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John Baez has been shared in 334 public circles

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AuthorFollowersDateUsers in CircleCommentsReshares+1Links
Atanas Georgiev Atanasov1,950SCIENCE AND TECHNOLOGY + FRIENDS : CIRCLE V.8; maintained by +Atanas Georgiev Atanasov  ; You can learn more about my personal scientific research at: http://www.ncbi.nlm.nih.gov/pubmed/25083916   To be added to the circle, please +1/reshare/comment/add me to your circles. I would be happy to connect on other networks too:http://about.me/Atanas_At   Below I am just pasting some keyords/topics to improve the visibility of the circle and to make it more discoverable: #Science   #Research   #Technology   #NASA    #Space   #Innovation   #Engineering   #NIAC   #nutrition   #Entrepreneur   #Commercial #ScienceSunday   #Sundayscience   #Science   #Research   #Tech #GameTech   #GameTechnology   #Gaming   #VideoGaming   #Microsoft #MicrosoftResearch    #innovation   #Inflammation   #Brain #mindcontrol   #photography   #tech   #socialmedia   #googleplus #naturalproduct   2014-11-17 05:24:223625111
Atanas Georgiev Atanasov1,633SCIENCE AND TECHNOLOGY + FRIENDS : CIRCLE V.7; maintained by +AtanasGeorgievAtanasov ; You can learn more about my personal scientific research at: http://www.ncbi.nlm.nih.gov/pubmed/25083916   To be added to the circle, please +1/reshare/comment/add me to your circles. I would be happy to connect on other networks too:http://about.me/Atanas_At   Below I am just pasting some keyords/topics to improve the visibility of the circle and to make it more discoverable: #Science   #Research   #Technology   #NASA    #Space   #Innovation   #Engineering   #NIAC   #nutrition   #Entrepreneur   #Commercial #ScienceSunday  #Sundayscience   #Science   #Research   #Tech #GameTech  #GameTechnology   #Gaming   #VideoGaming   #Microsoft #MicrosoftResearch   #innovation   #Inflammation   #Brain #mindcontrol  #photography   #tech   #socialmedia   #googleplus #naturalproduct  #artists   #foodies   #cars2014-11-13 05:47:40346101
Atanas Georgiev Atanasov1,490SCIENCE and TECHNOLOGY + FRIENDS of it : CIRCLE V.6; maintained by +AtanasGeorgievAtanasov ; You can learn more about my personal scientific research here: http://www.ncbi.nlm.nih.gov/pubmed/25083916   To be added to the circle, please +1/reshare/comment/add me to your circles. I would be happy to connect on other networks too:http://about.me/Atanas_At   Below I am just pasting some keyords/topics to improve the visibility of the circle and to make it more discoverable: #Science   #Research   #Technology   #NASA    #Space   #Innovation   #Engineering   #NIAC   #nutrition   #Entrepreneur   #Commercial #ScienceSunday  #Sundayscience   #Science   #Research   #Tech #GameTech  #GameTechnology   #Gaming   #VideoGaming   #Microsoft #MicrosoftResearch   #innovation   #Inflammation   #Brain #mindcontrol  #photography   #tech   #socialmedia   #googleplus #naturalproduct  #artists   #foodies   2014-11-10 06:22:16330011
Atanas Georgiev Atanasov1,228SCIENCE and TECHNOLOGY + FRIENDS of it : CIRCLE V.5; maintained by +AtanasGeorgievAtanasov ; You can learn more about my personal scientific research here: http://www.ncbi.nlm.nih.gov/pubmed/25083916   To be added to the circle, please +1/reshare/comment/add me to your circles. I would be happy to connect on other networks too:http://about.me/Atanas_At   Below I am just pasting some keyords/topics to improve the visibility of the circle and to make it more discoverable: #Science  #Research #Technology #NASA  #Space #Innovation #Engineering #NIAC #nutrition #Entrepreneur #Commercial#ScienceSunday #Sundayscience #Science #Research #Tech#GameTech #GameTechnology #Gaming #VideoGaming +Microsoft#MicrosoftResearch  #innovation #Inflammation #Brain#mindcontrol #photography #tech #socialmedia #googleplus#naturalproduct #artists #foodies #cars #sharingiscaring  #Liver#sharingmeansthankyou #socialmedia  #sports #Smartphones#tablets 2014-11-06 08:10:34369051
Sharon Caroline1,629Hello my friends, good morning/evening for you all!Boost Your visibility On Google+!Shared and be shared. :)Thank you for sharing and promoting this.#circleshare #sharedcircles #sharingcircles #sharedcircleoftheday2014-11-05 08:31:58463000
Atanas Georgiev Atanasov1,228SCIENCE and TECHNOLOGY + FRIENDS of it : Circle V.3, maintained by +AtanasGeorgievAtanasov To be added to the circle, please +1/reshare/comment/add me to your circles. I would be happy to connect on other networks too:http://about.me/Atanas_At   You can learn a bit more about my personal scientific research from these links: https://plus.google.com/115938908270684192009/posts/MGt3zvEtTgq  http://www.ncbi.nlm.nih.gov/pubmed/25083916   Below I am just pasting some keyords/topics to improve the visibility of the circle and to make it more discoverable: #Science  #Research #Technology #NASA  #Space #Innovation  #Engineering #NIAC #nutrition #Entrepreneur #Commercial #ScienceSunday #Sundayscience #Science #Research #Tech #GameTech #GameTechnology #Gaming #VideoGaming +Microsoft #MicrosoftResearch  #innovation #Inflammation #Brain #mindcontrol #photography #tech #socialmedia #googleplus #naturalproduct #artists #foodies 2014-11-05 07:02:42362101
