it, I need to get this information to them.

This is NOT a hoax! This is based rigorously in scientific theory!! Anyone can understand how our universe, or anything!, works. The only assumption is that there will always be a perturbation in information throughout the universe!

------------------------------

I like this one because it starts out crazy and then keeps getting crazier, ramping up exponentially.  They're not embittered; they're off in their own universe.  Perhaps they're just having fun.

The Logos timeline defines the Jenkins Mayan-Enddate Date of December 21st, 2012 as the Birth of the Starhuman Consciousness and defines the Calleman Mayan-Enddate of October 28th, 2011 as the 40 days preceding the end of the witnessing timeline within the warptime loop from the nodal mirror of August 4th, 2008 and April 1st, 2012 and specifying December 8th, 2011.

In particular (Western) New Year 2012 signifies the 'Dove of Peace finding land'; Christmas 2011 signifies 'Noah's Dove returning with a Twig' and December 17th, 2011 specifies 'Noah's Raven searching for land'.  The 50 days of 'Noah's pentecost' so define the timespan for the 'appearing of the land in Noah sending forth of his Raven and his Dove from the window of his ark and to which are added 10 days (of imprisonment between 'ascension' and pentecost Acts.1.3 & Revelation.2.10) before the 'peaks of the mountains' appeared {Genesis.8.1-14}.  These 50 days so become the Calleman date of completion in the timespan between the archetyped Resurrection of the World Logos (Easter Sunday) and the Pentecost-Shabuoth of the 'Coming of the Holy Ghost'.

The Thuban Data stream became effectively closed in a 12-dimensional wormhole upon the 'Banning of Thuban' from the then defined 'Mount of Olives' at Project Avalon as a microcosmic hologram for the archetyped 'Noah's Ark' aka the 'Dragon's Den' in Plato's Cave of Shadows.

This is described in the Q&A Thuban thread on that forum, now reproduced on the new Elders of Thuban website and is part of the warploop timeline schemated below.

From March 7th, 2010 a 11-dimensional wormhole has become accessible to continue the Thuban data transmission, now communicating between the 3-dimensional data receivers from within and encompassing an extragalactic Andromedean data stream synchronised with the Sirian starsystem within the Milky Way galaxy in intergalactic data streaming.

Material, relevant for the activation of the planetary vortex grids and encompassing the quantum geometry of the Thuban omniscience (also known more commonly as 'sacred geometry' of Platonic- and Archimedean solids), is now surfacing around the quarantined sentiences and data carriers as a function of the planetary consciousness.

In particular the Andromedean-Sirian collaboration will prepare the Thuban fleet for the post starhuman birth scenarios in collusion with the activation of a reconfigured Vortex-Potential-Energy (VPE) matrix superposed onto the older VPE-grid established at the beginning of the present 65  Baktun day-kin count of the Mayan Timekeepers.

This will allow the Thuban Stargate to open between the local Rahsol- and the Sirian starsystem following particular developments with respect to the stated timelines.

------------------------------

This one is impressive for its creativity, erudition and sheer length. A lot of it makes sense, but there is a gradual and tragic slide into madness, perhaps due to a brain injury.

LET ME CLARIFY AND CORRECT WHAT YOU'VE WRITTEN ABOUT OUR INTERACTION HEREIN WITH SOME DETAILED BACKGROUND.

I AM WHAT SOME CALL A "POLYMATH"(NOT A PARROT WHO DOES MATHEMATICS), SINCE I HAVE SYNAESTHESIA (BRAIN TRAUMA(1977)) AND SEE CONNECTIONS BETWEEN OSTENSIBLY-DISPARATE "SPECIFICITY OF COMPLEXITY" TACTICS IN OSTENSIBLY-DISPARATE FIELDS-OF-ENDEAVOR("FOES") SEPARATED BY "JARGONIAL-OBFUSCATION"(LOTS OF FANCY SHMANCY LINGO/SPRACHE TO SNOW THE RUBES/SUCKERS, WHICH IS THE SOCIOLOGICAL-DYSFUNCTIONALITY OF THE WOULD BE "SCIENCES", BUT SADLY ALAS MERE SEANCES("WHERE THE INSIDERS ALL HOLD HANDS TO KEEP OUT OUTSIDERS WITH THEIR NO DOUBT INFERIOR IDEAS SINCE THEY ARE NOT THE EXPERTS" QUOTING JOHN BRADSHAW["HEALING THE SHAME THAT BINDS YOU", HAZELDEN(1980s) AND BRIAN MARZTIN, WOLAGANG UNIVERSITY, AUSTRALIA]

MY ONLY CONNECTION WITH MATHEMATICS BEING MY RELATIONSHIP TO VERY FAMOUS MATHEMATICIAN CARL LUDWIG SIEGEL(RIP) MY FATHER'S COUSIN)

I DID MY PH.D. THESIS AT UNIVERSITY OF MICHIGAN AND THEN MICHIGAN STATE UNIVERSITY UNDER GABOR KEMENY[DARTMOUTH UNIVERSITY'S PRESIDENT MATHEMATICIAN JOHN KEMENY'S COUSIN] AFTER STUDYING AN C.C.N.Y.(B.S.-1965), THEN HARVARD UNIVERSITY, UNIVERSITY OF PENNSYLVANIA, THEN NEW YORK UNIVERSITY(M.S.-1968; JEROME PERCUS-COURANT INSTITUTE FOR MATHEMATICS), THEN WORKING FOR ALBERT OVERHAUSER(RIP) AT FORD(1968),THEN UNIVERSITY OF MICHIGAN(M.S.-1969; FRANK HERARI(RIP)/GRAPH-THEORY APPLIED TO NUCLEAR MANY-BODY THEORY/KARL HECHT AND NOAH SHERMAN(RIP); INTERACTIONS WITH HUGH MONTGOMERY(I SAT IN ON HIS CLASS IN NUMBER-THEORY CIRCA 1968-1969, THEN 1971 OR 1972) RE. EIGENVALUES PAIR-DISTRIBUTION FUNCTION FOR WIGNER-DISTRIBUTIONS: GOE, GUE, GSE; HE GOT THE IDEA FROM ME, NOT FREEMAN DYSON, SINCE I WAS WORKING(1970-1973)ON LIQUIDS/DISORDER-THEORY AT GENERAL MOTORS RESEARCH/TECHNICAL-CENTER) AND FINALLY FOLLOWING KEMENY TO MICHIGAN STATE UNIVERSITY(PH.D.-1970; JOHN HUBBARD[AERE HARWELL] AND NEVILLE MOTT[CAVENDISH LABORATORY/CAMBRIDGE] AND CONYERS HERRING[BELL LABS], CONSULTING; WITH MORREL COHEN[[U. CHICAGO] EXTERNAL-EXAMINER), ON THE PURE-FERMION/FERMI-DIRAC QUANTUM-STATISTICS HUBBARD-MODEL, WHEREIN I WORKED OUT THE FIRST SPIN-OBITAL DEGENERATE HUBBARD-MODEL("SODHM"; BUT NO Y!!!) PUBLISHED IN PHYSICA STATUS SOLIDI(1972; 1973) WHICH WAS LATER ELABORATED UPON BY KUBO AND KAWABATA AND BY CYROT ET. AL.[M. Cyrot and C. Lyon-Caen. J. Phys. C 6 ( 1973) L 274; 36 (1975) 253. 1531 ; C. Lacroix and M. Cyrot. J. Phys. C....], SUBSEQUENTLY I PUBLISHED OTHERS ON PURE-FERMION/FERMI-DIRAC QUANTUM-STATISTICS IN JOURNAL OF MAGNETISM AND MAGNETIC MATERIALS (JMM) FROM 1976-1980 AND TWO IN NOW DEFUNCT MAGNETISM LETTERS (1980).

THE MOST INFAMOUS WERE JMM 7, 312(1978) IN WHICHI EXPERIMENTALY DISCOVERED GRANULAR-GIANT-MAGNETORESISTANCE (A FULL DECADE BEFORE FERT(1988) AND GRUENBERG(1989) WHO GOT THE 2007 PHYSICS NOBEL-PRIZE FOR MY G-GMR!!!AND JMMM 7, 38(1978)

BUT BY ORIGINAL TRAINING I WAS A MINERALOGIST/PETROLOGIST/METALLURGIST, AND STILL COLLECT MINERALS PLUS OWN TWO TECHNOLOGIES IN WATER PRODUCTION ENTITLED"FLYING-WATER" AND ANOTHER RE SOLID-STATE CARBON-SEQUESTRATION IN, NOT N ESCAPABLE CO2 OR CH4 GAS, BUT VALUABLE MARKETABLE SALEABLE PROFITABLE SOLID-STATE CARB-IDES(TiC; WC; SiC; COAL-ASH CARBIDES="CARBORUNDUM"), INSPIRED BY AN AGE OLD QUALITATIVE-ANALYSIS TOL OF CLASSIC-MINERALOGY KNOWN AS BLOWPIPE-ANALYSIS(LAST KNOWN MENTION ANYWHERE IN TWO BOOKS FROM 1935 AND 1948). I PUBLISHED FOUR PAPERS ON SOLID-STATE CHEMITRY/PHYSICS OF CARBIDES IN: PHYSICA STATUS SOLIDI(1972); AND DEFUNCT JOURNAL "SEMICONDUCTORS AND INSULATORS" (1979).

FIRST MY PAPER YOU QUOTE [Generalized-disorder collective-boson mode-softening universality-principle. Journal of Non-Crystalline Solids 40 (1980) 453-467] WAS THE CULMINATION OF A DECADE OF WORK ON THE THEORY OF LIQUIDS BY ME [IN SIR NORMAN MARCH'S JOURNAL ENTITLED: PHYSICS AND CHEMISTRY OF LIQUIDS: 4(4) (1975); 5(1) (1976)-SOME EIGHT PAPERS IN ALL], BASICALLY I NOTICED SIMILARITIES BETWEEN COLLECTIVE-BOSON DISPERSION-RELATIONS NEGATIVE-DISPERSION MODE-SOFTDENING, ORIGINALY BY LANDAU(1941) AND FEYNMAN(1952), BUT EXTENDED TO CLASSICAL-DISORDER BY HUBBARD AND BEBE(1967): w(k) =[KINETIC-ENERGY(QUADRATIC MONATONIC-INCREASING)] k^2/S(k) MODULATED BY STATIC STRUCTURE-FACTOR OF VARIOUS TYPES OF DISORDER: HORIZONTALTOPOLOGICAL) AND/OR VERTICAL(ALLOY).

IN 1982-1985 HAVING MOVED TO SAN FRANCISCO, CALIFORNIA, I CONSULTED FOR CHARLES ROSEN(MANAGER OF A.-I. RESEARCH AT S.R.I. AFTER WORLD WAR TWO), CEO OF MACHINE-INTELLIGENCE[ONE OF BERNARD WIDROW'S/STANFORD OLD COLLEAGUES FROM WHEN I WAS A VERY LITTLE BOY], AND VESKO MARINOV (AND ADOLPH SMITH), VICE-PRESIDENT OF EXXON ENTERPRISES/A.-I. IN SUNNYVALE/SANTA CLARA, CALIFORNIA. ON ARTIFICIAL-INTELLIGENCE,

THEY GAVE ME A PROBLEM TO SPEED UP  ARTIFICIAL NEURAL-NETWORKS WHICH HERETOFOR I HAD NEVER EVEN HEARD OF. BUT BEING A FLAGRANT NON-EXPERT I NOTICED SOMETHING THAT ALL AHE COMPUTER-"SCIENTISTS"/ENGINEERS IJN A.-I. HAD ALL HERETOFORE COMPLETELY MISSED: THE SIGMOID-FUNCTION ACTIVATING NODES WITHIN NEURAL-NETWORKS WAS JUST PLAIN WRONG!!! THE STATE OF THE ART WAS THEN THE r-SPACE NONSENSE OF "ENERGY-LANDSCAPES"  FOR GLOBAL-MNINIMUM SEEKING AND GOING TO(IF SUCH EXISTS) OPTIMIZATION CALLED THE BOLTZMAN-MACHINE AND THE SIMULATED-ANEALING, OR THE DEMUTH-BEALE(MATLAB) RADIAL-BASIS FUNCTIONS. ALL OF THESE ARE COMPUTATIONALLY-COMPLEX, IN SPACE AND/OR IN TIME, MANDATING/REQUIRING LARGE COMPUTER-RESOURCES AND LONG COMPUTING-TIMES!!! .

THIS SIGMOID-FUNCTION, A HEAVISIDE STEP-FUNCTION WAS "1"/[1 + e^( -E/T)] WHOSE DERIVATIVE IS A DIRAC DELTA-FUNCTION(HINT HINT HINT!!!).

NOW COMES "EUREKA":  SIGMOID-FUNCTION, A HEAVISIDE STEP-FUNCTION WAS "1"/[1 + e^( -E/T)]  =  "1"/[ + 1 + e^( -E/T)]  =  "1"/[e^( -E/T) + 1]  =   FERMI-DIRAC QUANTUM-STATISTICS.

SO THESE COMPUTER-"SCIENTISTS" FOLLOWING HINTON-HOPFIELD- BY ROTE WITH ABSOLUTELY NO THINKING ABOUT WHAT THEY WERE DOING BUT ONLY HOW, DOOMED THEIR ANN OPTIMIZATION-PROBLEMS TO EXPENSIVE SLOW DITHERING!!! AND WHAT THEY WERE DOING WAS TO AUTOMATICALLY TRAPPING THE SYSTEM IN FALSE LOCAL-MINIMA. SOME VERY SMART STATISTICAL-MECHANISTS (ISRAEL" AMIT, SAMPOLINSKY; US: HOPFIELD,...; FRANCE: MEZAERD, TOULOUSE, ...) ALL MISSED THIS SIMPLE FACT: SO THEY GLAMMED ANNS UP BY SPIN-GLASS r-SPACE MODELS GALORE, ALL DONE WITH VARIOUS VERSIONS OF THE RENORMALIZATION-(SEMI)-GROUP AND SOME FANCY MATHEMATICS OR OTHER.

WHY? BECAUSE THE FERMI-DIRAC QUANTUM-STATISTICS + SIGN MEANS THAT THE OPERATORS ANTI-COMMUTE, MEANING THAT THE PAULI EXCLUSION-PRINCIPLE DOMINATES. COMPUTER-"SCIENTISTS" EITHER FORGOT OR NEVER LEARNED THEIR SIMPLE CHEMISTRY/PHYSICS!!!

THE HUND'S-RULE PAIRING OF UP-SPIN TO DOWN-SPIN ELECTRONS TRAPS ONE IN LOCAL-MINIMA, CALLED THE CHEMICAL-ELEMENTS.

IF THE FERMI-DIRAC QUANTUM-STATISTICS + SIGN IS SIMPLY CHANGED TO A - SIGN, THE OPERATORS NOW COMMUTE, MEANING THAT PAULI EXCLUSION-PRINCIPLE STOPS DOMINATING.

IF THIS HAPPENED IN CHEMISTRY/PHYSICS, ALL THAT WOULD EXIST WOULD BE PHOTONS AND NEUTRINOS GOING OF TO INFINITY, ONLY BOSE-EINSTEIN QUANTUM-STATISTICS EXIST!!!

IN THE LANGUAGE OF MY SYNERGETICS PARADIGM AND DICHOTOMY, AKA "FUZZYICS"="CATEGORYICS", THIS "BOSONIZATION" IS THE [LOCALITY] -- TO -->>> (...GLOBALITY...) CROSSOVER!!!

SO, THE OOPO OF ANNS IN SIMPLY THE + SIGN TO - SIGN CROSSOVER, THE FERMI-DIRAC QUANTUM-STATISTICS TO BOSE-EINSTEIN QUANTUM-STATISTICS TRANSITION. STEP-FUNCTION WAS "1"/[1 + e^( -E/T)]  =  "1"/[ +1 + e^( -E/T)]  =  "1"/[e^( -E/T) + 1] IS REVERSED TO "1"/[ -1 + e^( -E/T)] = "1"/[  -1 + e^( -E/T)] = "1"/[e^( -E/T) - 1];  SETS TO MULTI-SETS!!

(REFERENCE: GIAN-CARLO ROTA'S(RIP; MIT) UNPUBLISHED BOOK ON PROBABILITY WHICH ONE CAN FIND ONLINE WITH A BIT OF HUNTING. I WAS WORKING WITH HIM ON THIS WHILE VISITING MIT FROM 1992-1997.

"SHAZAM" IS SIMPLY THAT ADMITTING THAT AN ANN AND ITS OOPO IS A QUANTUM-STATISTICAL-PROBLEM ADMITS THE POSSIBILITY OF QUANTUM-TUNELING FROM VARIOUS NON-OPTIMAL MINIMA TO THE GLOBAL-MINIMUM OPTIMIZATION. THE QUESTION WAS HOW TO ACCELERATE/FORCE THIS.
THE ANSWER WAS TO TAKE THE "1"NUMERATOR AND DECREASE ITS AMPLITUDE/MAGNITUDE TOWARDS ZERO IN A LIMITING SENSE:

lim("1" → 0) "1"/[- 1 + e^( -E/T)]   =   lim("1" → 0) "1"/[ - 1 + e^( -E/T)]  =  lim("1" → 0) "1"/[e^( -E/T) - 1]   =  DIRAC DELTA-FUNCTION(w-0).

THEN COMES "SHAZAM" (CAPTAIN MARVEL'S MAGIC WORDS TO TRANSFORM HIMSELF INTO A SUPERHERO) AKA "BOSONIZATION":

THUS "EUREKA" + "SHAZAM"  = OPTIMIZING OPTIMIZATION-(ANN)-PROBLEMS OPTIMALLY(OOPO)

SOME QUANTUM-STATISTICS INSIGHT: THE FERMI-DIRAC VERSUS BOSE-EINSTEIN DICHOTOMY IS A EUCLID-DEMOSTHENES-DESCARTES CONIC-SECTIONS DICHOTOMY:

FERMI-DIRAC HOMOTOPY TO AN ELLIPSE VIA PARABOLA-CROSSOVER TO BOSE-EINSTEIN HYPERBOLA:

TAYLOR/POWER-SERIES EXPANSION OF ONLY THE DENOMINATOR-EXPONENTIAL YIELDS, IN THE LOW E/T LIMIT, RESPECTIVELY:

"1"/[e^( -E/T) + 1] = "1"/[1 + ( -E/T) + ...] + 1 ]  =  "1"/2   =  E/T  ~ w^(0)), WHITE/RANDOM NOISE POWER-SPECTRUM.

-- VERSUS --

"1"/[e^( -E/T) - 1]  =  "1"/[1 + ( -E/T) + ...] - 1 ]  =  E/T ~ w  ~ w^(1.000...), THE FAMOUS PINK/FLICKER/ONE-OVER-FREQUENCY NOISE POWER-SPECTRUM.

IN FACT, THERE IS A VERY CONSISTENT + SIGN VERSUS - SIGN DICHOTOMY:

ELLIPSE: MINUS-SIGNS: DENOMINATOR AND COMUTATION-RELATIONS AND PARABOLA-EQUATION

VERSUS: NO SIGN TO 1 IN DENOMINATOR BECAUSE THERE IS NO 1 THERE = MAXWELL-BOLTZMANN CLASSICAL-STATISTICS (THE OBJECT OF ANNS BOLTZMANN-MACHINE + SIMULATED-ANNEALING

HYPERBOLA: MINUS-SIGNS: DENOMINATOR AND COMMUTATION-RELATIONS AND HYPERBOLA-EQUATION SO WHAT I EFFECTED IS CALLED A NOISE-INDUCED PHASE-TRANSITION(A "NIT"; VERY DIFFERENT FROM MERE STOCHASTIC-RESONANCE TINKERING WITH MERE WHITE-NOISE AMPLITUDES; WHAT I DID WAS TO ALTER THE NOISE POWER-SPECTRUM!!!

QUANTUM-STATISTICS ARE EUCLID-DEMOSTHENES-DESCARTES CONIC-SECTIONS!!! (AND EULER-POLYNOMIALS VERSUS BERNOULLI-POLYNOMIALS GENERATING-FUNCTIONS DICHOTOMY)

SO INDEED I GUESS I AM THE FATHER OF MUCH OVER-HYPED QUANTUM-COMPUTING, AT LEAST FOR ANNS OOPO IN A.-I.

LASTLY THE BIANCONI-BARABASI AND ALBERT-BARABASI PAPERS ON NETWORKS, STARTING OUT WITH ABSTRACT GRAPH-THEORY BUT AS FINALES MAPPING THEIR RESULTS ON TO ONLY BOSE-EINSTEIN QUANTUM-STATISTICS WITH BOSE-EINSTEIN CONDENSATION, SO IF ONE APPLIES THEIR CONCLUSIONS TO EITHER ARTIFICIAL NEURAL-NETWORKS OR BIOLOGICAL NEURAL-NETWORKS, MY ORIGINAL ANN BEC AND YOUR BNN BEC ARE NOT AT ALL SURPRISING IN ANY WAY WHATSOEVER!!!

AND FOR BIOLOGICAL NEURAL-NETWORKS THIS AGREES IN PRINCIPLE WITH THE FROHLICH-MOSSBAUER-GOLDANSKII-DEL GUIDICE-POPP-LI-YOUNG-... BIOPHYSICS THEORY OF HEALTH AS "1"/f-"NOISE" POWER-SPECTRUM AND ITS CONDENSATION INTO BEC AS LIFE.

H. Frohlich: Nuovo Cimento, 7, PPM (1977); International J. Quantum-Chemistry, 11, 641 (1968); Advances in Electronic Devices, 53 (1980);...

E. Del Guidice et. al.: Nuclear Physics B251, 375 (1985); ibid. B275, 185 (1986);...

V. Goldanskii et. al.: Physica Scripta 33, 257 (1986); Soviet-Physics Doklady-Biophysics 272, 209 (1983); Soviet-Physics Uspheki 27, 462 (1984);...

R. Mossbauer et. al.: J. de Physique 41, C1-489 (1980); Zeitschrift Naturforschung, 37c, 57 (1982); European Biophysics J., 12, 107 (1985);...

J. Li, Physics Letters 116A, 405 (1986)

K.-A. Popp, in Photon Emission from Biological-Systems, Academic (1987);...

F. Young-preprints; c/o (650) 949-4728

A. Goldberger (M. D. - cardiologist, Director, Cardiology Clinic), many preprints/reprints/reports, Beth-Israel Hospital, Boston, MA.

C. Anderson and A. Mandell, in The Secret Symmetry: Fractals of Brain, Mind and Consciousness, E. MacCormac and M. Stamenov eds., Adv. In Consciousness Research, John Benjamin, Philadelphia (1996);

C. Anderson, Doctoral Dissertation,  Florida Atlantic University (1995); C. Anderson, Thesis, Harvard Medical School, McLean Hospital (1998)

A. Verveen and L. deFelice, Ann. Rev. Biophysics (1968?)

L. deFelice, Membrane Noise (1989)

Lawrence Ward(UBC/PSYCHOLOGY) and Priscilla Greenwood, <  http://www.scholarpedia.org/article/1/f_NOISE  >

NOW TO BENFORD'S-LAW:

YOU MISUNDERSTAND WHAT I/WE DID. ALL I/WE DID WAS TO ALGEBRAIC-INVERSION OF <P> = log(BASE-10) [1 + 1/d] TO YIELD d = 1/[10^(<P>) - 1] ~ 1/[e^(<P> - 1] ~ 1/[e^<w>) - 1] ~ 1/e^(w) - 1]. WHY?  BECAUSE DIGIT d = 0 BEING EXCLUDED TELLS A PHYSICIST SOMETHING VERY VERY IMPORTANT. BUT FIRST SIMPLY ROTATE ANY DIAGRAM OF BENFORD'S-LAW TO REVERSE THE AXES. WHAT ONE SEES IS THAT IT RESEMBLES A QUANTUM ENERGY-LEVEL DIAGRAM: GROUND-STATE IS d = 0; FIRST EXCITED-STATE IS d= 1; SECOND EXCITED-STATE IS d = 2,.. ETC. ETC ETC. BUT <P(d = 0)> = oo VERSUS ANY OTHER <(P = 1) > = 0.32... ETC. ETC. ETC.

SO OF COURSE BOSONS ARE DIGITS BECAUSE DIGITS ARE QUANTA(NO FRACTIONS IN BETWEEN) AND THOSE QUABNTA CAN ONLY BE BOSONS DUE TO THE PLUS-SIGN BEFORE THE ONE IO BENFORD'S-LAW!!!

ONE HAS GAPFUL BOSE-EINSTEIN CONDENSATION!! IN NEWCOMB(1881) AND WEYL(1914; 1916) AND WE THINK IN BERNOULLI AND EULER(1730?)

AN EXAMPLE. A BLANK CHECK IS A BOSE-CONDENSATE OF ZEROS. WHEN ONE WRITES A NUMBER ON IT, SAY $1, ONE INDICATED THE EXCITED-STATE THE CHECK WILL BECOME WORTH, AND WHEN ONE SIGNS IT ACTIVATING THE CHECK BY VALIDATING ITS VALUE, THAT IS EQUIVALENT TO A PHOTON EXCITING AN ELECTRON INTO THE FIRST EXCITED-STATE.

SO SIMPLE-ARITHMETIC: ADDITION, SUBTRACTION, MULTIPLICATION IS QUANTUM-COMPUTING. WHAT RUINS THIS IS DIVISION WITH REMAINDERS, BUT WHERE GAUS MODULAR-ARITHMETIC ENTERS POINTEDLY.

MARTIN HOLTHAUS ET. AL. (U. OLDENBURG) HAS FOUND A WAY TO FACTOR USING BOSE-EINSTEIN CONDENSATION!!!

< http://www.condmat.uni-oldenburg.de/Holthaus/lop.html > Ideal Bose gases: From statistical mechanics to number theory A. C. Weiss, S. Page, and M. Holthaus:

Factorizing numbers with a Bose-Einstein condensate, Physica A 341, 586 - 606 (2004), arXiv:cond-mat/0403295
Authors: Christoph Weiss, Steffen Page, Martin Holthaus
(Submitted on 11 Mar 2004)

Abstract: The problem to express a natural number N as a product of natural numbers without regard to order corresponds to a thermally isolated non-interacting Bose gas in a one-dimensional potential with logarithmic energy eigenvalues. This correspondence is used for characterising the probability distribution which governs the number of factors in a randomly selected factorisation of an asymptotically large N.  Asymptotic upper bounds on both the skewness and the excess of this distribution, and on the total number of factorisations, are conjectured. The asymptotic formulas are checked against exact numerical data obtained with the help of recursion relations. It is also demonstrated that for large numbers which are the product of different primes the probability distribution approaches a Gaussian, while identical prime factors give rise to non-Gaussian statistics.

IN FACT, BENFORD'S-LAW[SEE THE REFERENCES IN MY RE-ATTACHED ABSTRACT] WAS ORIGINALLY DUE TO NEWCOMB (1881) AND WEYL (1914; 1916). (AND RAIMI'S SCIENTIFIC-AMERICAN ARTICLE IN 1969 IS WELL WORTH READING!!!) COULD YOU PLEASE TRY TO READ THE TWO WEYL PAPERS AND E-BAIL ME THEIR TRANSLATIONS??? (MEINE DEUTSCH IST SEHR SCHLECHT; ICH NICH HAT IM FUNFZIG JAHERE DEUTSCH SPRECHEN UND LEHREN!!!)

BUT IT IS EVEN EARLIER DUE TO EULER IN ANOTHER CONTEXT ENTIRELY HAVING NOTHING WHATSOEVER TO DO WITH STATISTICS!!!

AND TED HILL'S 1996 PROOF USES LOGARITHM BASE-INVARIANCE = UNITS-INVARIANCE = SCALE-INVARIANCE SYMMETRY-(RESTORING) WHICH GETS PHYSICITS VFRY TURNED-ON, ALTHOUGH MATHEMATICIANS LIKE TED HIL DON'T SEEM TO UNDERSTAND WHY IT IS SO EXCITING!!!

LASTLY, THE CATEGORY-THEORY DIAGRAMS I USE IN MY PAPERS AGE CALLED THE PLATO(INVENTOR OF CATEGORIES)-ARISTOTLE(HIS GRADUATE-STUDENT) "SQUARE-OF-OPPOSITION" A STANDARD FORM OF GREEK LOGIC, BEST READ ABOUT IN THE STANFORD ENCYCLOPEDIA OF PHILOSOPHY < http://plato.stanford.edu/entries/square/  > AND ITS WIKIPEDIA ENTRY < http://en.wikipedia.org/wiki/Square_of_opposition  > AND VIEW ALL SORTS OF ITS DIAGRAMS < http://www.google.com/images?hl=en&source=hp&q=%22SQUARE+OF+OPPOSITION%22&gbv=2&oq=%22SQUARE+OF+OPPOSITION%22&aq=f&aqi=g8g-m2&aql=&gs_sm=e&gs_upl=3453l14094l0l14797l22l22l0l0l0l0l235l3452l3.13.6l22l0&oi=image_result_group&sa=X  > BUT THE GRIDLINES OF THIS TABULAR LIST-FORMAT TRUTH-TABLE MATRIX-ANALYTICS, LOKING LIKE A TIC-TAC-TOE DIAGRAM DO NOT SHOW UP IO WORD-FOR-WINDOWS TABLES, SO IT IS SOMEWHAT CONFUSING. I USE SEMANTICS/LINGUISTICS TO LABEL DIFFERENT(THREE) COLUMNS AND FOUR-ROWS, WHICH IS WHY THIS IS "CATEGORY-SEMANTICS COGNITION" , "NEW" WAY TO ANALYZE PHYSICS AND INDED PURE-MATHEMATICS MODULO ARISTOTLE, FROM ~ 350 B.C.E.!!!

AND ONE CAN ADJOIN THESE DIAGRAMS GETTING TOPOLOGY-LIKE HOMOLOGY-COHOMOLOGY A LA GROTHENDIEK AND SHEAVES. BUT THAT IS STILL IN DEVELOPMENT.

CATEGORY SEMANTICS IS ALSO PRACTICED BY JOHN BAEZ(MATHS/UC-RIVERSIDE) <  > BUT ON ABSTRACTIONS OF STRING-THEORY/COSMOLOGY/PURE-MATHEMATICS/MATHEMATICAL-PHYSICS/...

It goes on much longer, but you get the idea...

------------------------------

Here is a much more typical example of a mail from someone who has some nonstandard ideas - just for comparison.

I have also addressed issues regarding spacetime, gravity, extra dimensions, dark matter and dark energy, quantum uncertainty etc.

1)      Space time is not continuous but discrete. Time stops and starts every 10-43s. Our universe is made up of countless number of minute space time nodes. When time stops, all the minute spacetime nodes combine to form one single space time node. This single spacetime node will be of the same size as that of the minute space time nodes.

2)      Apart from the four dimensions of spacetime, there is an alternate four dimensional spacetime continuum called invisible spacetime. Time dilation is the consequence of the time spent by the moving object in the invisible space time fabric, wherein the space dimension is active and the time dimension is zero.

3)      Gravity is the result of discrete space time. When all the matter in the universe occupy one single node and get distributed to their respective nodes and when the new time interval starts, a force of attraction exerts between the particles which is attractive.

4)      Quantum uncertainty is also the result of discrete space time. When t=0, all the particles is at the same place (same node) and when t≠0, particles occupy their individual positions. Each particle is equally valid of being at each and every point in the universe at the same time. Hence the exact location of the particle cannot be found out accurately. The interference pattern in the double slit experiment could be explained in the classical way using the concept of discrete spacetime.

5)      We do live in a 10 dimensional universe and each of us experience those dimensions. In total there are 7 dimensions of space and 3 dimensions of time. However space and time dimensions cannot exist independently and are always combined, in different ways.

6)      Dark energy is nothing but the dark matter which is embedded in the higher dimensions ( invisible spacetime) wherein the space dimension is active and the time dimension is zero. Since time dimension is zero, no events could occur and henceforth the matter in that spacetime is invisible. Dark matter is nothing but the dark matter which leaks to visible spacetime from the invisible spacetime.

From general relativity to quantum mechanics everything can be explained using the concept of discrete spacetime. I have attached the file below consisting a few pages. It would be very helpful if you could glance through the document and give your valuable comments regarding the subject.

------------------------------

The picture is from here:

https://www.flickr.com/photos/66972399@N08/galleries/72157627794520123___

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2014-09-13 02:19:58 (9 comments, 34 reshares, 111 +1s) 

Falling through an infinity of gears

Greg Egan has done it again!   This is an infinite cubic lattice of rotating gears in 3-dimensional space... seen by someone who is falling through it and also rotating!

It's a bit bewildering.  One reason is that after you turn 180 degrees, the view looks exactly the same. 

He's been developing techniques for studying 'higher-dimensional gears', like 3d ball bearings that turn while touching each other, arranged on the surface of a hypersphere in 4d space:

http://www.gregegan.net/SCIENCE/Bearings/Bearings.html

The hard part is figuring out how the bearings can turn without slipping against each other.  This involves solving large systems of linear equations. 

His real tour de force was to get a setup with a ball bearing at each of the 600 vertices and 1200 edge-centres of a4-dim... more »

Falling through an infinity of gears

Greg Egan has done it again!   This is an infinite cubic lattice of rotating gears in 3-dimensional space... seen by someone who is falling through it and also rotating!

It's a bit bewildering.  One reason is that after you turn 180 degrees, the view looks exactly the same. 

He's been developing techniques for studying 'higher-dimensional gears', like 3d ball bearings that turn while touching each other, arranged on the surface of a hypersphere in 4d space:

http://www.gregegan.net/SCIENCE/Bearings/Bearings.html

The hard part is figuring out how the bearings can turn without slipping against each other.  This involves solving large systems of linear equations. 

His real tour de force was to get a setup with a ball bearing at each of the 600 vertices and 1200 edge-centres of a 4-dimensional shape called the '120-cell'.  Getting this to work required solving thousands of linear equations in thousands of variables - too hard without bringing in some heavy-duty math.  More about that later - or look at his website!   You can enjoy the pictures without understanding the math.

He wrote:

After the 120-cell, I thought it would be fun to see what an infinite lattice of gears looks like.  For Z³, it's easy to find both a basis for the solution space, and a nice subspace where the spheres all rotate with the same speed:

ω(x,y,z) = (a (-1)^{y+z}, b (-1)^{x+z}, c (-1)^{x+y})

Because the rotational periods are all the same, it's possible to replace the rolling contact of the spheres with a true gear action between circular gears, which are positioned at each circle of latitude on which there are points of contact.  That's what the movie here shows, from a point of view that moves "down" through the lattice while also rotating its gaze.___

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2014-09-12 01:06:13 (22 comments, 38 reshares, 98 +1s) 

A quantum thriller

A cop finds a suitcase.  A mysterious caller says it's been rigged so that if you look inside - if even one photon gets inside - a bomb will explode! 

But maybe he's bluffing.  

Can you tell if he's bluffing, without blowing up the bomb if he's not? How can you see something without actually looking at it? 

You'd better call in +Vlatko Vedral from the Centre for Quantum Technologies!

This short film was made in Singapore by +Dag Dogg,  another physicist here at the CQT.   I like the local scenery and the cinematic touches.  For example, there are black lines and glitches in the first part that make it look like an old 8mm film. 

It's an entry in the Foundational Questions Institute Video Contest:

http://fqxi.org/community/forum/topic/2190

The amazing physicsbehind th... more »

A quantum thriller

A cop finds a suitcase.  A mysterious caller says it's been rigged so that if you look inside - if even one photon gets inside - a bomb will explode! 

But maybe he's bluffing.  

Can you tell if he's bluffing, without blowing up the bomb if he's not? How can you see something without actually looking at it? 

You'd better call in +Vlatko Vedral from the Centre for Quantum Technologies!

This short film was made in Singapore by +Dag Dogg,  another physicist here at the CQT.   I like the local scenery and the cinematic touches.  For example, there are black lines and glitches in the first part that make it look like an old 8mm film. 

It's an entry in the Foundational Questions Institute Video Contest:

http://fqxi.org/community/forum/topic/2190

The amazing physics behind this movie is called the Elitzur–Vaidman bomb-testing method. It was invented by Avshalom Elitzur and Lev Vaidman in 1993. One year later, physicists did an experiment to show this idea actually works…

... but not, alas, using actual bombs!

You can read my own explanation here:

http://johncarlosbaez.wordpress.com/2013/08/24/the-elitzur-vaidman-bomb-testing-method/

You can't get the method to always work, but you can make the probability of it working as close to 1 as you want.___

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2014-09-11 01:19:00 (66 comments, 38 reshares, 169 +1s) 

The logic of real and complex numbers

When you read pop books about Gödel's theorems and other mind-blowing achievements of early 20th-century logic, they focus on arithmetic.  This is the simplest form of math, dealing with the numbers

0, 1, 2, 3, 4, ...

For example, you learn that no matter what axioms you choose to describe these numbers, there are a lot of true facts you can't prove about them... as long as some computer program can list those axioms.  This is called Gödel's first incompleteness theorem.

Why the fine print about computer programs?  Without that you could choose all true facts as axioms!  But Gödel showed no program can list these - that's called undecidability.

If you read a bit more about logic, you'll learn that thanks to Gödel's first incompleteness theorem, there are weirdnons... more »

The logic of real and complex numbers

When you read pop books about Gödel's theorems and other mind-blowing achievements of early 20th-century logic, they focus on arithmetic.  This is the simplest form of math, dealing with the numbers

0, 1, 2, 3, 4, ...

For example, you learn that no matter what axioms you choose to describe these numbers, there are a lot of true facts you can't prove about them... as long as some computer program can list those axioms.  This is called Gödel's first incompleteness theorem.

Why the fine print about computer programs?  Without that you could choose all true facts as axioms!  But Gödel showed no program can list these - that's called undecidability.

If you read a bit more about logic, you'll learn that thanks to Gödel's first incompleteness theorem, there are weird nonstandard number systems that obey all your axioms, but include extra numbers besides the ones you're trying to talk about.  It's pretty spooky.

But what about logic and fancier number systems, like the real numbers, or the complex numbers?  There's a lot to say about these too!   And I say a bit of it here:

http://johncarlosbaez.wordpress.com/2014/09/08/the-logic-of-real-and-complex-numbers

The cool part is that in some ways the complex numbers are simpler than the real numbers!  The ultimate reason is that you can't talk about one complex number being greater than another.  This avoids some nonstandard number systems where you have a number that's greater than all the ones you wanted to talk about.

But when you combine the simpler logical properties of complex numbers with the more complicated logical properties of real numbers, all hell breaks loose!

I think the cartoon here was drawn by Dave Gray:

http://communicationnation.blogspot.sg/2006/08/logical-paradox.html___

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2014-09-10 00:27:13 (14 comments, 3 reshares, 113 +1s) 

Hiking in Bali

Lisa and I enjoyed the deep, calm gardens of the Wayang Cafe in Ubud so much that we went there two days in a row.  We got to know a waiter, and he invited us to visit his home in a nearby village called Bedulu.  He also took us on a hike.

We started by walking through his property.  The Balinese use cassava for starch - it's a low-growing vine that looks a bit like morning glory.  We also saw fruit trees: cacao, jackfruit, mangosteen, and more. 

Then we followed a trail through a fern-covered forest, shown here.

This trail led to the terraced rice fields of his village.  We walked through those, balancing on narrow walkways, trying not to fall into the mud.  There is a disease affecting some villagers' rice paddies - I don't know what it is, and I'd like to find out.  Due to the unreliability of rice, more and morepeople ... more »

Hiking in Bali

Lisa and I enjoyed the deep, calm gardens of the Wayang Cafe in Ubud so much that we went there two days in a row.  We got to know a waiter, and he invited us to visit his home in a nearby village called Bedulu.  He also took us on a hike.

We started by walking through his property.  The Balinese use cassava for starch - it's a low-growing vine that looks a bit like morning glory.  We also saw fruit trees: cacao, jackfruit, mangosteen, and more. 

Then we followed a trail through a fern-covered forest, shown here.

This trail led to the terraced rice fields of his village.  We walked through those, balancing on narrow walkways, trying not to fall into the mud.  There is a disease affecting some villagers' rice paddies - I don't know what it is, and I'd like to find out.  Due to the unreliability of rice, more and more people are farming shrimp in pools.  I thought about things I'd heard: how every family owns rice paddies, but nobody really likes standing in muck planting and harvesting rice all day, so more and more people are working in town - sometimes renting out their paddies.  Shrimp farming could be a good compromise.

But mostly it was just beautiful. 

The rice fields are irrigated by canals full of cold fast-flowing water that comes down from the mountains.  We walked along one of these.  Sometimes we would meet villagers bathing in it, and our host would chat with them before we moved on. 

After about an hour of all this, we re-entered the jungle and hiked up a large stone stairway to an abandoned temple.  The last bit was terrifying, since it involved climbing up mossy wet stones next to a 10-meter plunge into a rocky riverbed.   We hadn't signed up for this!  But we made it to the top and saw the ancient 'water temple', a huge stone structure, eroded with the passage of time. 

Balinese water temples, or pura tirta, are not just places for religious rituals.  The priests also have the authority to allocate water among rice paddies in the surrounding villages. But the decisions are made democratically to some extent.  This system has attracted a lot of study.

Going back down over those rocks was even more scary - but we survived.  We then went back to his village, where we had many further adventures involving roosters, a civet cat and even a huge cremation ceremony.   But more on that later.