Atanas Georgiev Atanasov1,127Science and Technology +Friends: Circle 2014 V.2 To be added to the circle, please +1/reshare/comment. I would be happy to connect on other networks too:http://about.me/Atanas_At   You can learn a bit more about my personal scientific research from these links: https://plus.google.com/115938908270684192009/posts/MGt3zvEtTgq  http://www.ncbi.nlm.nih.gov/pubmed/25083916   Below I am just pasting some keyords/topics to improve the visibility of the circle and to make it more discoverable: #Science  #Research #Technology #NASA  #Space #Innovation  #Engineering #NIAC #nutrition #Entrepreneur #Commercial #ScienceSunday #Sundayscience #Science #Research #Tech #GameTech #GameTechnology #Gaming #VideoGaming +Microsoft #MicrosoftResearch  #innovation #Inflammation #Brain #mindcontrol #photography #tech #socialmedia #googleplus #naturalproduct #artists #foodies #cars 2014-11-04 06:51:44407101
Atanas Georgiev Atanasov982Science and Technology Circle 2014 To be added to the circle, please +1/reshare/comment. I would be happy to connect on other networks too:http://about.me/Atanas_At   You can learn a bit more about my personal scientific research from these links: https://plus.google.com/115938908270684192009/posts/MGt3zvEtTgq  http://www.ncbi.nlm.nih.gov/pubmed/25083916   Below I am just pasting some keyords/topics to improve the visibility of the circle and to make it more discoverable: #Science  #Research #Technology #NASA   #Space #Innovation   #Engineering #NIAC #nutrition #Entrepreneur #Commercial #ScienceSunday #Sundayscience #Science #Research #Tech #GameTech #GameTechnology #Gaming #VideoGaming #Microsoft #MicrosoftResearch   #innovation #Inflammation #Brain #mindcontrol #photography #tech #socialmedia #googleplus #naturalproduct #artists 2014-11-02 08:37:19453101
Becky Collins17,500Top Active Engager's 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 ?2014-10-13 05:05:40478002
Refurio Anachro4,745September engagers circle: This month's buzz has been boosted by curiosity about Hamiltonians - welcome you all and nice to meet you! I found to really like Hamiltonian mechanics, and i'll sure come back to what i meant by describing them as "intriguing like postmodern psychedelic sculpture". Stay tuned, it wouldn't be the same without you!These people are physicists and mathematicians, research scientists, teachers, and enthusiasts. By adding us to your stream you'll find yourself learning about the universe and looking at the beauty of maths.Last month's finds:+Liz Krane found this cool video demonstrating how to mine bitcoins by hand!https://plus.google.com/+RefurioAnachro/posts/ERJpN6vLypGRaytraced spheroidal billiards: A set of high res views, and animated iteration depth. Since then i've been naming some of the features in the comments below, you sure you didn't miss any?https://plus.google.com/+RefurioAnachro/posts/4hDyHdYwmjMThe physical ellipse is the application i had in mind for Hamiltonians. It seems i should be posting about elliptic integrals and their inverses soon.https://plus.google.com/+RefurioAnachro/posts/Q2nDr5phZfQAnother tiger toroid animation. Look in the comments for a link to yet another view, and to meet an expert:https://plus.google.com/+RefurioAnachro/posts/RVCnJ5rH8kCOn Hamiltonians, my first piece about them, a quick introduction. It left me with the desire to dive deeper.https://plus.google.com/+RefurioAnachro/posts/DQZZvMBVPafDiagram 19: "The 59 icosahedra" is a book about the stellations of the icosahedron_...https://plus.google.com/+RefurioAnachro/posts/AHVv1JGLZy8"Dear august engagers", here's previous month's circle:https://plus.google.com/+RefurioAnachro/posts/X6pQCjNR6FiIn that post i claimed to post "impressions of the mandelbrot set" on +Spherical Reflections. Well, i lied, at that time i had just posted a phoenix. It's a different formula! Where are you, Mandelbrot experts?https://plus.google.com/b/115434895453136495635/117866562756294273963/posts/Qmy98YMjuwc+Spherical Reflections' page, stuff like the above and circle shares.https://plus.google.com/b/117866562756294273963/117866562756294273963/postsYou're in this circle because you reshared, plussed or commented on one of my posts (possibly via +2014-10-01 09:20:53161000
Kenneth Nicholson3,806Active users on Google+. Circle Share. If you received a notification, please reshare to your circles If you’d like to be added to the next circle share: • +1 this circle • Share this circle to PUBLIC • Include me in your circles • Comment on this post*More you share more you get! :)Thanks!*#awesomecircle #circleme #sharedpoint #sharewithyou #ShareYourCircle #epicengagers #davidromaphotography #addcircle #addpeople #affiliate #awesome #awesomecircles #awesomepeople #besocial #bestengagers #bestsharedcircle #circle #circlefriday #circlemonday2014-09-25 13:06:13485455179
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 15:25:57463224129234
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 05:34:0348613618
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 21:46:51451213121238
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 12:58:0746211960146
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 05:31:36476209123224
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 20:33:554709446122
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 05:07:06489214128231
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 21:03:52371111320
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 04:49:32487185131233
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

Activity

Average numbers for the latest posts (max. 50 posts, posted within the last 4 weeks)

47
comments per post
29
reshares per post
114
+1's per post

2,804
characters per posting

Top posts in the last 50 posts

Most comments: 169

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2014-11-03 18:18:00 (169 comments, 16 reshares, 151 +1s)Open 

Suspiciously cute

Last week many of my old posts started getting +1ed by beautiful women.

One theory is that at finally, at the age of 53 and happily married, I've become an irresistible babe magnet.  Maybe it's my skill at explaining advanced concepts in math and science.  Maybe it's my devilish good looks.

It's time to look for another theory.

Obviously, the other theory is that these are entities posing as beautiful women.  I wasn't born yesterday!   This type of thing has been happening for a long time on G+. 

But here's the interesting part.   Each old post is getting +1ed just once - each by a different entity.  Each of these entities does not merely post ads for some company, the way a typical fake does.  They post about a variety of things - but with a lot of photos of themselves riding horses, dressed upin Hall... more »

Most reshares: 111

posted image

2014-10-17 16:06:27 (123 comments, 111 reshares, 203 +1s)Open 

The moving sofa problem

You've probably tried to move a sofa around a bend in a hallway.  It's annoying.   But it leads to some fun math puzzles.  Let's keep things simple and work in 2 dimensions.  Then the moving sofa problem asks:

What is the shape of largest area that can be maneuvered through an L-shaped hallway of width 1? 