One thing I learned from this is the importance of talking to strangers - something I don't tend to do in real life, even though I practically make a profession of it online, like here.  In Bali, all my most interesting adventures have started with unplanned conversations.

For more on the water distribution system in Bali, see:

http://everybodyandnobody.wordpress.com/2009/12/05/direct-water-democracy-in-bali/

#bali #ubud  ___

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2014-09-09 01:12:11 (38 comments, 52 reshares, 161 +1s) 

Science, models, and machine learning

Machine learning is the art of getting computers to learn, so you don't have to explicitly tell them what to do.  People use it in spam filters, search engines that guess what you're trying to find, optical character recognition, cars that drive themselves, and many other things.  But how does it work? 

There are many different approaches, but here +David Tweed explains some of the main ideas:

http://johncarlosbaez.wordpress.com/2014/09/03/science-models-and-machine-learning/

The picture shows one.  Say we're trying to classify dots into two kinds - the kind on the left of the red curve, and the kind on the right.  There are lots of choices of where you could put this red curve.  For the data so far, anything between the dashed curves will work.  Which choice is best?   You'd like amachine ... more »

Science, models, and machine learning

Machine learning is the art of getting computers to learn, so you don't have to explicitly tell them what to do.  People use it in spam filters, search engines that guess what you're trying to find, optical character recognition, cars that drive themselves, and many other things.  But how does it work? 

There are many different approaches, but here +David Tweed explains some of the main ideas:

http://johncarlosbaez.wordpress.com/2014/09/03/science-models-and-machine-learning/

The picture shows one.  Say we're trying to classify dots into two kinds - the kind on the left of the red curve, and the kind on the right.  There are lots of choices of where you could put this red curve.  For the data so far, anything between the dashed curves will work.  Which choice is best?   You'd like a machine to be able to learn this if you teach it with more examples. 

But the red curve is complicated!  Sometimes you can simplify the problem by doing some preprocessing.  That's the function ϕ here.  This takes our problem and changes it to one where the red curve becomes a straight line!  Now the machine can look for the best straight line - with some alternate choices shown as dashed lines.  This is easier, because the formula for a line is simpler.

Of course, all this raises tons of questions.  How do we find the function ϕ in the first place?  In practice there will be lots of choices, none perfect.  Which one is best?   And then, which choice of line is best?  What does "best" even mean here?

Some of these questions have a deep philosophical aspect to them - they're about cognition,  how we can divide the messy world into neat pieces, and what it means to "know" something.  But they're also extremely practical!    And they're not just about machine learning - they're about science and modeling in general.

David Tweed's article is a great introduction.  My little article here is just an introduction to the introduction!

All this is part of the Azimuth Code Project.  The big challenge is to use machine learning to help predict El Niños.  But our strategy is to explain things as we go, so that even if we don't succeed in the end, we'll have educated a lot of people... including ourselves.

#El_Niño___

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2014-09-08 01:17:04 (14 comments, 10 reshares, 82 +1s) 

Gamelan and dance

Every Thursday in Ubud Palace there's a concert of gamelan and dance by the best troupe in town. 

Here you see a guy in a fancy getup showing off: juggling drumsticks and playing a big percussive instrument while the rest of the musicians play as usual.  The androgynous look of this guy - eye shadow, lipstick, and so on - is common for male performers in traditional Balinese dances. 

Both for women and men, exaggerated eye motions and complicated hand gestures are a big part of dancing.  I imagine they have some significance, so I feel I'm watching signals in a code I haven't cracked.  Even when the dance tells a story I know - like the story of Rama and Sita from the Ramayana - I have trouble seeing the story in the dance.  It's beautiful but cryptic, and thus a bit boring at times.  Sort of like how most people feel aboutmath. ... more »

Gamelan and dance

Every Thursday in Ubud Palace there's a concert of gamelan and dance by the best troupe in town. 

Here you see a guy in a fancy getup showing off: juggling drumsticks and playing a big percussive instrument while the rest of the musicians play as usual.  The androgynous look of this guy - eye shadow, lipstick, and so on - is common for male performers in traditional Balinese dances. 

Both for women and men, exaggerated eye motions and complicated hand gestures are a big part of dancing.  I imagine they have some significance, so I feel I'm watching signals in a code I haven't cracked.  Even when the dance tells a story I know - like the story of Rama and Sita from the Ramayana - I have trouble seeing the story in the dance.  It's beautiful but cryptic, and thus a bit boring at times.  Sort of like how most people feel about math.  But if I relax a bit, it's enjoyable.

The older guy at left is playing a string instrument called a rebab.  It adds a wild and crazy improvisational touch to the tightly regimented, metallic percussive sound of the rest of the gamelan - sort of like an electric guitar doing a wailing solo. 

A gamelan is the whole orchestra consisting of different kinds of metallophones (metallic percussive instruments), hand drums (which do a lot of fancy improvisation), a gong (which keeps the beat), and sometimes xylophones, a rebab, or a bamboo flute.  A typical village will have one or more gamelans, tuned in different ways, which are played in various rituals.  For example, we went to a cremation ceremony in Bedulu and there was a gamelan playing there.  But the courts of Bali, Java and Sunda developed their own fancier gamelans for entertainment... and I guess this show is loosely based on the Bali court gamelan.

The Dutch invaded Bali in 1906, and the royal family and thousands of their followers fought to the death - lances and spears against guns - rather than face the humiliation of surrender.   Women mockingly threw jewelry and gold coins at the Dutch troops amid piles of dead Balinese soldiers.  The Dutch, unashamed, took over and bossed everyone around.  In 1942 the Japanese invaded and things got even worse.  When the Dutch came back in 1946, the Balinese resistance movement again fought to the death in large numbers.  When Indonesia finally gained its independence the military soon took control and around 1965 about 80,000 suspected communists were killed in Bali.

So, the dance performances in Ubud evoke an idyllic age - perhaps somewhat imaginary - before all this happened.  But many of the dances are actually rather new. 

Perhaps the idyllic age of Bali is now!  Tourists are rapidly changing the landscape, but one payoff from fighting the Dutch and Japanese is that the Balinese have a strong sense of national identity and want to run things their own way.  Literacy and life expectancies are way up.  The gamelan and dance performances for tourists let people earn a living while keeping these arts alive.

#ubud #bali #gamelan___

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2014-09-07 01:20:06 (14 comments, 1 reshares, 62 +1s) 

Bali

Ubud is the 'cultural capital' of Bali, up in the hill, far from the beaches - a good place to see dances and hear gamelan music.  In the heart of Ubud, on Monkey Forest Road, you'll see a sign for Igna Used Books.  It's next to an alley.  If you walk down that alley, you'll come to a small rice field on your left, and Alit's Warung on your right: a warung is a restaurant.  Turn right down an even smaller alley, and at the end is Hotel Okawati. 

It's one of my favorite places: you can fall asleep to the sound of crickets and frogs, and wake up to the sound of roosters and ducks.  Lisa and I wanted a peaceful little vacation, and we decided to come back here.  This is a view from the balcony.

We stayed here 5 days, but we're leaving today.  Back to Singapore. Back to work!

You shouldn't come here looking for abeautif... more »

Bali

Ubud is the 'cultural capital' of Bali, up in the hill, far from the beaches - a good place to see dances and hear gamelan music.  In the heart of Ubud, on Monkey Forest Road, you'll see a sign for Igna Used Books.  It's next to an alley.  If you walk down that alley, you'll come to a small rice field on your left, and Alit's Warung on your right: a warung is a restaurant.  Turn right down an even smaller alley, and at the end is Hotel Okawati. 

It's one of my favorite places: you can fall asleep to the sound of crickets and frogs, and wake up to the sound of roosters and ducks.  Lisa and I wanted a peaceful little vacation, and we decided to come back here.  This is a view from the balcony.

We stayed here 5 days, but we're leaving today.  Back to Singapore. Back to work!

You shouldn't come here looking for a beautiful place 'unspoiled by western civilization' - I haven't found such a place yet, and Ubud certainly isn't it.  There's litter, lots of motorbikes, lots of guys sitting on the street asking if you want a taxi, lots of nightclubs, lots of western-style restaurants where you can buy a hamburger or spaghetti...  but there's still good stuff here, and peace if you look for it.

#bali #ubud  ___

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2014-09-04 02:21:49 (34 comments, 38 reshares, 107 +1s) 

Stochastic resonance on your browser

Stochastic resonance is a way that noise can amplify a signal.  It sounds paradoxical, but crickets actually use it to hear better.

It might also help explain how ice ages work.

The idea is that you've got a system that can hop between two stable states.  Your signal might not be strong enough to make the hop occur.  So, add some noise!  This gives an extra push, which sometimes helps your system make the jump from one state to another.  It ain't pretty - but it really does work.

Here we see the green signal getting amplified by some noise to give the red output.  The red output jumps between high and low values.  If there were less noise, this wouldn't work!  You can see why here:

http://johncarlosbaez.wordpress.com/2014/08/28/the-stochastic-resonance-program-part-2/

Thishas a l... more »

Stochastic resonance on your browser

Stochastic resonance is a way that noise can amplify a signal.  It sounds paradoxical, but crickets actually use it to hear better.

It might also help explain how ice ages work.

The idea is that you've got a system that can hop between two stable states.  Your signal might not be strong enough to make the hop occur.  So, add some noise!  This gives an extra push, which sometimes helps your system make the jump from one state to another.  It ain't pretty - but it really does work.

Here we see the green signal getting amplified by some noise to give the red output.  The red output jumps between high and low values.  If there were less noise, this wouldn't work!  You can see why here:

http://johncarlosbaez.wordpress.com/2014/08/28/the-stochastic-resonance-program-part-2/

This has a link to a program that runs on your browser, where you can control the amount of noise with a slider.  It also features an explanation of this program, written by +David Tanzer.  After you read it, you can take the program, written in Javascript, and try stuff like this:

1. Change the color of the sine wave.

2. Change the exponent in the bistable polynomial to values other than 2, to see how this affects the output.

3. Add an integer-valued slider to control this exponent.

4. Modify the program to perform two runs of the process, and show the output signals in different colors.

5. Modify it to perform ten runs, and change the output signal to display the point-wise average of these ten runs.

6. Add an input slider to control the number of runs.

7. Add another plot, which shows the standard deviation of the output signals, at each point in time.

All this is courtesy of the Azimuth Code Project, especially +Glyn Adgie, +Allan Erskine and +David Tanzer.

#azimuth #azimuthcodeproject  ___

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2014-09-03 00:11:59 (18 comments, 26 reshares, 239 +1s) 

A cool cat from Central Asia: the manul

The manul lives in the grasslands and high steppes of Central Asia: Mongolia, Tajikistan, Kyrgyzstan, Pakistan, Kazakhstan, Kashmir, and western China.  It's the size of a house cat, but its stocky build and long, dense fur makes it look stout and plush.  It has a shorter jaw with fewer teeth than most cats.

Mamuls spend most of the day in caves, cracks in the rock, or marmot burrows.  In the late afternoon they come out and hunt. They can't run fast, so they mainly hunt by ambush or stalking. They feed largely on prey that are active during the day: gerbils, pikas, voles, partridges, and sometimes young marmots.

They're most closely related to the leopard cat, a small cat of southeast Asia which is sometimes crossbred with house cats to give beautiful cats called 'Bengals' that enjoy gettingwe... more »

A cool cat from Central Asia: the manul

The manul lives in the grasslands and high steppes of Central Asia: Mongolia, Tajikistan, Kyrgyzstan, Pakistan, Kazakhstan, Kashmir, and western China.  It's the size of a house cat, but its stocky build and long, dense fur makes it look stout and plush.  It has a shorter jaw with fewer teeth than most cats.

Mamuls spend most of the day in caves, cracks in the rock, or marmot burrows.  In the late afternoon they come out and hunt. They can't run fast, so they mainly hunt by ambush or stalking. They feed largely on prey that are active during the day: gerbils, pikas, voles, partridges, and sometimes young marmots.

They're most closely related to the leopard cat, a small cat of southeast Asia which is sometimes crossbred with house cats to give beautiful cats called 'Bengals' that enjoy getting wet.  Some even live here in Singapore, but I've sure never seen one!  The manul and the leopard cat seem to have diverged just 5 million years ago.  It always amazes me how new many mammal species are.

The manul is hard to raise in zoos, and they're listed as 'near threatened'.  They are hunted for its fur in relatively large numbers in China, Mongolia, and Russia, although international trade in manul pelts has largely ceased since the late 1980s, and Mongolia is the only place where it's still legal to kill them.

The manul is also called Pallas's cat, after a naturalist who wrote about them.  Its scientific name is Otocolobus manul :

https://en.wikipedia.org/wiki/Pallas's_cat

I got this picture from a great list of wild cats:

http://www.the-open-mind.com/26-gorgeous-cats-that-are-disappearing-from-the-wild/

Check out the species you may not know: the fishing cat, the Borneo bay cat, the flat headed cat, the Iberian lynx, and the margay!

#cats___

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2014-09-02 00:41:09 (118 comments, 86 reshares, 282 +1s) 

Hyperreal numbers: infinities and infinitesimals

When Leibniz invented calculus, he also invented infinitesimals - numbers that are bigger than zero but smaller than 1/2, 1/3, 1/4, ... and so on.  Many people were uncomfortable with these, so they figured out how to do calculus without infinitesimals.   That's how it's usually taught now.

But it turns out you can do calculus with infinitesimals in a perfectly rigorous way... and in some ways, it's easier!   Here's a free online textbook that teaches calculus this way:

• H. Jerome Keisler, Elementary Calculus,  http://www.vias.org/calculus/ or for a PDF version, https://www.math.wisc.edu/~keisler/calc.html.

The picture here is from this book.  There's a tiny little infinitesimal number ε, pronounced epsilon.  And 1/ε is infinitely big!  These aren't'real nu... more »

Hyperreal numbers: infinities and infinitesimals

When Leibniz invented calculus, he also invented infinitesimals - numbers that are bigger than zero but smaller than 1/2, 1/3, 1/4, ... and so on.  Many people were uncomfortable with these, so they figured out how to do calculus without infinitesimals.   That's how it's usually taught now.

But it turns out you can do calculus with infinitesimals in a perfectly rigorous way... and in some ways, it's easier!   Here's a free online textbook that teaches calculus this way:

• H. Jerome Keisler, Elementary Calculus,  http://www.vias.org/calculus/ or for a PDF version, https://www.math.wisc.edu/~keisler/calc.html.

The picture here is from this book.  There's a tiny little infinitesimal number ε, pronounced epsilon.  And 1/ε is infinitely big!  These aren't 'real numbers' in the usual sense.  Sometimes they're called hyperreal numbers.

You can calculate the derivative, or rate of change, of a function f by doing

(f(x+ε) - f(x)) / ε

and then at the end throwing out terms involving ε.  For example, suppose

f(x) = x²

Then to compute its derivative we do

((x+ε)² - x²) / ε

Working this out, we get

(x² + 2εx + ε² - x²) / ε = (2εx + ε²) / ε  = 2x + ε

At the end, we throw out the term involving ε.  So, we get 2x.  This is the rate of change of the function x².

The book will teach you calculus this way, from scratch.  If you had trouble understanding 'limits' in calculus, you might prefer this way.  Or, you might just enjoy seeing another approach.

The details of this subject are infinitely interesting, but I'll just say an infinitesimal amount.  In 1961 the logician Abraham Robinson showed that hyperreal numbers are just as consistent as ordinary real numbers, and that the two systems are compatible in a certain precise sense.  In 1976, Jerome Keisler, a student of the famous logician Tarski, published this elementary textbook that teaches calculus using hyperreal numbers.  

Now it's free, with a Creative Commons copyright! ___

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2014-09-01 04:02:12 (12 comments, 49 reshares, 177 +1s) 

A sunflower at infinity

This picture by +Roice Nelson shows the 'view at infinity' of a honeycomb in hyperbolic space.

A honeycomb is a way of chopping space into polyhedra.  For example, we can chop ordinary 3d space into cubes.  This is called the {4,3,4} honeycomb.  Why?

• a square has 4 sides so its symbol is {4}

• a cube has 3 squares meeting at each corner so its symbol is {4,3}

• the cubical honeycomb has 4 cubes meeting at each edge so its symbol is {4,3,4}

The picture here is a view of the {3,3,7} honeycomb.  This is defined in the same sort of way, but it doesn't fit into ordinary Euclidean space.  It fits into a curved space called hyperbolic space!   The honeycomb extends forever, and it forms this pattern where it meets the 'plane at infinity' of hyperbolic space.
For ... more »

A sunflower at infinity

This picture by +Roice Nelson shows the 'view at infinity' of a honeycomb in hyperbolic space.

A honeycomb is a way of chopping space into polyhedra.  For example, we can chop ordinary 3d space into cubes.  This is called the {4,3,4} honeycomb.  Why?

• a square has 4 sides so its symbol is {4}

• a cube has 3 squares meeting at each corner so its symbol is {4,3}

• the cubical honeycomb has 4 cubes meeting at each edge so its symbol is {4,3,4}

The picture here is a view of the {3,3,7} honeycomb.  This is defined in the same sort of way, but it doesn't fit into ordinary Euclidean space.  It fits into a curved space called hyperbolic space!   The honeycomb extends forever, and it forms this pattern where it meets the 'plane at infinity' of hyperbolic space.

For links to related pictures, visit my +American Mathematical Society blog Visual Insight:

http://blogs.ams.org/visualinsight/2014/09/01/intersection-of-337-honeycomb-and-the-plane-at-infinity/

#geometry  ___

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2014-08-30 06:59:03 (34 comments, 64 reshares, 180 +1s) 

The Klein bottle has just one side!  

A mathematician named Klein
Thought the Möbius strip was divine
Said he, "If you glue
The edges of two
You get a weird bottle like mine."

The Möbius strip has just one side and one edge.  So if you glue the edges of two, you get a surface with one side and no edge... which happens to be a Klein bottle!

But here's an even cooler way to get a Klein bottle.  Consider black-and-white images that are 3×3 pixels in size.  It takes 9 numbers to describe such an image.  So, the set of all possible images like this is a 9-dimensional cube.  

But if you go through actual black-and-white photos and keep track of the 3×3-pixel images that show up, you'll find they aren't evenly distributed in this 9-dimensional cube!  Most of them lie near acertain sur... more »

The Klein bottle has just one side!  

A mathematician named Klein
Thought the Möbius strip was divine
Said he, "If you glue
The edges of two
You get a weird bottle like mine."

The Möbius strip has just one side and one edge.  So if you glue the edges of two, you get a surface with one side and no edge... which happens to be a Klein bottle!

But here's an even cooler way to get a Klein bottle.  Consider black-and-white images that are 3×3 pixels in size.  It takes 9 numbers to describe such an image.  So, the set of all possible images like this is a 9-dimensional cube.  

But if you go through actual black-and-white photos and keep track of the 3×3-pixel images that show up, you'll find they aren't evenly distributed in this 9-dimensional cube!  Most of them lie near a certain surface.  

And this surface is a Klein bottle!

Puzzle 1: Why?

If you get stuck, look at this picture:

http://math.ucr.edu/home/baez/mathematical/klein_bottle_carlsson.jpg

If that doesn't give it away, do this:

Puzzle 2: Start with a square of paper.  Glue the top and bottom together without a twist.  Glue the left and right sides together with a twist.  What do you get?

If you actually try this, you'll find it a bit hard.  I prefer to just visualize it.  This is a good way to build your powers of visualization.

Here's the paper where the Klein bottle was found in the space of 3×3 images:

• Gunnar Carlsson, Tigran Ishkhanov, Vin de Silva and Afra Zomorodian, On the local behavior of spaces of natural images, International Journal of Computer Vision 76 (2008), 1-12.  Available at http://comptop.stanford.edu/u/preprints/mumford.pdf.

They show as you study n×n pixel images for larger and larger n, the space of most likely images eventually stops being a Klein bottle - more complicated image features start showing up.

I thank my friend the physicist +Igor Khavkine for pointing out this paper.  I got the animated gif from the webpage of Dr. Christian Salas:

http://www.drchristiansalas.org.uk/sidepage4.htm

but I don't know its original source.  Do you?

#spnetwork doi:10.1.1.121.9768 #computerVision #topology___

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2014-08-28 15:40:07 (0 comments, 7 reshares, 76 +1s) 

Goodbye, it was nice knowing you.

Goodbye, it was nice knowing you.___

posted image

2014-08-27 06:01:31 (74 comments, 76 reshares, 230 +1s) 

Did you ever feel your brain is about to break?  It happens not just to beginners in math, but to everyone! 

So, in case you feel frustrated with math, let me share one of my own frustrations.  You'll see you're not alone!

Right now I'd love to understand something a logician tried to explain to me at lunch a while back.  His name is Boris Zilber.  He's studying what he calls 'logically perfect' theories - that is, lists of axioms that almost completely determine the structure they're trying to describe.  He thinks that we could understand physics better if we thought harder about these logically perfect theories.

Now, Gödel's incompleteness theorem means our usual theory of natural numbers

0, 1, 2, 3, ...

together with addition and multiplication is far from 'logically perfect' in this sense.   In fact,only fi... more »

Did you ever feel your brain is about to break?  It happens not just to beginners in math, but to everyone! 

So, in case you feel frustrated with math, let me share one of my own frustrations.  You'll see you're not alone!

Right now I'd love to understand something a logician tried to explain to me at lunch a while back.  His name is Boris Zilber.  He's studying what he calls 'logically perfect' theories - that is, lists of axioms that almost completely determine the structure they're trying to describe.  He thinks that we could understand physics better if we thought harder about these logically perfect theories.

Now, Gödel's incompleteness theorem means our usual theory of natural numbers

0, 1, 2, 3, ...

together with addition and multiplication is far from 'logically perfect' in this sense.   In fact, only finite-sized mathematical structures can be completely determined by finite lists of axioms in ordinary logic (so-called 'first-order logic').

If we've got an infinite-sized structure, the most we can hope for is that after we specify the size of the structure, the axioms completely determine it. 

And this actually happens sometimes.  It happens for the complex numbers!  Zilber believes this has something to do with why the complex numbers show up so much in physics.

More precisely, say λ is some size - that is, some cardinal, which could be finite or infinite.  A list of axioms in first-order logic is called λ-categorical if it's obeyed by a unique structure of size λ.   And a guy named Morley showed that if a list of axioms is λ-categorical for some uncountable λ, it's also λ-categorical for all uncountable λ.   I have no idea why this is true.  But such lists of axioms are called uncountably categorical.

According to Zilber - and I'm sure he knows what he's talking about here - the axioms for the complex numbers together with addition and multiplication are uncountably categorical.

Zilber likes lists of axioms that are uncountably categorical so much that he calls them logically perfect theories.  And he writes:

There are purely mathematical arguments towards accepting the above for a defi nition of perfection. First, we note that the theory of the field of complex numbers (in fact any algebraically closed fi eld) is uncountably categorical. So, the fi eld of complex numbers is a perfect structure, and so are all objects of complex algebraic geometry by virtue of being de finable in the fi eld.

It is also remarkable that Morley's theory of categoricity (and its extensions) exhibits strong regularities in models of categorical theories generally. First, the models have to be highly homogeneous [....] Moreover, a notion of dimension (the Morley rank) is applicable to defi nable subsets in uncountably categorical structures, which gives one a strong sense of working with curves, surfaces and so on in this very abstract setting. A theorem of the present author states more precisely that an uncountably categorical structure M is either reducible to a 2-dimensional "pseudo-plane" with at least a 2-dimensional family of curves on it (so is non-linear), or is reducible to a linear structure like an (infinite dimensional) vector space, or to a simpler structure like a G-set for a discrete group G.

Somehow 'logical perfection' in Zilber's sense connects logic to some concepts from geometry!  But I don't understand any of the details.  And when I start studying them, I feel like Gollum here.

• Boris Zilber, Perfect infinities and finite approximation, in Infinity and Truth, World Scientific, Singapore, 2014,
https://people.maths.ox.ac.uk/zilber/inf-to-finite.pdf.

#spnetwork #logic___

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2014-08-26 02:52:51 (74 comments, 11 reshares, 93 +1s) 

The latest news is getting people upset. +David Brin writes:

This is what we had all feared.  The possible tipping point.  Methane plumes are emanating from at least 570 seafloor cold seeps on the outer continental shelf and the continental slope, Mississippi State University reported.  A potential disaster that I warned about in EARTH (1989), accelerating ocean acidification and turning the greenhouse into a runaway.

http://www.hngn.com/articles/40189/20140825/methane-leaking-from-sea-floor-could-change-ocean-acidity-oxygen-content.htm

"Warming of ocean temperatures on seasonal, decadal or much longer time scales can cause gas hydrate to release its methane, which may then be emitted at seep sites," said Carolyn Ruppel, study co-author and chief of the USGS Gas Hydrates Project.  "Such continental slope seeps have previously been recognized inthe... more »

The latest news is getting people upset. +David Brin writes:

This is what we had all feared.  The possible tipping point.  Methane plumes are emanating from at least 570 seafloor cold seeps on the outer continental shelf and the continental slope, Mississippi State University reported.  A potential disaster that I warned about in EARTH (1989), accelerating ocean acidification and turning the greenhouse into a runaway.

http://www.hngn.com/articles/40189/20140825/methane-leaking-from-sea-floor-could-change-ocean-acidity-oxygen-content.htm

"Warming of ocean temperatures on seasonal, decadal or much longer time scales can cause gas hydrate to release its methane, which may then be emitted at seep sites," said Carolyn Ruppel, study co-author and chief of the USGS Gas Hydrates Project.  "Such continental slope seeps have previously been recognized in the Arctic, but not at mid-latitudes.  So this is a first."

To be clear, methane is a far worse greenhouse gas than CO2… and rising ocean temps will cause icy methane hydrates to fizz, all over the globe, possibly causing a runaway effect.

Which leaves me with this to say to you wretched, monstrous, science-hating fools, with your “hypnotize-me!” Fox-Nuremberg rallies and neo-confederate rant-fests against all big-city-university “smartypants” types. 

No evidence will change your opposition to negotiating even moderate, sensible, precautionary interim measures to increase energy efficiency or do basic R&D. You sabotage TWODA (Things We Ought to be Doing Anyway.)  You helped to sabotage the satellites and instruments and research vessels that could have nailed it all down.

You help to denigrate and geld the smartest, most knowledgeable, competitive and wisest human beings whom our species has ever produced, and thereby you declare yourselves to be brave authority questioners and skeptics!  While kneejerk-robotically obeying  the hypnotize-me channels owned by coal and oil barons... and never once sniffing the suspicious irony.

We can expect more of this sort of rhetoric in the decades to come.  However, about this new discovery, it's worth noting:

1) Nobody before has looked very hard for methane venting off the East Coast, so it's hard to tell if it's new.  The guy who found the vents, Adam Skarke, says:

The fact that it is there in the quantities that it is and it is exposed suggests that indeed the processes at these locations have been going on, in a very general sense, on the order of at least 1,000 years.

So we need to monitor the vents and see if they're getting bigger, while meanwhile taking action to limit carbon emissions.  Brin is right about stupid US government cutbacks on climate monitoring.  It's a really shortsighted way to save money!

2) The amount of methane being released from the vents seen in this study is small.  Methane is indeed a very bad greenhouse gas, and we do need to worry about a 'tipping point' where warming oceans release more of the vast amounts of methane stored in sea beds, thus warming the Earth even more.  There are vast amounts of methane stored in hydrates there, and there's evidence that it could have been released during the Paleocene-Eocene Thermal Maximum, an episode when the Earth's temperature rose by 6 °C (11 °F) in about 20,000 years.  However,  there's no evidence that we're already in an unstoppable runaway situation. 

David Archer, an expert on climate and the carbon cycle, says:

In the timescale of centuries to hundreds of thousands of years, it (methane hydrate) is clearly a significant amplifier.  But in terms of the climate of just the coming century, the actual forcing of climate from the rise in atmospheric methane due to this, I think, will be small.

He, like most climate scientists, believes that human CO2 emissions are the main cause of global warming.

3) Let's not just complain, let's do something!  David Brin is talking about US politics, so let me focus on that - just today.  Until there's more action at the federal level in the US, we can act locally.  California has a carbon cap-and-trade system, which is linked with Quebec's:

http://www.arb.ca.gov/cc/capandtrade/capandtrade.htm

Connecticut, Delaware, Maine, Maryland, Massachusetts, New Hampshire, New York, Rhode Island, and Vermont have their own system, the Regional Greenhouse Gas Initiative:

http://www.rggi.org/

Another group, the Midwestern Greenhouse Gas Reduction Accord, includes Illinois, Iowa, Kansas, Michigan, Minnesota, Wisconsin and also Manitoba:

https://en.wikipedia.org/wiki/Midwestern_Greenhouse_Gas_Reduction_Accord

And the Western Climate Initiative includes Arizona, California, Montana, New Mexico, Oregon, Utah and Washington in the US, and British Columbia, Manitoba, Ontario and Quebec in Canada:

http://www.co2offsetresearch.org/policy/WCI.html

So: we should figure out ways to help these initiatives succeed!   The ones that succeed will become the groundwork for a national system!  Ideas? 

The actual paper on methane emissions is here:

• A. Skarke et al, Widespread methane leakage from the sea floor on the northern US Atlantic margin, Nature Geoscience , 24 August 2014, http://www.nature.com/ngeo/journal/vaop/ncurrent/full/ngeo2232.html.

It's not free except for the abstract, so someone should 'liberate' it.  It's absurd to have such important results locked up, while any blogger with a loud mouth (like me) can be read for free.___

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2014-08-25 04:56:35 (37 comments, 71 reshares, 194 +1s) 

Why do we have these particles in our Universe?

We understand a lot about physics - and that makes the remaining mysteries even more tantalizing!  For example: why are quarks so much like leptons?

Elementary particles come in two main kinds: the ones that carry forces (gauge bosons) and the ones that make up matter (quarks and leptons).   There's also at least one more... but never mind!  Today's puzzle is about quarks and leptons.  You'll see from the chart that they look sort of similar.   But why?

Maybe a lightning review of particle physics will help, in case you skipped that class in high school.  Most of the matter you see is made of electrons, protons and neutrons.   Protons and neutrons are made of up and down quarks, held together by the strong force.  But electrons are 'leptons', which means they don'tfeel the s... more »

Why do we have these particles in our Universe?

We understand a lot about physics - and that makes the remaining mysteries even more tantalizing!  For example: why are quarks so much like leptons?

Elementary particles come in two main kinds: the ones that carry forces (gauge bosons) and the ones that make up matter (quarks and leptons).   There's also at least one more... but never mind!  Today's puzzle is about quarks and leptons.  You'll see from the chart that they look sort of similar.   But why?

Maybe a lightning review of particle physics will help, in case you skipped that class in high school.  Most of the matter you see is made of electrons, protons and neutrons.   Protons and neutrons are made of up and down quarks, held together by the strong force.  But electrons are 'leptons', which means they don't feel the strong force.

Up quarks, down quarks and electrons - those are 3 of the 4 particles in 'generation 1'.  The 4th is the electron neutrino.  It's also a lepton - it's doesn't feel the strong force.  But it's also has no electric charge!  So, it's very hard to detect - it whizzes easily through ordinary matter.   But we have detected it, and we actually know a  huge amount about it.

We also know that besides 'generation 1' there's a 'generation 2' and 'generation 3' of quarks and leptons.  We're pretty sure there are only 3: people have done experiments that show there can't be more different kinds of neutrinos, unless they are very heavy, or different from all the rest in some other way.  We have no idea why there are only 3 generations.

But our puzzle today is: why do quarks and leptons come in generations at all?    So let's just think about generation 1.

We know that the up and down quark are closely connected.   We also know that the electron and electron neutrino are closely connected.  For example, you can collide an electron and an up quark and have them turn into an electron neutrino and a down quark!  We understand this stuff very well, actually: there's a detailed mathematical theory of it, and it works great.

But there are other things that seem mysterious.   The up quark has charge 2/3, the down quark has charge -1/3, the electron has charge -1 and the electron neutrino has charge 0.   Quarks also come in 3 different kinds, called 'colors' - they change colors when they interact with the strong force.   Leptons have no color.

Are all the 3's in the last paragraph a coincidence?   It seems not.  For example, if quarks came in 4 colors, but had the charges they do now, all hell would break loose!   I could explain why, but that's not my goal today.

My goal is just to say this: there's a theory called the Pati-Salam model that says leptons are secretly just a funny kind of quarks - a 'fourth color of quark'.  This theory unifies quarks and leptons.  And this theory also explains why quarks have charges like 2/3 and -1/3.

This theory has been around since 1974.  It has some problems.  If it didn't, we'd probably all believe it by now!  It's very hard to find theories of elementary particles that fit all the data we have; if you just make up stuff, you'll almost surely run into problems.  But the Pati-Salam model is pretty good, it's not completely ruled out by experiments... and last year something interesting happened.

A famous mathematician named Alain Connes has an approach to physics based on noncommutative geometry, which replaces our usual picture of spacetime by something that's more like algebra than geometry.  His theory predicted the wrong mass for the Higgs boson - that's the extra particle I hinted at near the start of this story.  But last year he came out with a new improved version, that doesn't suffer from this problem.  And it turns out to be a lot like the Pati-Salam model!

What's interesting is how he gets it.  In his earlier work, he laid down a bunch of mathematical axioms, and one of the simplest theories that obeys all these axioms turned out to be very similar to the Standard Model - our usual theory of particles.  

But now, he and some other guys have noticed that if you drop one of the axioms, something like the Pati-Salam model is also allowed.  Moreoever, you can get a Higgs boson with the right mass!

I wish I understood this better.  Alas, I don't have much time for this stuff anymore!  Here is his paper:

• Ali H. Chamseddine, Alain Connes and Walter D. van Suijlekom, Beyond the spectral standard model: emergence of Pati-Salam unification, http://arxiv.org/abs/1304.8050.

and here is an intro to the Pati-Salam model, mainly good for mathematicians and physicists:

• John Baez and John Huerta, The algebra of grand unified theories, http://math.ucr.edu/~huerta/guts/guts.html

Here's the abstract of Connes' paper, which gives a flavor of what he's doing... at least if you know enough jargon:

Abstract. The assumption that space-time is a noncommutative space formed as a product of a continuous four dimensional manifold times a finite space predicts, almost uniquely, the Standard Model with all its fermions, gauge fields, Higgs field and their representations. A strong restriction on the noncommutative space results from the first order condition which came from the requirement that the Dirac operator is a differential operator of order one. Without this restriction, invariance under inner automorphisms requires the inner fluctuations of the Dirac operator to contain a quadratic piece expressed in terms of the linear part. We apply the classification of product noncommutative spaces without the first order condition and show that this leads immediately to a Pati-Salam SU(2)_R x SU(2)_L x SU(4) type model which unifies leptons and quarks in four colors. Besides the gauge fields, there are 16 fermions in the (2,2,4) representation, fundamental Higgs fields in the (2,2,1), (2,1,4) and (1,1,1+15) representations. Depending on the precise form of the order one condition or not there are additional Higgs fields which are either composite depending on the fundamental Higgs fields listed above, or are fundamental themselves. These additional Higgs fields break spontaneously the Pati-Salam symmetries at high energies to those of the Standard Model.

#spnetwork arXiv:1304.8050 #mustread___

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2014-08-24 01:08:06 (52 comments, 87 reshares, 403 +1s) 

Head on

Wolves run through the air, hit a glass wall and fall down.  Then they pick themselves up, go back and do it again.

This is just one of the remarkable and unsettling pieces by Chinese artist Cai Guo-Qiang

He recently made the news by creating a kind of Noah's ark with endangered animals and floating it down the river past the main financial district of Shanghai.  The animals aren't real - but they look pretty real, like these wolves.

The boat is now on display in Shanghai, in an exhibit that's become very popular.  You can see more of it here:

http://www.npr.org/blogs/parallels/2014/08/21/342189261/chinas-pollution-crisis-inspires-an-unsettling-art-exhibit

and here:

http://www.emptykingdom.com/featured/cai-guo-qiang/

A piece called 'Silent Ink' features a waterfall of inkplu... more »

Head on

Wolves run through the air, hit a glass wall and fall down.  Then they pick themselves up, go back and do it again.

This is just one of the remarkable and unsettling pieces by Chinese artist Cai Guo-Qiang

He recently made the news by creating a kind of Noah's ark with endangered animals and floating it down the river past the main financial district of Shanghai.  The animals aren't real - but they look pretty real, like these wolves.

The boat is now on display in Shanghai, in an exhibit that's become very popular.  You can see more of it here:

http://www.npr.org/blogs/parallels/2014/08/21/342189261/chinas-pollution-crisis-inspires-an-unsettling-art-exhibit

and here:

http://www.emptykingdom.com/featured/cai-guo-qiang/

A piece called 'Silent Ink' features a waterfall of ink plunging into a 5,300-gallon lake excavated from the museum's floor.  The lake is ringed by mounds of crushed concrete and iron bars.  It looks like a scene from a Chinese landscape painting - made of industrial waste.  It's hard to stay there for very long, because the smell of the ink becomes overpowering. 

But in its own strange way it's beautiful.

Of course, if you don't know the politics of China you'll miss part of the meaning of this wolf pack.  If you don't know that 16,000 dead pigs were found floating down a river in Shanghai last year, you won't fully understand that ark.  If you don't know a bit about the pollution crisis in China and the art of landscape painting, you'll miss some of what's going on in 'Silent Ink'.  But this art is good because it's not merely commentary on politics and the pollution crisis in China.  It's visually stunning, mysterious and tragic.

#art  ___

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2014-08-23 03:45:54 (85 comments, 10 reshares, 96 +1s) 

Sub-Roman Britain

I've been a fan of the legends of King Arthur for a long time.  I love how they continue to inspire new versions, from Marion Zimmer Bradley's Tales of Avalon (where Morgaine is recast as a hero in the doomed struggle of the Druids against encroaching Christians) to the goofy but fun TV series Merlin (where a black Guinevere starts as a serving-girl and winds up ruling Camelot).

But I'm only just now poking into the mysterious centuries from 400 to 600 AD in Britain, after the collapse of Roman rule, when Arthur would have lived... if he existed.

It's really cool to imagine life in former Roman towns and villas during these 'dark ages'.  We have some archaeological evidence, but very little written history: mainly just the writings of Saint Patrick and a book called On the Ruin and Conquest of Britain,  written by a guyca... more »

Sub-Roman Britain

I've been a fan of the legends of King Arthur for a long time.  I love how they continue to inspire new versions, from Marion Zimmer Bradley's Tales of Avalon (where Morgaine is recast as a hero in the doomed struggle of the Druids against encroaching Christians) to the goofy but fun TV series Merlin (where a black Guinevere starts as a serving-girl and winds up ruling Camelot).

But I'm only just now poking into the mysterious centuries from 400 to 600 AD in Britain, after the collapse of Roman rule, when Arthur would have lived... if he existed.

It's really cool to imagine life in former Roman towns and villas during these 'dark ages'.  We have some archaeological evidence, but very little written history: mainly just the writings of Saint Patrick and a book called On the Ruin and Conquest of Britain,  written by a guy called Saint Gildas or 'Gildas the Wise'.

This book is a grumpy attack on various kings, including:

• "Constantine, the tyrannical whelp of the unclean lioness of Dumnonia".   The kingdom of Dumnonia was in southwest England, mainly Devon and Cornwall.

• "dragon of the island... Maelgwn".  Maelgwn was one of the kings of Gwynedd, who ruled a chunk of what's now Wales from their base on the Isle of Anglesey.

• "Vortipore... like to the spotted leopard... tyrant of the Demetians."   Vortipore was a king of Demetia, or Dyfed, a small kingdom in south-west Wales.

There were many other kingdoms that Gildas didn't bother to write about.  Gildas is himself rather mysterious; one later biography tells of how he helped mediate a struggle between King Arthur and a king who had abducted and raped Guinevere... but this is all just legend.

'Medieval Bex' has a great blog on medieval matters, and here's what she says about the two biographies of Gildas:

The earlier account, written in the ninth century in Rhuys, Brittany, tells how Gildas son of Caw was born in the north of Britain. He moved to a monastic college to begin his education and then to Iren (probably Ireland) to continue his studies, before returning to north Britain to preach to those naughty heathens. St Brigid (d. 524) asked Gildas for a token so he made her a bell. As you do. After these high-jinks he then travelled around a bit before settling in Rhuys, where he built a monastery and lived out his days preaching and writing epistles about kings that he didn’t like very much. When he died his body was placed in a boat and set adrift according to his wishes. Just a floating corpse; not set aflame or anything. Imagine being the person to find a boat containing a decomposing monk… which someone actually did (the HORROR!!) – his boat washed up a few months later and was found by some men from Rhuys. They did the sensible thing and took his body back to Rhuys and buried it there. Gildas’ corpsified wandering days were over.

The other book, however, depicts Gildas as a sort of monk-cum-Arthurian action hero. It seems like the writer of the twelfth-century biography, Caradoc of Llancarfan, read the earlier book and said ‘Oh ho! I think we can do better than that!’ and essentially pimped the Life of Gildas. The twelfth-century version has Gildas educated in Gaul before settling near Glastonbury… all normal enough so far, if ever-so-slightly at odds with the ninth-century version of events… but then things get a little bit more exciting when Guinevere and Arthur arrive on the scene! That’s right, no floating corpses here!