This movie shows one attempt to solve this problem.  It's called the Hammersley sofa, since it was discovered by John Hammersley.  It has an area of

π/2 + 2/pπ = 2.20741609916...

But it's not the best known solution!  In 1992, Joseph Gerver found a shape of area 2.2195 that works.

On the other hand, Hammersley showed that any solution has area at most

2 sqrt(2) = 2.82842712475...

So, the moving sofa problem remains unsolved.  Anothereasily stat... more »

Most plusones: 303

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2014-10-09 20:07:55 (146 comments, 58 reshares, 303 +1s)Open 

Hollow Earth

There have been lots of theories saying the Earth is hollow, but I know only one that could be true.

Edmond Halley, the guy who discovered the famous comet, had a theory where Earth consists of a hollow shell about 800 kilometers thick, two smaller shells nested inside, and a ball in the middle - all separated by atmospheres and rotating at different speeds!  It sounds nutty, but Halley was trying to explain the Earth's rather complicated magnetic fields: each of his shells was magnetic.

People sometimes accuse Leonhard Euler, the famous mathematician and physicists, of believing the Earth was hollow.  But that's not true.  In fact, all Euler did was propose a famous thought experiment:

Puzzle: If you could drill a hole all the way through the Earth, and drop a stone in, what would happen?

With the passage of time,sci... more »

Latest 50 posts

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2014-11-22 17:43:26 (33 comments, 60 reshares, 164 +1s)Open 

The virus has landed

This is a virus called a T4 bacteriophage.  It has landed on a bacterium.   Now it's getting ready to lower its tail, puncture the bacterium's cell wall, and inject its DNA.

When this happens:

1.  It immediately stops the bacterium's own genes from being expressed.

2.  In 5 minutes, its DNA starts synthesizing enzymes needed to make new copies of the virus.

3.  In 10 minutes, its DNA starts replicating.

4.  In 12 minutes, new copies of the virus start being formed.

5.  In 30 minutes, the bacterium bursts, releasing 100 to 150 new copies of the virus!

This deadly machine is only 0.2 micrometers tall.  Its DNA - the instruction book that makes everything work - is contained in the head, which is shaped like an icosahedron.  The DNA is 169,000 base pairs long, and itcodes for... more »

The virus has landed

This is a virus called a T4 bacteriophage.  It has landed on a bacterium.   Now it's getting ready to lower its tail, puncture the bacterium's cell wall, and inject its DNA.

When this happens:

1.  It immediately stops the bacterium's own genes from being expressed.

2.  In 5 minutes, its DNA starts synthesizing enzymes needed to make new copies of the virus.

3.  In 10 minutes, its DNA starts replicating.

4.  In 12 minutes, new copies of the virus start being formed.

5.  In 30 minutes, the bacterium bursts, releasing 100 to 150 new copies of the virus!

This deadly machine is only 0.2 micrometers tall.  Its DNA - the instruction book that makes everything work - is contained in the head, which is shaped like an icosahedron.  The DNA is 169,000 base pairs long, and it codes for 289 proteins.  Biologists understand it quite well now.

https://en.wikipedia.org/wiki/Enterobacteria_phage_T4

This picture is not a photograph; it was made by Mike Smith for a company called Xvivo Scientific Animation.  You can see other pictures by them here:

http://www.xvivo.net/wallpaper/___

posted image

2014-11-21 16:07:28 (20 comments, 4 reshares, 51 +1s)Open 

Heptagons - worst of all?

This picture by Toby Hudson shows the densest known packing of the regular heptagon. Of all convex shapes, the regular heptagon is believed to have the lowest maximal packing density.

Let's see how the competition stacks up!

You can pack equal-sized circles with a density of at most

0.9068996…

For regular octagons, the best you can do is

0.9061636…

If you smooth the corners of these octagons in a certain way, the best you can do is

0.902414…

This smoothed octagon is conjectured to be the worst shape for dense packing that's both convex and centrally symmetric. Convex means that if you draw a line segment between two points in your shape, the whole line segment is in the shape.  Centrally symmetric means that if you reflect the shape vertically andthen h... more »

Heptagons - worst of all?