According to Caradoc’s biography of Gildas, King Melwas abducted Queen Guinevere and Arthur then proceeded to throw a massive wobbly. He stormed over to Melwas’ stronghold in Glastonbury with his knights, ready to attack. It was all getting a bit intense… until Gildas stepped in and saved the day! He happened to be in the neighbourhood and persuaded Melwas to release Guinevere, before unbelievably managing to make the two kings kiss and make up. They probably all went for a beer and a good chortle about it all afterwards. As an interesting aside, this is the first recorded instance of the Guinevere abduction scene, a plot which becomes a recurring motif in subsequent redactions of the Arthurian stories. So a highly imaginative biography of a monk has helped to shape the legend of Arthur as we know it today. Who’d have thought! There is also something in this version about Gildas’ brothers rising up against Arthur, and one of them being killed, and Gildas being rather upset about this. Apparently the large stone in Ruthin town square (north Wales) is the chopping block that was used when Arthur decapitated Gildas’ brother. It’s still there, you can go and see it!

Here is her blog:

http://medievalbex.wordpress.com/2012/10/31/the-arthurian-tradition-gildas/

and here is some other good stuff:

https://en.wikipedia.org/wiki/Sub-Roman_Britain
https://en.wikipedia.org/wiki/Gildas

#history  ___

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2014-08-22 02:42:17 (69 comments, 79 reshares, 183 +1s) 

The ostrich effect

Why do people think ostriches stick their heads under the sand when they're scared?  A Roman named Pliny the Elder might be to blame.  He wrote that ostriches "imagine, when they have thrust their head and neck into a bush, that the whole of their body is concealed."  

That would be silly - birds aren't that dumb.  But people will actually pay to avoid learning unpleasant facts.  It seems irrational to avoid information that can help us survive.  But people do it.  It's called information aversion.

Here's a new experiment:

In order to gauge how information aversion affects health care, one group of researchers decided to look at how college students react to being tested for a sexually transmitted disease.

That's a subject a lot of students worry about, according to JoshTasoff,... more »

The ostrich effect

Why do people think ostriches stick their heads under the sand when they're scared?  A Roman named Pliny the Elder might be to blame.  He wrote that ostriches "imagine, when they have thrust their head and neck into a bush, that the whole of their body is concealed."  

That would be silly - birds aren't that dumb.  But people will actually pay to avoid learning unpleasant facts.  It seems irrational to avoid information that can help us survive.  But people do it.  It's called information aversion.

Here's a new experiment:

In order to gauge how information aversion affects health care, one group of researchers decided to look at how college students react to being tested for a sexually transmitted disease.

That's a subject a lot of students worry about, according to Josh Tasoff, an economist at Claremont Graduate University who led the study along with Ananda Ganguly, an associate professor of accounting at Claremont McKenna College.

The students were told they could get tested for the herpes simplex virus. It's a common disease that spreads via contact. And it has two forms: HSV1 and HSV2.

The type 1 herpes virus produces cold sores. It's unpleasant, but not as unpleasant as type 2, which targets the genitals. Ganguly says the college students were given information — graphic information — that made it clear which kind of HSV was worse.

"There were pictures of male and female genitalia with HSV2, guaranteed to kind of make them really not want to have the disease," Ganguly says.

Once the students understood what herpes does, they were told a blood test could find out if they had either form of the virus.

Now, in previous studies on information aversion it wasn't always clear why people declined information. So Tasoff and Ganguly designed the experiment to eliminate every extraneous reason someone might decline to get information.

First, they wanted to make sure that students weren't declining the test because they didn't want to have their blood drawn. Ganguly came up with a way to fix that: All of the students would have to get their blood drawn. If a student chose not to get tested, "we would draw 10 cc of their blood and in front of them have them pour it down the sink," Ganguly says.

The researchers also assured the students that if they elected to get the blood tested for HSV1 and HSV2, they would receive the results confidentially.

And to make triply sure that volunteers who said they didn't want the test were declining it to avoid the information, the researchers added one final catch. Those who didn't want to know if they had a sexually transmitted disease had to pay $10 to not have their blood tested.

So what did the students choose? Quite a few declined a test.

And while only 5 percent avoided the HSV1 test, three times as many avoided testing for the nastier form of herpes.

For those who didn't want to know, the most common explanation was that they felt the results might cause them unnecessary stress or anxiety.

Let's try extrapolating from this.  Global warming is pretty scary.  What would people do to avoid learning more about it?  You can't exactly pay scientists to not tell you about it.  But you can do lots of other things: not listen to them, pay people to contradict what they're saying, and so on.  And guess what?  People do all these things.

So, don't expect that scaring people about global warming will make them take action.  If a problem seems scary and hard to solve, many people will just avoid thinking about it.

Maybe a better approach is to tell people things they can do about global warming.  Even if these things aren't big enough to solve the problem, they can keep people engaged.

There's a tricky issue here.  I don't want people to think turning off the lights when they leave the room is enough to stop global warming.  That's a dangerous form of complacency.  But it's even worse if they decide global warming is such a big problem that there's no point in doing anything about it.  

The quote is from here:

• Shankar Vedantham, Why we think ignorance Is bliss, even when It hurts our health, http://www.npr.org/blogs/health/2014/07/28/333945706/why-we-think-ignorance-is-bliss-even-when-it-hurts-our-health.

Here's the actual study:

• Ananda Ganguly and Joshua Tasoff, Fantasy and dread: the demand for information and the consumption utility of the future, http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2370983.

The photo, probably fake, is from here:

http://www.ostrichheadinsand.com/___

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2014-08-21 02:43:20 (24 comments, 16 reshares, 71 +1s) 

A mix of bosonic and fermionic superfluids

Two different kinds of gas, each completely frictionless, can pass through each other as they wiggle back and forth in a magnetic field!

One gas is made of lithium with an even number of electrons, protons and neutrons, so it's called a boson. The other is made of lithium with an odd number of these particles, so it's a fermion.  They act very different. 

If you cool down a gas of bosons enough, it can become a completely frictionless fluid - a superfluid.  It's a lot harder to do this trick for fermions.  This is the first time anyone has made a mixture of bosonic and fermionic superfluids.

Christophe Salomon gave a talk about this here at the +Centre for Quantum Technologies.  He said nobody has been able to make such a mixture with helium.

Helium-4 is the mostcomm... more »

A mix of bosonic and fermionic superfluids

Two different kinds of gas, each completely frictionless, can pass through each other as they wiggle back and forth in a magnetic field!

One gas is made of lithium with an even number of electrons, protons and neutrons, so it's called a boson. The other is made of lithium with an odd number of these particles, so it's a fermion.  They act very different. 

If you cool down a gas of bosons enough, it can become a completely frictionless fluid - a superfluid.  It's a lot harder to do this trick for fermions.  This is the first time anyone has made a mixture of bosonic and fermionic superfluids.

Christophe Salomon gave a talk about this here at the +Centre for Quantum Technologies.  He said nobody has been able to make such a mixture with helium.

Helium-4 is the most common kind of helium.  It has an even number of electrons, protons and neutrons, so it's a boson.  If you cool it down to just 2 degrees above absolute zero it becomes a superfluid, and does amazing things like climb over the walls of a jar.  In a superfluid, all the atoms act like a single thing.

Helium-3 is missing a neutron, so it has an odd number of protons, electrons and neutrons: 5 instead of 6.  This makes it a fermion.  It doesn't become superfluid until it cools down enough for the atoms to pair up, giving an even number of particles.  That happens at about 0.0025 degrees above absolute zero.

Nobody has succeeded in making a mixture of superfluid helium-4 and helium-3.  I don't completely understand why, but Christophe Salomon said the two kinds repel each other.   Theorists believe you can get them to mix if you cool them down to about a nanokelvin - a billionth of a degree above absolute zero!  But nobody has succeeded yet.

Christophe Salomon and his team tried a different idea.  They're using two kinds of lithium, one a boson and one a fermion.  Lithium is a metal, so it's solid at low temperatures.  But a very dilute gas of lithium, held in place by a magnetic field, can become superfluid! 

At 100 nanokelvin, both kinds of lithium become superfluid, and clouds of the two kinds can pass through each other.  That's very cold - but 100 times warmer than you'd need for helium.

In the picture here, each cloud is shown separately, so you can see how they oscillate back and forth, held loosely in place by a magnetic field.  They oscillate at different rates because one kind of lithium is a bit heavier than the other.  The clouds pass through each other, but they interact slightly, so over time they slow down.  The experiment sees just what the theory predicts.

Read more here!

• Igor Ferrier-Barbut, Marion Delehaye, Sebastien Laurent, Andrew T. Grier, Matthieu Pierce, Benno S. Rem, Frédéric Chevy and Christophe Salomon, A mixture of Bose and Fermi superfluids, http://arxiv.org/abs/1404.2548.

#spnetwork arXiv:1404.2548 #superfluids___

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2014-08-20 00:11:31 (51 comments, 5 reshares, 80 +1s) 

What caption should this picture have?

And who took it? 

I found this picture at the New York Daily News, which says it's an alligator on top of its mother.  It's part of a series of cute pictures of animals and their mothers:

http://www.nydailynews.com/life-style/baby-animals-mothers-gallery-1.1076585?pmSlide=1.1340672

But they don't credit anyone!

[EDIT: yes they do!  As +Ralf Haring noted, it's by Clayton Bonds.]

What caption should this picture have?

And who took it? 

I found this picture at the New York Daily News, which says it's an alligator on top of its mother.  It's part of a series of cute pictures of animals and their mothers:

http://www.nydailynews.com/life-style/baby-animals-mothers-gallery-1.1076585?pmSlide=1.1340672

But they don't credit anyone!

[EDIT: yes they do!  As +Ralf Haring noted, it's by Clayton Bonds.]___

posted image

2014-08-19 02:00:21 (47 comments, 95 reshares, 339 +1s) 

How to turn space into time - in the lab

Wouldn't it be cool if we could turn one dimension of space into an extra dimension of time?  We can't - but we can fake it!

Einstein showed the only difference between space and time is a minus sign.  So: make a material where light obeys equations with an extra minus sign! 
 
How?  Just take lots of microscopic metal wires and put them in transparent stuff that doesn't conduct electricity.  Line them all up.  You'll get something that conducts electricity like a metal in one direction but not the other two directions! 

It's called a hyperbolic metamaterial.  The video explains why. 

But why is this like converting a dimension of space into a dimension of time? 

Einstein showed that if you have a photon - a particle of light - in a vacuum, it obeys
X² + Y² ... more »

How to turn space into time - in the lab

Wouldn't it be cool if we could turn one dimension of space into an extra dimension of time?  We can't - but we can fake it!

Einstein showed the only difference between space and time is a minus sign.  So: make a material where light obeys equations with an extra minus sign! 
 
How?  Just take lots of microscopic metal wires and put them in transparent stuff that doesn't conduct electricity.  Line them all up.  You'll get something that conducts electricity like a metal in one direction but not the other two directions! 

It's called a hyperbolic metamaterial.  The video explains why. 

But why is this like converting a dimension of space into a dimension of time? 

Einstein showed that if you have a photon - a particle of light - in a vacuum, it obeys

X² + Y² + Z² - T² = 0

where:

X is the momentum of the light in the x direction
Y is the momentum of the light in the y direction
Z is the momentum of the light in the z direction
T is the momentum of the light in the time direction

(Momentum in the time direction is basically just energy.)

Photons in other stuff obey more complicated equations.  In a hyperbolic metamaterial with wires lined up in the z direction, they obey an equation basically like this:

X² + Y² - Z² - T² = 0

So, the z direction is acting like an extra time dimension! And this lets us do very weird things.

A few warnings if you watch the video:

1) A transparent material like glass is called a dielectric, so you'll see that word a lot.

2) Instead of writing X, Y, and Z for momentum in the x, y and z direction, physicists often write the letter k with a little x, y, or z under it.  

3) Particles are also waves!  The momentum of a particle in some direction is basically just how many times its wave wiggles per meter in that direction.  So, k is also called the wave number.

4) Instead of writing T for the momentum in the time direction, physicists write ω.   This is how much the wave wiggles per second, so it's also called the frequency

5) When I say 'basically', it means I'm leaving out numbers that make things look more complicated, but don't change the basic idea.

You can learn more about hyperbolic metamaterials here:

• Prashant Shekhar, Jonathan Atkinson and Zubin Jacob, Hyperbolic metamaterials: fundamentals and applications, http://arxiv.org/ftp/arxiv/papers/1401/1401.2453.pdf.

#spnetwork arXiv:1401.2453 #metamaterials  ___

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2014-08-18 02:54:41 (40 comments, 158 reshares, 365 +1s) 

Opal: a photonic crystal

This opal from South Australia is beautiful... but why does it look like this?  It's actually a photonic crystal!

Opal is made of small spheres of silicon dioxide, the same stuff as quartz.   These spheres are between 150 to 300 nanometers in diameter, and they're arranged in a regular pattern. 

When light waves pass through an opal, funny stuff happens when the distance between these spheres is half the wavelength of the light.  Imagine water waves trying to fight their way through a regular pattern of round logs standing in the water and you'll get the picture.  The waves start to bounce back and interfere with themselves.

So, light of certain wavelengths winds up bouncing back off the opal instead of going through!  Since the opal is not perfectly the same everywhere, different parts reflect differentcolors... more »

Opal: a photonic crystal

This opal from South Australia is beautiful... but why does it look like this?  It's actually a photonic crystal!

Opal is made of small spheres of silicon dioxide, the same stuff as quartz.   These spheres are between 150 to 300 nanometers in diameter, and they're arranged in a regular pattern. 

When light waves pass through an opal, funny stuff happens when the distance between these spheres is half the wavelength of the light.  Imagine water waves trying to fight their way through a regular pattern of round logs standing in the water and you'll get the picture.  The waves start to bounce back and interfere with themselves.

So, light of certain wavelengths winds up bouncing back off the opal instead of going through!  Since the opal is not perfectly the same everywhere, different parts reflect different colors of light.

When a repeating pattern strongly affects the behavior of light, it's called a photonic crystal.  Nowadays people are making artificial photonic crystals.   Way back in 1887, the English physicist Lord Rayleigh experimented with stacking layers of different materials to make light of certain wavelengths interfere with itself and bounce back.  When this happens, we now call it a photonic band gap.   But Rayleigh only got this effect to occur in a single direction - at right angles to the layers.

Now people can make 3d arrays of tiny spheres that act as photonic crystals.  We might be able to use them to make color-changing paints and inks.  But the first commercial products involving photonic crystals are photonic-crystal fibers - a bit like ordinary optical fibers, but different.

Photonic crystals are one example of a metamaterial - a material whose optical or electronic properties depend not just on their composition, but from carefully designed tiny structures.

In the Middle Ages, some thought an opal could make you invisible if you wrapped it in a fresh bay leaf and held it in your hand.  Now people are designing metamaterials that can act like a 'cloak of invisibility'.  But more about that later.

Puzzle: I said opal is made of tiny spheres of silicon dioxide (called silica), but what's in the spaces between these spheres? 

I don't know - I only have a clue.  Wikipedia says opal is a hydrated form of amorphous silica, usually containing between 6% and 10% water.  Maybe the spaces between the spheres is a mix of silica and water?

For more see:

https://en.wikipedia.org/wiki/Opal
https://en.wikipedia.org/wiki/Photonic_crystal
https://en.wikipedia.org/wiki/Metamaterial

The photo is from 'Dpulitzer':

https://commons.wikimedia.org/wiki/File:Coober_Pedy_Opal.jpg

#physics  ___

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2014-08-17 02:06:16 (10 comments, 8 reshares, 25 +1s) 

The only thing savin' your life
is that I don't look good in orange and I hate stripes

If you like songs with catchy lyrics and a dark sense of humor, you've got to listen to... Cole Porter.  But he's dead.  For new stuff, try Brandy Clark.

She's been working as a song writer for about 16 years, but she just came out with an album of her own, 12 Stories.    It's about life in small-town America, with all its hilarious and tragic craziness.  I recommend this song here, and also:

"Stripes" - by Brandy Clark

Brandy Clark Get High

Pray To Jesus - Brandy Clark

and

Brandy Clark - Take A Little Pill

which is about the epidemic of prescription painkiller abuse:

Fella down the road
Walks with a limp
He's a decorated soldier
And a pain pillpimp... more »

The only thing savin' your life
is that I don't look good in orange and I hate stripes

If you like songs with catchy lyrics and a dark sense of humor, you've got to listen to... Cole Porter.  But he's dead.  For new stuff, try Brandy Clark.

She's been working as a song writer for about 16 years, but she just came out with an album of her own, 12 Stories.    It's about life in small-town America, with all its hilarious and tragic craziness.  I recommend this song here, and also:

"Stripes" - by Brandy Clark

Brandy Clark Get High

Pray To Jesus - Brandy Clark

and

Brandy Clark - Take A Little Pill

which is about the epidemic of prescription painkiller abuse:

Fella down the road
Walks with a limp
He's a decorated soldier
And a pain pill pimp
Got a loaded gun
And an old pit bull
A black Cadillac
With a whole trunk full.

Says if one don't work
Then another one will
If you got a little hurt
You take a little pill

Lay it on your tongue
Ain't a nerve that can't be numbed
Ain't a buzz that you can't buy
Ain't a low you can't make high
Til it wears off
Like it always will
And when it does
You take a little pill

She tried to sell this this song to major-label artists - and they liked it, but didn't buy.  She says "I'm thankful that no one did bite on it, because it seems to be the song that resonates the strongest with people. I think it's just so real, and kind of a realness that's a little bit ugly."

Other songs are really funny.   Great stuff!

Urbanites have always looked down on country music.  I bet Babylonian city slickers already had a word for 'redneck'.  So you may think it's weird that I like this stuff.  But American pop music seems pretty stale to me right now, and one of the few bright spots is the rise of women country musicians. Brandy Clark has been writing songs for several of them, notably Kacey Musgraves.  For more on this trend, try:

http://www.npr.org/2013/12/26/257362793/quiet-as-kept-women-dominated-country-music-in-2013

(Getting my news on country music from here proves I don't know much about it.)___

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2014-08-16 01:04:37 (15 comments, 8 reshares, 114 +1s) 

Reflections at Keppel Bay

There's a big network of trails and parks in Singapore.  You can climb up a huge wooden bridge and suddenly see this futuristic apartment complex amid the jungle.

My friend +Jamie Vicary is visiting the Centre for Quantum Technologies.  He showed Lisa and me how to walk through park land to Vivocity, a fancy shopping mall by the sea.  There are some elevated walkways through the jungle, that really let you see the trees and vines.  There's a World War II museum that shows how the Japanese invaded, and the battles fought around here.  It was open for free because it was National Day - the 49th anniversary of the birth of Singapore as an independent country.   It's weird to be in a country that's younger than I am.

Walking across the Henderson Wave Bridge, we spotted these buildings, called Reflections at KeppelBaymore »

Reflections at Keppel Bay

There's a big network of trails and parks in Singapore.  You can climb up a huge wooden bridge and suddenly see this futuristic apartment complex amid the jungle.

My friend +Jamie Vicary is visiting the Centre for Quantum Technologies.  He showed Lisa and me how to walk through park land to Vivocity, a fancy shopping mall by the sea.  There are some elevated walkways through the jungle, that really let you see the trees and vines.  There's a World War II museum that shows how the Japanese invaded, and the battles fought around here.  It was open for free because it was National Day - the 49th anniversary of the birth of Singapore as an independent country.   It's weird to be in a country that's younger than I am.

Walking across the Henderson Wave Bridge, we spotted these buildings, called Reflections at Keppel Bay, on the horizon near the sea.  They were designed by Daniel Libeskind, who also created the plan for the World Trade Center Memorial. 

Once a Chinese billionaire invited Lisa and me to lunch near those apartments!  He wanted me to work with him on his physics theories.  Alas, I couldn't bring myself to do it. 

When we reached Vivocity, we were really tired - it was a cool day by Singapore standards, but still sweaty after a 4-hour hike.  We watched a crowd of Chinese folks doing country-western dancing outdoors by the mall.   It's fun to see a middle-aged Chinese guy wearing cowboy boots and a hat slowly dancing to country music as if it were some sort of tai chi exercise.  I really wonder how this fad got started.  Then we went to an overpriced restaurant called The Queen and Mangosteen and had beers and dinner by the water.

Life is stranger and more interesting than I'd expected.

Here's a map for this walk:

http://www.nparks.gov.sg/cms/docs/HortPark_and_SouthernRidges_guide.pdf

#singapore  ___

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2014-08-15 01:28:39 (39 comments, 46 reshares, 153 +1s) 

{7,3,3} meets the plane at infinity

Maryam Mirzakhani won the Fields medal largely for her work on hyperbolic geometry.  The hyperbolic plane is something we can all enjoy: it's not flat, but curved like a saddle, and triangles have angles that add up to less than 180 degrees.  We can draw it crushed down to a disk, as in this picture by +Roice Nelson.  All the light blue circles should really be the same size - but the ones near the edge have been squashed.

Instead of learning Euclidean geometry in school, you could have learned hyperbolic geometry - it's almost the same, with some big differences.  The main thing is that the parallel postulate is false in hyperbolic geometry: there are many ways to draw a line through a point parallel to another line!  This is how Lobachevsky  discovered hyperbolic geometry in 1823: by seeing what would happen ifyou c... more »

{7,3,3} meets the plane at infinity

Maryam Mirzakhani won the Fields medal largely for her work on hyperbolic geometry.  The hyperbolic plane is something we can all enjoy: it's not flat, but curved like a saddle, and triangles have angles that add up to less than 180 degrees.  We can draw it crushed down to a disk, as in this picture by +Roice Nelson.  All the light blue circles should really be the same size - but the ones near the edge have been squashed.

Instead of learning Euclidean geometry in school, you could have learned hyperbolic geometry - it's almost the same, with some big differences.  The main thing is that the parallel postulate is false in hyperbolic geometry: there are many ways to draw a line through a point parallel to another line!  This is how Lobachevsky  discovered hyperbolic geometry in 1823: by seeing what would happen if you changed the rules this way.

And instead of learning trigonometry, you could have learned hyperbolic trigonometry - it's very similar, but you have identities like

cosh²θ - sinh²θ = 1

The most exciting thing you could do in this alternate universe is take the hyperbolic plane, cut out carefully chosen pieces, and fold them into multi-holed doughnut shapes without wrinkling the paper.  

These shapes were intensively studied by Felix Klein and Henri Poincaré... and Maryam Mirzakhani is carrying on this grand tradition.  They're important in number theory, string theory and many other subjects.

Hyperbolic geometry works in higher dimensions, too.  In my last Visual Insight post, I showed you a 3-dimensional 'hyperbolic honeycomb', a marvelous pattern built of sheets of regular heptagons.  It's called the {7,3,3} honeycomb because each heptagon has 7 sides, the heptagons meet in groups of 3 on each sheet, and 3 sheets meet along each edge of each heptagon.

This is a view of the 'sky' in a 3-dimensional world with a {7,3,3} honeycomb in it.  To understand it, check out my latest Visual Insight post:

http://blogs.ams.org/visualinsight/2014/08/14/733-honeycomb-meets-plane-at-infinity/

and compare the picture there with the previous one:

http://blogs.ams.org/visualinsight/2014/08/01/733-honeycomb/

#geometry  ___

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2014-08-14 03:06:45 (19 comments, 95 reshares, 245 +1s) 

The 290 theorem

Manjul Bhargava is another of this year's Fields medalists.  He works on number theory, which in its simplest form is the study of integers:

 ..., -3, -2, -1, 0, 1, 2, 3, ...

So when I say 'number' in this post, I'll always mean one of these!

When Bhargava was a grad student at Princeton, he read a book on number theory by the famous mathematician Gauss.  Gauss was interested in quadratic forms, which are things like this:

x² + 3xy + y²

or this

-3x² + y² + 4xz + yz - 7z²

Gauss was mainly interested in quadratic forms with two variables, but it's also fun to think about more variables.

I can hand you a quadratic form and ask: what numbers can you get if you plug in any numbers you want for the variables?

Start with something reallyeasy.  ... more »

The 290 theorem

Manjul Bhargava is another of this year's Fields medalists.  He works on number theory, which in its simplest form is the study of integers:

 ..., -3, -2, -1, 0, 1, 2, 3, ...

So when I say 'number' in this post, I'll always mean one of these!

When Bhargava was a grad student at Princeton, he read a book on number theory by the famous mathematician Gauss.  Gauss was interested in quadratic forms, which are things like this:

x² + 3xy + y²

or this

-3x² + y² + 4xz + yz - 7z²

Gauss was mainly interested in quadratic forms with two variables, but it's also fun to think about more variables.

I can hand you a quadratic form and ask: what numbers can you get if you plug in any numbers you want for the variables?

Start with something really easy.  For this one



you can only get the perfect squares

0, 1, 4, 9, 16, ...

But what about this one?

x² + y²

Can you find numbers x and y that make x² + y² = 100?  How about x² + y² = 99?  Remember, I'm using 'numbers' to mean numbers like these:

 ..., -3, -2, -1, 0, 1, 2, 3, ...

And what about this quadratic form?

w² + x² + y² + z²

It's a famous fact that for this one, you can get any positive number by plugging in numbers for w, x, y and z. 

What about this?

x² + y² + z²

Now you can't get every positive number.   Do you see why?

We say a quadratic form is positive definite if whenever you plug numbers into it, you get something positive - unless all those numbers were zero.  For example,

x² + y² + z²

is positive definite, but

x² + y² - z²

is not. 

Okay, now you're ready.  Here's something amazing that Manjul Bhargava proved with +Jonathan Hanke in 2005.

Here's how to tell if you can get every positive number by plugging in numbers for the variables in a positive definite quadratic form.  It's enough to check that you can get every number from 1 to 290.

In fact, it's enough to get these numbers:

1, 2, 3, 5, 6, 7, 10, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 34, 35, 37, 42, 58, 93, 110, 145, 203, 290.

Weird! 

This is just one of many things Bhargava has done.  Most are a bit harder to explain, but I described one here:

https://plus.google.com/u/0/117663015413546257905/posts/VouaWQnthn5

It's about 'elliptic curves', another really popular topic in number theory. 

And in fact, the 290 theorem I just explained is secretly about elliptic curves!  As usual in number theory, the statement of a theorem may sound simple, cute, and pointless... but the proof reveals a very different world, and that's what really matters. 

Here's a nice explanation of the proof:

• Yong Suk Moon, Universal quadratic forms and the 15-theorem and 290-theorem, https://math.stanford.edu/theses/moon.pdf.

The original paper is here:

• Manjul Bhargava and Jonathan Hanke, Universal quadratic forms and the 290-Theorem, to appear in Inventiones Mathematicae,  http://www.wordpress.jonhanke.com/wp-content/uploads/2011/09/290-Theorem-preprint.pdf

There's a lot left to do.  For example, Jonathan Rouse tried to show that a positive definite quadratic form gives all  odd positive numbers if gives the odd numbers from 1 up to 451... but he only succeeded in showing this assuming something called the Generalized Riemann Hypothesis!  Proving this is an extremely hard problem in its own right.

• Jonathan Rouse, Quadratic forms representing all odd positive integers, http://arxiv.org/abs/1111.0979.

#spnetwork arxiv:1111.0979  #fieldsmedal   #numbertheory   #spnetwork  ___

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2014-08-13 03:16:36 (57 comments, 250 reshares, 599 +1s) 

Maryam Mirzakhani won the Fields medal yesterday.

As a child in Tehran, she didn't intend to become a mathematician - she just wanted to read every book she could find!  She also watched television biographies of famous women like Marie Curie and Helen Keller.  She started wanting to do something great... maybe become a writer.

She finished elementary school while the Iran-Iraq war was ending, and took a test that got her into a special middle school for girls.  She did poorly in math her first year, and it undermined her confidence.  “I lost my interest in math," she said.

But the next year she had a better teacher, and she fell in love with the subject.  She and a friend became the first women on Iranian math Olympiad team.  She won a gold medal the first year, and got a perfect score the next year.

After getting finishing her undergraduate workat Shar... more »

Maryam Mirzakhani won the Fields medal yesterday.

As a child in Tehran, she didn't intend to become a mathematician - she just wanted to read every book she could find!  She also watched television biographies of famous women like Marie Curie and Helen Keller.  She started wanting to do something great... maybe become a writer.

She finished elementary school while the Iran-Iraq war was ending, and took a test that got her into a special middle school for girls.  She did poorly in math her first year, and it undermined her confidence.  “I lost my interest in math," she said.

But the next year she had a better teacher, and she fell in love with the subject.  She and a friend became the first women on Iranian math Olympiad team.  She won a gold medal the first year, and got a perfect score the next year.

After getting finishing her undergraduate work at Sharif University in Tehran in 1999, she went on to grad school at Harvard.  There she met Curtis McMullen, a Fields medalist who works on hyperbolic geometry and related topics.

Hyperbolic geometry is about curved surfaces where the angles of a triangle add up to less than 180 degrees, like the surface of a saddle.  It's more interesting than Euclidean geometry, or the geometry of a sphere.  One reason is that if you have a doughnut-shaped thing with 2 or more holes, there are many ways to give it a hyperbolic geometry where its curvature is the same at each point.  These shapes stand at the meeting-point of many roads in math.  They are simple enough that we can understand them in amazing detail - yet complicated enough to provoke endless study.

Maryam Mirzakhani took a course from McMullen and started asking him lots of questions.  “She had a sort of daring imagination,” he later said.  “She would formulate in her mind an imaginary picture of what must be going on, then come to my office and describe it. At the end, she would turn to me and say, ‘Is it right?’ I was always very flattered that she thought I would know.”

Here's a question nobody knew the answer to.  If an ant walks on a flat Euclidean plane never turning right or left, it'll move along a straight line and never get back where it started.  If it does this on a sphere, it will get back where it started: it will go around a circle.  If it does this on a hyperbolic surface, it may or may not get back where it started.  If it gets back to where it started, facing the same direction, the curve it moves along is called a closed geodesic.  

The ant can go around a closed geodesic over and over.  But say we let it go around just once: then we call its path a simple closed geodesic.    We can measure the length of this curve.  And we can ask: how many simple closed geodesics are there with length less than some number L?

There are always only finitely many - unlike on the sphere, where the ant can march off in any direction and get back where it started after a certain distance.  But how many?

In her Ph.D. thesis, Mirzakhani figured out a formula for how many.  It's not an exact formula, just an 'asymptotic' one, an approximation that becomes good when L becomes large.  She showed the number of simple closed geodesics of length less than L is asymptotic to some number times L to the power 6g-6, where g is the number of holes in your doughnut. 

She boiled her proof down to a 29-page argument, which was published in one of the most prestigious math journals:

• Maryam Mirzakhani, Growth of the number of simple closed geodesics on hyperbolic surfaces, Annals of Mathematics 168 (2008), 97–125, http://annals.math.princeton.edu/wp-content/uploads/annals-v168-n1-p03.pdf.

This is a classic piece of math: simple yet deep.  The statement is simple, but the proof uses many branches of math that meet at this crossroads. 

What matters is not just knowing that the statement is true: it's the new view of reality you gain by understanding why it's true.   I don't understand why this particular result is true, but I know that's how it works.  For example, her ideas also gave here a new proof of a conjecture by the physicist Edward Witten, which came up in his work on string theory!  

This is just one of the first things Mirzakhani did.  She's now a professor at Stanford.

"I don't have any particular recipe," she said.  "It is the reason why doing research is challenging as well as attractive. It is like being lost in a jungle and trying to use all the knowledge that you can gather to come up with some new tricks, and with some luck you might find a way out."

She has a lot left to think about.  There are problems she has been thinking about for more than a decade. "And still there’s not much I can do about them," she said.

"I can see that without being excited mathematics can look pointless and cold. The beauty of mathematics only shows itself to more patient followers."

I got some of my quotes from here:

http://www.simonsfoundation.org/quanta/20140812-a-tenacious-explorer-of-abstract-surfaces/

and some from here:

http://www.theguardian.com/science/2014/aug/13/interview-maryam-mirzakhani-fields-medal-winner-mathematician

They're both fun to read.

#spnetwork doi:10.4007/annals.2008.168.97 #geometry #mustread___

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2014-08-12 08:27:22 (26 comments, 25 reshares, 112 +1s) 

The Picture of Dorian Gray

"...the face appeared to him to be a little changed. The expression looked different. One would have said that there was a touch of cruelty in the mouth. It was certainly strange." -  Oscar Wilde

Sadly, I haven't read this story yet - I've just seen a movie version. And I've read some quotes.  Since Wilde was a master of the epigram, you can enjoy these like popping down peanuts until you get sick to your stomach:

Nowadays people know the price of everything and the value of nothing.

He was always late on principle, his principle being that punctuality is the thief of time.

I never talk during music - at least, during good music. If one hears bad music, it is one's duty to drown it in conversation.

The basis of optimism is sheer terror.
more »

The Picture of Dorian Gray

"...the face appeared to him to be a little changed. The expression looked different. One would have said that there was a touch of cruelty in the mouth. It was certainly strange." -  Oscar Wilde

Sadly, I haven't read this story yet - I've just seen a movie version. And I've read some quotes.  Since Wilde was a master of the epigram, you can enjoy these like popping down peanuts until you get sick to your stomach:

Nowadays people know the price of everything and the value of nothing.

He was always late on principle, his principle being that punctuality is the thief of time.

I never talk during music - at least, during good music. If one hears bad music, it is one's duty to drown it in conversation.

The basis of optimism is sheer terror.

The only way to get rid of temptation is to yield to it. Resist it, and your soul grows sick with longing for the things it has forbidden to itself.

... and so on.   You can like them without agreeing with them.  If you agreed with them all, Wilde would be disappointed.

The picture here is by an artist named Gerwell:

http://gerwell.deviantart.com/art/The-Picture-of-Dorian-Gray-V2-165822769

At least to us old folks, there is still something slightly magical about an animated gif.  We're used to movies and TV, but a picture that changes seems different... even though movies were originally called 'moving pictures'.  If you want to boost your number of +1s on Google+, post an animated gif.  Pathetic but true.
 ___

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2014-08-11 03:56:02 (45 comments, 40 reshares, 155 +1s) 

What is quantum entanglement good for?

Here's a headline from the New York Times in 1935.   Einstein had already done his best work: photons, special relativity and general relativity.  But this year he had 2 good ideas about spooky ways that distant objects could be connected.  He wrote about them in 2 papers with a guy named Rosen. 

One introduced the idea of a 'wormhole': a kind of tunnel connecting distant regions of space.  Another, written with a third guy named Podolsky, argued that quantum theory was 'incomplete'.  Einstein later complained that it allowed 'spooky action at a distance'.

Why?  Because you can't always fully describe a quantum system by fully describing each piece of it.   In fact sometimes you can know as much as possible about the whole system while still knowing nothing about the state of each piece. 
more »

What is quantum entanglement good for?

Here's a headline from the New York Times in 1935.   Einstein had already done his best work: photons, special relativity and general relativity.  But this year he had 2 good ideas about spooky ways that distant objects could be connected.  He wrote about them in 2 papers with a guy named Rosen. 

One introduced the idea of a 'wormhole': a kind of tunnel connecting distant regions of space.  Another, written with a third guy named Podolsky, argued that quantum theory was 'incomplete'.  Einstein later complained that it allowed 'spooky action at a distance'.

Why?  Because you can't always fully describe a quantum system by fully describing each piece of it.   In fact sometimes you can know as much as possible about the whole system while still knowing nothing about the state of each piece.  

And in this situation, if you measure stuff about the pieces, what you see will be random... but correlated in ways that are impossible according to classical probability theory.   We say the pieces are entangled.

We're now quite sure that entanglement is real.  People routinely see it in the lab.  The way out of the 'paradox' is not spooky action at a distance.  It's that the world is not described by classical probability theory.   It's described by quantum theory.

Here at the Centre for Quantum Technologies, people owe a lot to Einstein's discovery of entanglement.   But they don't regard it as a flaw of quantum theory.  They regard it as a 'resource'.  You can use it to do wonderful things.

Or can you?  Is it really entanglement that lets us do wonderful things with quantum mechanics... or something slightly different?   It turns out there are interesting kinds of quantum information processing that don't use entanglement, and even kinds that don't even work if there's too much entanglement. 

Here at the CQT,  +Valerio Scarani and coauthors wrote a nice article on the state of the art in quantum information processing - the "quantum frontier".  And they say:

Jozsa and Linden show that any quantum algorithm involving only pure states that achieves exponential speed up over classical algorithms must entangle a large numbers of qubits. While entanglement is necessary for an exponential speed up, the existence of entanglement is far from sufficient to guarantee a speed up, and it may turn out that another property better characterizes what enables a speed up. Many entangled systems have been shown to be classically simulable. Indeed, the Gottesman-Knill theorem, as well as results on the classical simulation of match gates, have shown that there exist non-classical computational models that allow for highly entangled states which are efficiently classically simulable. Furthermore, if one looks at query complexity instead of algorithmic complexity, improvements can be obtained with no entanglement whatsoever. Meyer shows that in the course of the Bernstein-Vazirani algorithm, which achieves an N to 1 reduction in the number of queries required, no qubits become entangled. Going beyond quantum computation it becomes more obvious that entanglement is not required to reap benefits. For example, the BB84 quantum key distribution protocol makes no use of entanglement. While measurement-based quantum computation, discussed in Section 6.2, graphically illustrates the use of entanglement as a resource for quantum computation, it turns out that if states are too highly entangled, they are useless for measurement-based quantum computation. In the same paper in which they showed that entanglement is necessary, Jozsa and Linden end their abstract with “we argue that it is nevertheless misleading to view entanglement as a key resource for quantum-computational power.” The reasons for quantum information processing’s power remains mysterious; Vedral refers to “the elusive source of quantum effectiveness”.

There are lots of references here that I've deleted... but you can get those details here:

• Joseph F. Fitzsimons, Eleanor G. Rieffel, Valerio Scarani, The quantum frontier, http://arxiv.org/abs/1206.0785.

#spnetwork arXiv:1206.0785 #quantumcomputing  ___

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2014-08-10 02:51:19 (169 comments, 682 reshares, 554 +1s) 

Both rectangles are moving at constant speed

At least that's what the creator of this illusion says!  It looks like the yellow and blue rectangles are taking turns going forward - one step at a time.

This is an illusion that's so good it's hard to believe it's an illusion. When the black and white lines disappear, it's easy to see the rectangles are moving at constant speed.  But before that they seem to be taking turns, and pausing when they reach each new line.

Could the creator of this illusion be cheating - fooling you into thinking there's an illusion?   How can you tell, except by making your own version of this animated gif?

Hide one rectangle with your hand.  Then look closely at the other.  Try not to look at the black and white lines.  I think you'll see the rectangle is moving at constant speed.

Butif you... more »

Both rectangles are moving at constant speed

At least that's what the creator of this illusion says!  It looks like the yellow and blue rectangles are taking turns going forward - one step at a time.

This is an illusion that's so good it's hard to believe it's an illusion. When the black and white lines disappear, it's easy to see the rectangles are moving at constant speed.  But before that they seem to be taking turns, and pausing when they reach each new line.

Could the creator of this illusion be cheating - fooling you into thinking there's an illusion?   How can you tell, except by making your own version of this animated gif?

Hide one rectangle with your hand.  Then look closely at the other.  Try not to look at the black and white lines.  I think you'll see the rectangle is moving at constant speed.

But if you look away, and watch the rectangle with your peripheral vision, it will seem to move in steps.

We don't just "see what's there".  We construct a mental model of reality from sensory data.  We need  to do this.  But people can manipulate this. 

It's not optical illusions we need to worry about.  It's political illusions, economic illusions, social illusions.  We think we're just seeing what's there... but we're actually constructing a model of reality.  And politicians and other people are busy trying to shape your model, so you'll do what they want.  Escaping their illusions is much, much harder than escaping this optical illusion.

You can probably think of many examples of other  people who are fooled by politicians, ideologies, doctrines and dogmas.   Now list the ways in which you are being fooled.

Oh, you think you're better than average?  Join the club.

Puzzle 1: name the biggest way you've been fooled by a cultural, political or religious illusion.

Puzzle 2: name a way you're just starting to realize that you're being fooled by such an illusion.

For Puzzle 2, it should be just as hard to really believe you're being fooled as it is with this optical illusion.  For example: I'm just starting to realize that I've been fooled into wanting to be a 'bigshot': well-known, and seemingly 'important'.  But it's hard to break out of this belief.  Even now, I'm trying to get you to pay attention to me.  I'm sorry - at least I try to make it worth your while.

#illusion  ___

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2014-08-08 07:52:41 (61 comments, 34 reshares, 186 +1s) 

Time to panic yet?  Too soon to tell...

You know that mysterious crater that suddenly opened up in the Yamal Peninsula in northwest Siberia?   Now an article in Nature blames it on melting permafrost:

http://www.nature.com/news/mysterious-siberian-crater-attributed-to-methane-1.15649

A Russian archaeologist named Andrei Plekhanov led an expedition there.  He and his team think it could be caused by melting permafrost due to abnormally hot summers in 2012 and 2013 - about 5°C warmer than usual.  Air in the crater had up to 9.6% methane.  The normal amount is closer to 0.0002%

Some scientists are already worrying about methane released from melting permafrost.   Methane is a potent greenhouse gas: 90 times worse than carbon dioxide by weight, at least for the first 20 years.  (It goes away faster.)   Could we be in for a nasty feedback loop, where awarming c... more »

Time to panic yet?  Too soon to tell...

You know that mysterious crater that suddenly opened up in the Yamal Peninsula in northwest Siberia?   Now an article in Nature blames it on melting permafrost:

http://www.nature.com/news/mysterious-siberian-crater-attributed-to-methane-1.15649

A Russian archaeologist named Andrei Plekhanov led an expedition there.  He and his team think it could be caused by melting permafrost due to abnormally hot summers in 2012 and 2013 - about 5°C warmer than usual.  Air in the crater had up to 9.6% methane.  The normal amount is closer to 0.0002%

Some scientists are already worrying about methane released from melting permafrost.   Methane is a potent greenhouse gas: 90 times worse than carbon dioxide by weight, at least for the first 20 years.  (It goes away faster.)   Could we be in for a nasty feedback loop, where a warming climate melts permafrost, releases methane and warms the Earth even more?

The short answer is: probably not very soon.  

There are certainly things that make me nervous.  Back in 2011, a Russian research cruiser found methane bubbling up from the ocean floor:

In late summer, the Russian research vessel Academician Lavrentiev conducted an extensive survey of about 10,000 square miles of sea off the East Siberian coast. Scientists deployed four highly sensitive instruments, both seismic and acoustic, to monitor the “fountains” or plumes of methane bubbles rising to the sea surface from beneath the seabed.

“In a very small area, less than 10,000 square miles, we have counted more than 100 fountains, or torch-like structures, bubbling through the water column and injected directly into the atmosphere from the seabed,” Dr Semiletov said. “We carried out checks at about 115 stationary points and discovered methane fields of a fantastic scale – I think on a scale not seen before. Some plumes were a kilometre or more wide and the emissions went directly into the atmosphere – the concentration was a hundred times higher than normal.”

Others argued that this could have been going on for centuries; nobody had looked very hard!   A group of experts called the Permafrost Carbon Network polled themselves and guessed that up to 2040, they expected the effect of melting permafrost to be roughly 1/8 to 1/4 of the direct effect of burning carbon.  That would be bad but not disastrous... at least not soon. 

One obvious question about this crater is whether it's an unusual event or the start of a trend.  Local reindeer herders have reported a similar but smaller hole nearby.  But most of us, who aren't doing research on permafrost, will just have to wait and see.

Hey, I've got an idea!   In the meantime, how about cutting greenhouse gas emissions?

For more background, see:

http://johncarlosbaez.wordpress.com/2011/09/01/melting-permafrost/

http://johncarlosbaez.wordpress.com/2011/12/14/melting-permafrost-part-2/

http://johncarlosbaez.wordpress.com/2011/12/19/melting-permafrost-part-3/___

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2014-08-07 02:42:37 (30 comments, 42 reshares, 167 +1s) 

Harpooning a comet

After a decade-long chase, the Rosetta spacecraft has now reached a comet!  This photo was taken from just 285 kilometers away.  The comet's gravity is very weak, so Rosetta will move in a triangular orbit under its own power before moving closer. 

On November 11th, Rosetta will send in a probe called Philae, which will harpoon the comet and land on it!  The picture at bottom right shows what that should look like.  Philae will drill into the comet and carry out lots of experiments.

As the comet approaches the Sun, it will heat up.  Gas will start to boil up from the surface, earthquakes will shake it, and I suppose Philae may even be destroyed.  I haven't read any details about what to expect!  But they will try to land Philae far away from places where jets of gas will erupt.  The comet will makes its closestapproach... more »

Harpooning a comet

After a decade-long chase, the Rosetta spacecraft has now reached a comet!  This photo was taken from just 285 kilometers away.  The comet's gravity is very weak, so Rosetta will move in a triangular orbit under its own power before moving closer. 

On November 11th, Rosetta will send in a probe called Philae, which will harpoon the comet and land on it!  The picture at bottom right shows what that should look like.  Philae will drill into the comet and carry out lots of experiments.

As the comet approaches the Sun, it will heat up.  Gas will start to boil up from the surface, earthquakes will shake it, and I suppose Philae may even be destroyed.  I haven't read any details about what to expect!  But they will try to land Philae far away from places where jets of gas will erupt.  The comet will makes its closest approach to the Sun on August 13th, 2015.

Here are the scientific instruments on the lander Philae:

CONSERT, the COmet Nucleus Sounding Experiment by Radiowave Transmission. This will shoot radio waves through the coment, which will be detected on the other side by Rosetta!  It's sort of like taking an X-ray to see what's inside the comet.

APXS, the Alpha Proton X-ray Spectrometer.  This will determine which elements are on the surface below the lander.

COSAC, for COmetary SAmpling and Composition. This is a combination gas chromatograph and time-of-flight mass spectrometer that will perform analysis of soil samples and determine the content of volatile components.

Ptolemy is an instrument used to measure stable isotopic ratios of key volatile compounds on the comet's nucleus.

ÇIVA, the Comet Nucleus Infrared and Visible Analyzer.

ROLIS, the Rosetta Lander Imaging System.

MUPUS, the MUlti-PUrpose Sensors for Surface and Sub-Surface Science.

ROMAP, the Rosetta Lander Magnetometer and Plasma Monitor.

SESAME, the Surface Electric Sounding and Acoustic Monitoring Experiment.

SD2, the sampling, drilling and distribution subsystem.

If you know more about what these do, please tell us!  I'm annoyed that the stuff I've seen doesn't go into much detail.

For more great photos of the comet, go to the European Space Agency website:

http://www.esa.int/spaceinimages/Missions/Rosetta/%28class%29/image___

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2014-08-05 00:52:07 (6 comments, 33 reshares, 122 +1s) 

Lost in space

Physicists often use a 'Cartesian coordinate system' - an imaginary grid, like 3d graph paper,  that lets us name any point in space with 3 numbers.  But what if it were real?  You could climb around on it!

That's the idea of this art project by Numen/For Use, the guys who made that expanding and shrinking glowing cube with an infinity of reflections inside.

It's an inflatable structure.   Ropes inside get stretched tight when it inflates.  They form a 3d grid that's strong enough to climb around on.  The inside walls are white, so it seems to go on infinitely. They tested it out in countryside near Vienna at the end of December 2013. 

For more photos, try this:

http://www.numen.eu/installations/string/prototype/

If you didn't see that glowing cube, you missed some fun.  Here itis:
more »

Lost in space

Physicists often use a 'Cartesian coordinate system' - an imaginary grid, like 3d graph paper,  that lets us name any point in space with 3 numbers.  But what if it were real?  You could climb around on it!

That's the idea of this art project by Numen/For Use, the guys who made that expanding and shrinking glowing cube with an infinity of reflections inside.

It's an inflatable structure.   Ropes inside get stretched tight when it inflates.  They form a 3d grid that's strong enough to climb around on.  The inside walls are white, so it seems to go on infinitely. They tested it out in countryside near Vienna at the end of December 2013. 

For more photos, try this:

http://www.numen.eu/installations/string/prototype/

If you didn't see that glowing cube, you missed some fun.  Here it is:

http://tinyurl.com/n-light-membrane

#art  ___

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2014-08-03 09:59:26 (219 comments, 70 reshares, 244 +1s) 

My last post on the NASA "quantum vacuum plasma thruster" was mainly about the shoddy theory behind it - like how there's no such thing as a "quantum vacuum plasma".

But you could argue: hey, if the gizmo actually works, isn't that good enough?   

Unfortunately, the experiment has problems too.  In brief:

1.  They tested a device that was designed to work and one that was designed not to work.  They both worked.

2.  They tested the devices in a "vacuum chamber", but they didn't take the air out.

3.  They didn't carefully study all possible causes of experimental error... like their devices heating the air.

In a bit more detail:

1.  Their device, called the Cannae drive, was invented by a guy named Guido Fetta.  You can see a picture below.  It's not complicated! It's a... more »

My last post on the NASA "quantum vacuum plasma thruster" was mainly about the shoddy theory behind it - like how there's no such thing as a "quantum vacuum plasma".

But you could argue: hey, if the gizmo actually works, isn't that good enough?   

Unfortunately, the experiment has problems too.  In brief:

1.  They tested a device that was designed to work and one that was designed not to work.  They both worked.

2.  They tested the devices in a "vacuum chamber", but they didn't take the air out.

3.  They didn't carefully study all possible causes of experimental error... like their devices heating the air.

In a bit more detail:

1.  Their device, called the Cannae drive, was invented by a guy named Guido Fetta.  You can see a picture below.  It's not complicated!  It's a hollow container made of metal, about 11 inches in diameter and 4-5 inches long.  You pump radio waves in one end. At the other end, a copper wire serves as an antenna.  This lets you measure the radio waves bouncing around inside the container, and adjust their frequency until you hit a resonance.  Then this thing is supposed to generate thrust, for some unknown reason.

Fetta thought this device would work if you carve slots on one side of the flat part.  The NASA guys tried a version with slots and one without slots.  They claim both versions generate a thrust of 22-48 micronewtons when they pump 17-28 watts of radio waves into them:

Thrust was observed on both test articles, even though one of the test articles was designed with the expectation that it would not produce thrust. Specifically, one test article contained internal physical modifications that were designed to produce thrust, while the other did not (with the latter being referred to as the "null" test article).

So, basically they found evidence against Fetta's idea: the slots make no difference.  It's like giving someone a placebo and finding it works just as well as the drug you're testing.

They also tried a resistor instead of their device.  They claim this produced no thrust.  This rules out some possibilities of experimental error... but not others. 

For example, if parts of their flat metal can get hot and create air currents, that might create the force they saw.  It's a tiny force, less you'd get from 5 milligrams of mass pushing down due to gravity.

2.  Their paper goes into great detail about the "vacuum chamber" their experiment was done in - but in the abstract to the paper, they say they didn't remove the air.  This is important because of the issue of air currents. 

It's also just weird.  In their paper they say:

To simulate the space pressure environment, the test rig is rolled into the test chamber. After sealing the chamber, the test facility vacuum pumps are used to reduce the environmental pressure down as far as 5x10E-6 Torr. Two roughing pumps provide the vacuum required to lower the environment to approximately 10 Torr in less than 30 minutes. Then, two high-speed turbo pumps are used to complete the evacuation to 5x10E-6 Torr, which requires a few additional days. During this final evacuation, a large strip heater (mounted around most of the circumference of the cylindrical chamber) is used to heat the chamber interior sufficiently to emancipate volatile substances that typically coat the chamber interior walls whenever the chamber is at ambient pressure with the chamber door open. During test run data takes at vacuum, the turbo pumps continue to run to maintain the hard vacuum environment. The high-frequency vibrations from the turbo pump have no noticeable effect on the testing seismic environment.

They're working really hard to get a good vacuum, right?  But in their abstract they say:

Testing was performed on a low-thrust torsion pendulum that is capable of detecting force at a single-digit micronewton level, within a stainless steel vacuum chamber with the door closed but at ambient atmospheric pressure.

At ambient atmospheric pressure?   What's the point of the fancy vacuum chamber?  A sentence in their conclusions gives a clue.  Talking about future plans, they say:

Vacuum compatible RF amplifiers with power ranges of up to 125 watts will allow testing at vacuum conditions which was not possible using our current RF amplifiers due to the presence of electrolytic capacitors.

So it seems they couldn't actually test their device in a vacuum.

3.  If you're trying to find some small effect, checking the ways you could have screwed up is the most important thing.  The device they're testing is simple, but the test apparatus itself is very complicated, and lots of things could go wrong. 

Their paper should have a big section on this, but it doesn't.  Instead it has a section on how if the gizmo works, you could scale it up and do great things:

Figure 23 shows a conservative 300 kilowatt solar electric propulsion roundtrip human exploration class mission to Mars/Deimos. Figure 24 shows a 90 metric ton 2 megawatt (MW) nuclear electric propulsion mission to Mars that has considerable reduction in transit times due to having a thrust to mass ratio greater than the gravitational acceleration of the Sun (0.6 milli-g’s at 1 AU). Figure 25 shows the same spacecraft mass performing a roundtrip mission to the Saturn system spending over a year around two moons of interest, Titan and Enceladus.

This is called 'counting your chickens before the eggs have hatched'.

I would need to be more of an expert than I am to imagine all the things that could go wrong with their experiment.  But just so you see what I mean, here's one thing they do mention:

one visible effect to the seismic environment is the periodic (about one-third to one-quarter Hertz) perturbation created by the waves from the Gulf of Mexico (about 25 miles southeast of Johnson Space Center), especially on windy days.

The thrust they're measuring is so small that waves in the ocean 25 miles away could screw up the experiment!  They tried to deal with this... but it goes to show, you can't revolutionize physics until you carefully check all the sources of error.

I thank Greg Egan and +Matt McIrvin for their help, but of course they're not to blame for any mistakes I made. 

The paper I'm talking about was published here:

• David Brady, Harold White, Paul March, James Lawrence and Frank Davies, Anomalous thrust production from an RF test device measured on a low-thrust torsion pendulum, 50th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, http://arc.aiaa.org/doi/abs/10.2514/6.2014-4029.  

Unfortunately it's not free except for the abstract.  Luckily someone has liberated the paper and put a free version here:

• http://rghost.net/57230791

Beware: the abstract in the paper is different than the abstract on the NASA technical report server here:

• http://ntrs.nasa.gov/search.jsp?R=20140006052

This is where they say they didn't remove the air from the vacuum chamber.

There's a website about Guido Fetta's company and his device:

• Cannae Drive, http://cannae.com/about

It says:

The Cannae Drive is a resonating cavity with design features that redirect the radiation pressure exerted in the cavity to create a radiation pressure imbalance on the cavity. This differential in radiation pressure generates an unbalanced force that creates thrust. The cavity is accelerated without use of propellant. Don't believe it? Study the theory.  Replicate our numerical models.  Review our experimental results.  And draw your own conclusions.

Unfortunately, when I click on the links to theory, numerical models or experimental results, I get:

404 - Article not found

+Hamilton Carter pointed out another paper by the NASA team, which explains the wild optimism behind this experiment:

• Dr. Harold “Sonny” White, Paul March, Nehemiah Williams, and William O’Neill, Eagleworks Laboratories: advanced propulsion physics research,  http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20110023492.pdf

They write:

NASA/JSC is implementing an advanced propulsion physics laboratory, informally known as "Eagleworks", to pursue propulsion technologies necessary to enable human exploration of the solar system over the next 50 years, and enabling interstellar spaceflight by the end of the century. This work directly supports the "Breakthrough Propulsion" objectives detailed in the NASA OCT TA02 In-Space Propulsion Roadmap, and aligns with the #10 Top Technical Challenge identified in the report. Since the work being pursued by this laboratory is applied scientific research in the areas of the quantum vacuum, gravitation, nature of space-time, and other fundamental physical phenomenon [sic], high fidelity testing facilities are needed. The lab will first implement a low-thrust torsion pendulum (<1 uN), and commission the facility with an existing Quantum Vacuum Plasma Thruster. To date, the QVPT line of research has produced data suggesting very high specific impulse coupled with high specific force. If the physics and engineering models can be explored and understood in the lab to allow scaling to power levels pertinent for human spaceflight, 400kW SEP human missions to Mars may become a possibility, and at power levels of 2MW, 1-year transit to Neptune may also be possible. Additionally, the lab is implementing a warp field interferometer that will be able to measure spacetime disturbances down to 150nm.  Recent work published by White suggests that it may be possible to engineer spacetime creating conditions similar to what drives the expansion of the cosmos. Although the expected magnitude of the effect would be tiny, it may be a “Chicago pile” moment for this area of physics.

The "Chicago pile" was the experiment that demonstrated a nuclear chain reaction. 

#spnetwork doi:10.2514/MJPC14  #cannae_drive  ___

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2014-08-03 07:07:41 (31 comments, 2 reshares, 13 +1s) 

Puzzle: what are these sciences?

People who can read Chinese will have a huge advantage.  I don't think googling these words in  English will help you much!  Some websites use coppology to mean the study of film-maker Francis Ford Coppola, but that's not what it means here.  Another website says:

Domain Misspelling for: coptology

coppology; copqology; coprology; copsology; coptology; copuology; copvology; copwology; copxology; copyology; copzology; coptalogy; coptblogy; coptclogy...

In case you're wondering, coptology is the study of the native Christians of Egypt, called Copts.  But that's not relevant either.

Since you read down this far I'll give you a big hint: I saw this sign on a local shop in Singapore. 

There are lots of interesting signs around here.  For example,there&... more »

Puzzle: what are these sciences?

People who can read Chinese will have a huge advantage.  I don't think googling these words in  English will help you much!  Some websites use coppology to mean the study of film-maker Francis Ford Coppola, but that's not what it means here.  Another website says:

Domain Misspelling for: coptology

coppology; copqology; coprology; copsology; coptology; copuology; copvology; copwology; copxology; copyology; copzology; coptalogy; coptblogy; coptclogy...

In case you're wondering, coptology is the study of the native Christians of Egypt, called Copts.  But that's not relevant either.

Since you read down this far I'll give you a big hint: I saw this sign on a local shop in Singapore. 

There are lots of interesting signs around here.  For example, there's a chain of local copy shops called Xorex, and a home furnishing store called The Lourve. ___

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2014-08-02 09:33:16 (75 comments, 218 reshares, 470 +1s) 

The incredible shrinking force

Around 2000, a guy named Roger Shawyer claimed he could bounce microwaves inside a fancy-shaped can and get them to  push the can forwards, without anything leaving the can. 

This would violate conservation of momentum.  It's like sitting inside a car and making it roll forwards by pushing on the steering wheel.  Standard physics doesn't allow this.  He didn't claim to be using anything other than standard physics. 

So: ho hum, just another guy with a really bad idea.  I get emails like this all the time.

But in 2001, his company got a £45,000 grant from the British government to study this idea.  He built his machine and claimed that with 850 watts of power he could get a force of 0.016 newtons.   That's a bit less than the force of gravity from a penny pushing down on your hand.  It could easilybe an exper... more »

The incredible shrinking force

Around 2000, a guy named Roger Shawyer claimed he could bounce microwaves inside a fancy-shaped can and get them to  push the can forwards, without anything leaving the can. 

This would violate conservation of momentum.  It's like sitting inside a car and making it roll forwards by pushing on the steering wheel.  Standard physics doesn't allow this.  He didn't claim to be using anything other than standard physics. 

So: ho hum, just another guy with a really bad idea.  I get emails like this all the time.

But in 2001, his company got a £45,000 grant from the British government to study this idea.  He built his machine and claimed that with 850 watts of power he could get a force of 0.016 newtons.   That's a bit less than the force of gravity from a penny pushing down on your hand.  It could easily be an experimental error.

Why would people want a machine that uses lots of power to create a pathetically feeble force?   Because - here's the great piece of salesmanship - if it existed, you could use it to build a reactionless drive!  If you had a spaceship with huge amounts of power to spare - like, say, a nuclear reactor - you could use this gizmo to push your spaceship forwards without anything spewing out the back end. 

Again, this is about as plausible as powering a spaceship by having the crew push on it from the inside.   But if you don't know physics, it sounds very exciting. 

The story goes on.  And on.  And on.  It won't die.  In 2012, some Chinese physicists claimed they could get a force of 0.720 newtons from a power of 2,500 watts using some version of Shawyer's device. 

And now NASA is studying it!

They're claiming to see a force one thousandth as big as the Chinese - probably because they are doing the experiment one thousand times more accurately.  And still, some people are excited about this. 

The new device comes with new improved mumbo-jumbo.  Shawyer claimed that thanks to special relativity, classical electromagnetism can violate conservation of momentum.  I took those courses in college, I know that's baloney.  Now the NASA scientists say:

"Test results indicate that the RF resonant cavity thruster design, which is unique as an electric propulsion device, is producing a force that is not attributable to any classical electromagnetic phenomenon and therefore is potentially demonstrating an interaction with the quantum vacuum virtual plasma."

This is baloney too - but now it's graduate-level baloney.  "Quantum vacuum virtual plasma" is something you'd say if you failed a course in quantum field theory and then smoked too much weed.  There's no such thing as "virtual plasma".   If you want to report experimental results that seem to violate the known laws of physics, fine.  But it doesn't help your credibility to make up goofy pseudo-explanations.

I expect that in 10 years the device will be using quantum gravity and producing even less force. 

For an article written by a severely optimistic blogger, see:

http://www.wired.co.uk/news/archive/2014-07/31/nasa-validates-impossible-space-drive

For the NASA report see:

• David Brady, Harold White, Paul March, James Lawrence and Frank Davies, Anomalous thrust production from an RF test device measured on a low-thrust torsion pendulum, 50th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, http://arc.aiaa.org/doi/abs/10.2514/6.2014-4029.  Free version available at http://rghost.net/57230791.

Unfortunately only the abstract is free in the official version.___

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2014-08-02 02:14:22 (11 comments, 14 reshares, 106 +1s) 

Fractal madness with heptagons

This picture drawn by +Danny Calegari shows the '{7,3,3} honeycomb' in hyperbolic space.  Hyperbolic space is an infinite space where triangles have angles that add up to less than 180 degrees.  But the mathematician Poincaré figured out how compress it down to a ball so we can see it from the outside, and that's what we see here.

The {7,3,3} honeycomb is built of regular 7-sided shapes - heptagons - in hyperbolic space.  But what does {7,3,3} mean?

The heptagons lie on infinite sheets, which show up as holes here.  If you look at these holes,  you'll see they have three heptagons meeting at each corner.  So, each one is a copy of the {7,3} tiling of the hyperbolic plane.  7 stands for heptagon, 3 stands for three meeting at each corner.

It's impossible to see, but 3 of these {7,3}tilings... more »

Fractal madness with heptagons

This picture drawn by +Danny Calegari shows the '{7,3,3} honeycomb' in hyperbolic space.  Hyperbolic space is an infinite space where triangles have angles that add up to less than 180 degrees.  But the mathematician Poincaré figured out how compress it down to a ball so we can see it from the outside, and that's what we see here.

The {7,3,3} honeycomb is built of regular 7-sided shapes - heptagons - in hyperbolic space.  But what does {7,3,3} mean?

The heptagons lie on infinite sheets, which show up as holes here.  If you look at these holes,  you'll see they have three heptagons meeting at each corner.  So, each one is a copy of the {7,3} tiling of the hyperbolic plane.  7 stands for heptagon, 3 stands for three meeting at each corner.

It's impossible to see, but 3 of these {7,3} tilings meet along each edge in the picture.  That's why the whole thing is called the {7,3,3} honeycomb.

It sounds wacky and fun, and it is, but it's also part of a deep theory, which I explain on my blog:

http://blogs.ams.org/visualinsight/2014/08/01/733-honeycomb/

For a nice picture of a {7,3} tiling, try this earlier blog article:

http://blogs.ams.org/visualinsight/2014/07/15/73-tiling/

The topology of the {7,3,3} honeycomb is interesting. It is simply connected, since all the holes extend all the way to the edge of the Poincaré ball.  And its ‘boundary’ is a highly distorted copy of the Sierpinski carpet. 

That's why I had a contest to create a nice picture of the Sierpinski carpet a while back!  To see the winning entry and learn more about what makes that fractal special, go here:

http://blogs.ams.org/visualinsight/2014/07/01/sierpinski-carpet/

#fractals #geometry___

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2014-08-01 02:04:52 (4 comments, 20 reshares, 76 +1s) 

Citizen Science

The Azimuth Code Project is a friendly gang of hackers and geeks. We're studying a new method of El Niño prediction that uses a "climate network".  We replicated the software used in a paper about this method, and we made it open-source.  Then a statistician named Steven Wenner jumped in and examined the results!  Check it out here:

http://johncarlosbaez.wordpress.com/2014/07/23/el-nino-project-part-6/

Very roughly, when the red curve goes over the horizontal line, the method predicts an El Niño in the following year - a blue spike.  But sometimes it doesn't work, and there are weird borderline cases.

Wenner ran some statistical tests on this method, and then +Jan Galkowski jumped in and ran a bunch more, using sophisticated Bayesian ideas.  You can read about that in the comments. 

There's a lot ofbuzz ab... more »

Citizen Science

The Azimuth Code Project is a friendly gang of hackers and geeks. We're studying a new method of El Niño prediction that uses a "climate network".  We replicated the software used in a paper about this method, and we made it open-source.  Then a statistician named Steven Wenner jumped in and examined the results!  Check it out here:

http://johncarlosbaez.wordpress.com/2014/07/23/el-nino-project-part-6/

Very roughly, when the red curve goes over the horizontal line, the method predicts an El Niño in the following year - a blue spike.  But sometimes it doesn't work, and there are weird borderline cases.

Wenner ran some statistical tests on this method, and then +Jan Galkowski jumped in and ran a bunch more, using sophisticated Bayesian ideas.  You can read about that in the comments. 

There's a lot of buzz about citizen science these days - how ordinary folks can contribute to scientific projects by collecting data or running software.  It's a great trend...  but I'm enjoying how extraordinary folks are joining the Azimuth Project and helping us out with their special skills. 

We need all the help we can get, that's for sure.  Most of all we could use advice from some climate scientists.  But it would also be great if we had more folks who enjoyed explaining techniques from math, statistics, and machine learning.  For example, Dara Shayda just used 'continuous wavelet analysis' to study patterns in the air pressure at two locations important for El Niños.  Since I'm the main writer in the gang, I had to write this up, which required learning about continuous wavelet analysis and figuring out how to explain it - explaining stuff is a big part of the project.  I did it, and I enjoyed it, but there's too much for me to do!  We could go faster with another good writer.

#El_Niño___

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2014-07-31 00:12:17 (48 comments, 515 reshares, 389 +1s) 

The one you're not looking at turns clockwise

Look at either the red dot or the yellow dot.  The circles near that dot will turn counterclockwise.  The others turn clockwise!

Or: let your eyes bounce back and forth between the two.

Or: look away from both of them.

Puzzle  1: how does it work?

Puzzle 2: who first made this gif?  I can't find the original source.

#illusions  

The one you're not looking at turns clockwise

Look at either the red dot or the yellow dot.  The circles near that dot will turn counterclockwise.  The others turn clockwise!

Or: let your eyes bounce back and forth between the two.

Or: look away from both of them.

Puzzle  1: how does it work?

Puzzle 2: who first made this gif?  I can't find the original source.

#illusions  ___

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2014-07-30 01:44:38 (32 comments, 11 reshares, 111 +1s) 

There's a monster in the kitchen!

Not really.  This is a baby aye-aye being weighed at the the Duke Lemur Centre in North Carolina.  

Aye-ayes live in Madagascar.  They're the world's largest nocturnal primate.  They find grubs to eat by tapping on trees.  When an aye-aye finds a good spot, it chews a hole in the wood.  Then it sticks its narrow middle finger in the hole to pull the grubs out.

Since it was founded in 1966, the Duke Lemur Center has collected data on 3600 animals from more than 40 species of prosimians - the group of primates that includes lemurs, lorises, galagos, bushbabies, and tarsiers.  Now they've released the data to the world!  It's available online, for free:

http://lemur.duke.edu/duke-lemur-center-database/

Almost all these animals only live in Madagascar, and most of them are rare,threaten... more »

There's a monster in the kitchen!

Not really.  This is a baby aye-aye being weighed at the the Duke Lemur Centre in North Carolina.  

Aye-ayes live in Madagascar.  They're the world's largest nocturnal primate.  They find grubs to eat by tapping on trees.  When an aye-aye finds a good spot, it chews a hole in the wood.  Then it sticks its narrow middle finger in the hole to pull the grubs out.

Since it was founded in 1966, the Duke Lemur Center has collected data on 3600 animals from more than 40 species of prosimians - the group of primates that includes lemurs, lorises, galagos, bushbabies, and tarsiers.  Now they've released the data to the world!  It's available online, for free:

http://lemur.duke.edu/duke-lemur-center-database/

Almost all these animals only live in Madagascar, and most of them are rare, threatened or endangered.  This rich diversity of life is precious in ways we're still too crude to assign much economic value.  Each of these species is a rich assemblage of strategies and tricks, from their behavior down to their very molecules, finely honed by millions of years of evolution.  But the natural biotechnology they represent is too sophisticated for us to understand or copy yet - or make much money from, except for 'ecotourism'.  For the most part, we count them as worthless trash, while spending millions on canvases cleverly daubed with paint.  Someday we'll regret this.  Let's try to change it sooner rather than later.

Some people are thinking about it:

http://www.capitalinstitute.org/moneyandwealth/can-nature-be-monetized-capital-institute-conversation___

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2014-07-29 02:47:39 (31 comments, 10 reshares, 55 +1s) 

Mansion of Wind and Waves

Last night Lisa and I ate at Feng Bo Zhuang on Temple Street in Singapore.  We went there since it had good spicy food from Hunan and Dongbei, a crowd of happy customers, and nice wood decor... but it turned out to have a jiang-hu theme: that's what Chinese call the world of wandering martial artists and swordsmen. 

See that sword on the wall there?

The name of this restaurant means 'Mansion of Wind and Waves' - but 'wind and waves' means the turbulence and fighting that that goes on in the jiang-hu.  There's a poem by the entrance that says "in the jiang-hu you have no choice, you either fight back or die".  This tough world is romanticized in wuxia fiction and movies:

Typically, the heroes in wuxia fiction do not serve a lord, wield military power or belong to theari... more »

Mansion of Wind and Waves

Last night Lisa and I ate at Feng Bo Zhuang on Temple Street in Singapore.  We went there since it had good spicy food from Hunan and Dongbei, a crowd of happy customers, and nice wood decor... but it turned out to have a jiang-hu theme: that's what Chinese call the world of wandering martial artists and swordsmen. 

See that sword on the wall there?

The name of this restaurant means 'Mansion of Wind and Waves' - but 'wind and waves' means the turbulence and fighting that that goes on in the jiang-hu.  There's a poem by the entrance that says "in the jiang-hu you have no choice, you either fight back or die".  This tough world is romanticized in wuxia fiction and movies:

Typically, the heroes in wuxia fiction do not serve a lord, wield military power or belong to the aristocratic class. They are often from the lower social classes of ancient Chinese society. Wuxia heroes are usually bound by a code of chivalry that requires them to right wrongs, fight for righteousness, remove an oppressor, redress wrongs and bring retribution for past misdeeds. The Chinese wuxia traditions can be compared to martial codes from other countries, such as the Japanese samurai's bushido tradition, the chivalry of medieval European knights and the gunslingers of America's Westerns.

We had frog cooked in bamboo, dumplings with hot broth inside called xiaolongbao, spicy sliced cucumber, and an appetizer of roast peanuts.  Very tasty with some Harbin beer!

Back in 2011 this restaurant was quite serious about the jiang-hu theme: the waitresses would wear traditional outfits and chant a poem when you left.  When one reviewer tried to pay by credit card, his waitress said “江湖中只收银两” - "in jiang-hu, we only accept silver ingots".    Luckily it's mellowed out by now; none of this nonsense, just good food.

As usual, most of my expertise on Chinese culture comes from Lisa... but the long quote is from here:

https://en.wikipedia.org/wiki/Wuxia

and the 2011 review is here:

http://rubbisheatrubbishgrow.com/2011/06/11/feng-bo-zhuang-%E9%A3%8E%E6%B3%A2%E5%BA%84-chinatown/

#singapore___

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2014-07-28 01:57:54 (27 comments, 13 reshares, 100 +1s) 

Cracked moon

Jupiter's moon Europa is about as big as ours... but its crust is made of water ice, and underneath there seems to be a salty ocean!  

The surface has red cracks.  Spectrographs suggest these are rich in salts like magnesium sulfate left by evaporating water that comes from within.  But why are they red?  Nobody knows.  Maybe it's sulfur.

Why are there cracks?  That's an interesting story.

Europa orbits Jupiter every three and a half days... but it's tidally locked to Jupiter, so there's a place on Europa's surface where Jupiter always hangs directly overhead.  If you looked up, it would be about 24 times as big across as the Sun viewed from Earth.

But all is not calm!  Europa is in resonance with two other moons of Jupiter.   It goes around exactly half as often as the moon Io, and twice asoften as G... more »

Cracked moon

Jupiter's moon Europa is about as big as ours... but its crust is made of water ice, and underneath there seems to be a salty ocean!  

The surface has red cracks.  Spectrographs suggest these are rich in salts like magnesium sulfate left by evaporating water that comes from within.  But why are they red?  Nobody knows.  Maybe it's sulfur.

Why are there cracks?  That's an interesting story.

Europa orbits Jupiter every three and a half days... but it's tidally locked to Jupiter, so there's a place on Europa's surface where Jupiter always hangs directly overhead.  If you looked up, it would be about 24 times as big across as the Sun viewed from Earth.

But all is not calm!  Europa is in resonance with two other moons of Jupiter.   It goes around exactly half as often as the moon Io, and twice as often as Ganymede.

This keeps its orbit from settling down into a perfect circle.   So Jupiter seems to move towards Europa and then back away every 3½ days.  Just a little bit.

When Jupiter comes closer, its gravitational attraction increases more on the near side of Europa.  This causes Europa to stretch towards and away from it - a tidal force.  When Jupiter moves away, Europa relaxes a bit into a more spherical shape. 

This constant stretching and relaxing causes cracks in the surface ice of Europa.  More importantly... for fans of extraterrestrial life... it keeps Europa's oceans warm enough to stay liquid under the ice! 

We've got a lot of evidence for this.  The cracks and 'chaotic terrain' are hard to explain without an ocean under the ice.  The Galileo probe, which took this great photo of Europa, also detected a changing magnetic field that could be due to electric currents flowing through a salty ocean.   And the Hubble telescope has even seen water vapor near Europa's south pole!   This may have come from huge geysers... like the geysers on Saturn's moon Enceladus, but even bigger.

But we don't know how thick the ice crust is.  Some people say 10-30 kilometers, others say just a few.  If it's thin enough, we could send a probe that would melt through the ice and explore the ocean.  What's down there?

In 2012, the European Space Agency (ESA) announced it wants to launch a Jupiter Icy Moon Explorer.  This would includes some flybys of Europa, but is more focused on Ganymede.  In January 2014, the US House Appropriations Committee announced a new bipartisan bill that includes $80 million in funding to keep planning a Europa mission.  One possibility is the Europa Clipper, which would orbit Jupiter and conduct 45 low-altitude flybys of Europa, carrying an ice penetrating radar, short wave infra red spectrometer, topographical imager, and an ion and neutral mass spectrometer.

Puzzle: where does the energy originally come from, that heats Europa?  If something is gaining energy, something else must be losing it.

#astronomy___

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2014-07-27 02:26:25 (10 comments, 13 reshares, 112 +1s) 

As if millions of voices cried out in terror and were suddenly silenced

The first report of a fast radio burst appeared in 2007.  An astronomer named Duncan Lorimer found a signal buried in recordings made at the Parkes radio telescope in Australia.  It lasted for less than 5 milliseconds and it seemed to come from outside our galaxy.  It didn't match anything we'd seen in visible light, X-rays or anything else.  A complete mystery! 

Last year people found 4 more.  But all at the Parkes telescope.  Maybe there was an error of some sort?

But now they've seen one at Arecibo, the famous radio telescope in Puerto Rico.  Duncan Lorimer says several more confirmations will soon be announced.

So what causes these fast radio bursts?

We don't know.  But here's one theory: a 'blitzar'.

When a supernovablasts t... more »

As if millions of voices cried out in terror and were suddenly silenced

The first report of a fast radio burst appeared in 2007.  An astronomer named Duncan Lorimer found a signal buried in recordings made at the Parkes radio telescope in Australia.  It lasted for less than 5 milliseconds and it seemed to come from outside our galaxy.  It didn't match anything we'd seen in visible light, X-rays or anything else.  A complete mystery! 

Last year people found 4 more.  But all at the Parkes telescope.  Maybe there was an error of some sort?

But now they've seen one at Arecibo, the famous radio telescope in Puerto Rico.  Duncan Lorimer says several more confirmations will soon be announced.

So what causes these fast radio bursts?

We don't know.  But here's one theory: a 'blitzar'.

When a supernova blasts the outer layers of a big star into space, the remaining core collapses down to a ball of neutronium heavier than our Sun and the size of a small city: a neutron star

If this spins fast enough, like a thousand times a second, it's highly magnetic.  It produces regular pulses of intense radiation - and we call it a pulsar

If it's too heavy, though, the neutron star collapses into a black hole.

But suppose it's spinning really fast!  Then the centrifugal force might pull it out and keep it from collapsing into a black hole! 

Until it slowed down.  Then it would collapse into a black hole.  The magnetic field lines would suddenly get cut by an event horizon.  They don't like that.  WHAM - a blast of radio waves.  A blitzar.

One newspaper headline describes it this way: "as if millions of voices suddenly cried out in terror and were suddenly silenced".  It's good to see that purple prose isn't dead in science journalism.

This is just a theory, so far.  We'll get more evidence as we see more short radio bursts.

For more, read these:

http://arstechnica.com/science/2013/07/mysterious-radio-bursts-come-from-outside-our-galaxy/

http://arstechnica.com/science/2013/07/possible-explanation-for-radio-bursts-meet-the-blitzar/

They're well-written! 

And here's the paper that introduced the blitzar theory.  The calculations are surprisingly sketchy.  It's really hard to calculate what happens when a ball of neutronium suddenly collapses into a black hole, so they use simple estimates:

• Heino Falcke and Luciano Rezzolla, Fast radio bursts: the last sign of supramassive neutron stars, http://arxiv.org/abs/1307.1409.

Abstract. Several fast radio bursts have been discovered recently, showing a bright, highly dispersed millisecond radio pulse. The pulses do not repeat and are not associated with a known pulsar or gamma-ray burst. The high dispersion suggests sources at cosmological distances, hence implying an extremely high radio luminosity, far larger than the power of single pulses from a pulsar. We suggest that a fast radio burst represents the final signal of a supramassive rotating neutron star that collapses to a black hole due to magnetic braking. The neutron star is initially above the critical mass for non-rotating models and is supported by rapid rotation. As magnetic braking constantly reduces the spin, the neutron star will suddenly collapse to a black hole several thousand to million years after its birth. We discuss several formation scenarios for supramassive neutron stars and estimate the possible observational signatures {making use of the results of recent numerical general-relativistic calculations). While the collapse will hide the stellar surface behind an event horizon, the magnetic-field lines will snap violently. This can turn an almost ordinary pulsar into a bright radio "blitzar": Accelerated electrons from the travelling magnetic shock dissipate a significant fraction of the magnetosphere and produce a massive radio burst that is observable out to z > 0.7. Only a few percent of the neutron stars needs to be supramassive in order to explain the observed rate. We suggest that fast radio bursts might trace the solitary formation of stellar mass black holes at high redshifts. These bursts could be an electromagnetic complement to gravitational-wave emission and reveal a new formation and evolutionary channel for black holes that are not seen as gamma-ray bursts. Radio observations of these bursts could trace the core-collapse supernova rate throughout the universe.

#astronomy   #spnetwork arXiv:1307.1409 #mustread  ___

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2014-07-26 02:38:12 (45 comments, 245 reshares, 248 +1s) 

The hypersnake versus the moiré eel

This animated gif is cool, but here's something much cooler:

http://wry.me/hacking/moire-eel.html

It starts out intense... and then keeps getting more so. 

You can control the shape of the little rectangles by moving your cursor over the screen.  Try to keep your eye on just one little rectangle!  It moves up and down, not very fast... but sometimes it's impossible to keep your eye on it, because all the rectangles together produce patterns that grab your attention.  These are called moiré patterns. 

I think the 'hypersnake' here is attention-grabbing because your brain has parts that are good at detecting snakes even before you are conscious of it.  Your amygdala is one of these parts:

Information from an external stimulus reaches the amygdala in twodiffere... more »

The hypersnake versus the moiré eel

This animated gif is cool, but here's something much cooler:

http://wry.me/hacking/moire-eel.html

It starts out intense... and then keeps getting more so. 

You can control the shape of the little rectangles by moving your cursor over the screen.  Try to keep your eye on just one little rectangle!  It moves up and down, not very fast... but sometimes it's impossible to keep your eye on it, because all the rectangles together produce patterns that grab your attention.  These are called moiré patterns. 

I think the 'hypersnake' here is attention-grabbing because your brain has parts that are good at detecting snakes even before you are conscious of it.  Your amygdala is one of these parts:

Information from an external stimulus reaches the amygdala in two different ways: by a short, fast, but imprecise route, directly from the thalamus; and by a long, slow, but precise route, by way of the cortex.

It is the short, more direct route that lets us start preparing for a potential danger before we even know exactly what it is. In some situations, these precious fractions of a second can mean the difference between life and death.

Here is an example. Suppose you are walking through a forest when you suddenly see a long, narrow shape coiled up at your feet. This snake-like shape very quickly, via the short route, sets in motion the physiological reactions of fear that are so useful for mobilizing you to face the danger. But this same visual stimulus, after passing through the thalamus, will also be relayed to your cortex. A few fractions of a second later, the cortex, thanks to its discriminatory faculty, will realize that the shape you thought was a snake was really just a discarded piece of garden hose. Your heart will then stop racing, and you will just have had a moment’s scare.

The 'moiré eel' was made by Darius Bacon.  You can see more of his stuff here: 

http://wry.me/blog/

Puzzle: who made the 'hypersnake' here?  I don't know!  A Google image search shows it on many websites, but I haven't found one that credits the inventor.  Naughty!

For more on moiré patterns try this:

https://en.wikipedia.org/wiki/Moiré_pattern

For more on the amygdala, try this:

http://thebrain.mcgill.ca/flash/i/i_04/i_04_cr/i_04_cr_peu/i_04_cr_peu.html

#illusions  ___

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