This picture by Toby Hudson shows the densest known packing of the regular heptagon. Of all convex shapes, the regular heptagon is believed to have the lowest maximal packing density.

Let's see how the competition stacks up!

You can pack equal-sized circles with a density of at most

0.9068996…

For regular octagons, the best you can do is

0.9061636…

If you smooth the corners of these octagons in a certain way, the best you can do is

0.902414…

This smoothed octagon is conjectured to be the worst shape for dense packing that's both convex and centrally symmetric. Convex means that if you draw a line segment between two points in your shape, the whole line segment is in the shape.  Centrally symmetric means that if you reflect the shape vertically and then horizontally, it is unchanged.

If you drop the requirement that your shape be convex, you can make its maximum packing density as close to zero as you want!

So, to keep the game interesting, let's drop the requirement that the shape be centrally symmetric... but still demand that it be convex.  Then we can use the regular heptagon - a 7-sided shape with all sides and all angles equal.  It's believed that this shape has maximal packing density

0.89269…

And it's believed that the heptagon is the worst convex shape for packing.   But nobody has proved either of these things!

Puzzle 1: what the highest density you can get for regular pentagons?

Puzzle 2: is there a way to smooth the corners of the regular heptagon, to make it even worse?

For more, read my blog article:

http://blogs.ams.org/visualinsight/2014/11/15/packing-regular-heptagons/

#geometry  ___

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2014-11-20 17:14:00 (40 comments, 37 reshares, 113 +1s)Open 

A secret code

This is the Golay code.  Each row in this picture shows a string of 24 bits.  There are 12 rows.  If you look at any two rows, you'll see they differ in at least 8 places.

Here's how to get the Golay code.  Take a 12 x 12 square of bits with all 0's except for 1's down the diagonal - you can see that at left here.  Take another 12 x 12 square of bits that tells you when two faces of a dodecahedron share an edge: 0 if they do, 1 if they don't.  Stick these squares together and you get the Golay code!

Some guys around here keep asking if the math I talk about is good for anything.  In this case it is! 

The Voyager 1 and 2 spacecraft needed to transmit hundreds of color pictures of Jupiter and Saturn in their 1979, 1980, and 1981 fly-bys.   They had very little bandwidth, so they needed a good error-correctingcode.  Th... more »

A secret code

This is the Golay code.  Each row in this picture shows a string of 24 bits.  There are 12 rows.  If you look at any two rows, you'll see they differ in at least 8 places.

Here's how to get the Golay code.  Take a 12 x 12 square of bits with all 0's except for 1's down the diagonal - you can see that at left here.  Take another 12 x 12 square of bits that tells you when two faces of a dodecahedron share an edge: 0 if they do, 1 if they don't.  Stick these squares together and you get the Golay code!

Some guys around here keep asking if the math I talk about is good for anything.  In this case it is! 

The Voyager 1 and 2 spacecraft needed to transmit hundreds of color pictures of Jupiter and Saturn in their 1979, 1980, and 1981 fly-bys.   They had very little bandwidth, so they needed a good error-correcting code.  They used the Golay code! 

The point is that we can use the rows of this picture as code words.  If we take some rows and add them - adding each entry separately, mod 2 - we get more code words.  We get a total of 2^12 = 4096 code words. 

These code words have a cool property: it takes at least 8 errors to turn any code word into any other.   So, we say the Hamming distance between any two code words is at least 8.   In fact, the Golay code is the only code with 24-bit code words where the Hamming distance between any two is at least 8.

There's a whole theory of codes like this, and this is an especially good one.  You can transmit 12 bits of data with 24 bits... but since the Hamming distance between code words is big, someone can understand what you meant even if there are lots of errors!  So, the Golay code is useful for transmitting data in a noisy environment.

But the reason I like the Golay code is that it has a big and important symmetry group.  Its symmetry group is called M24 - one of the amazing things called Mathieu groups.    It has

24 x 23 x 22 x 21 x 20 x 48 = 244,823,040

elements.   It's connected to many other symmetrical things in math: for example, it acts as symmetries of the Leech lattice, the densest way to pack balls in 24 dimensions.

To be more precise, this code here is called the extended binary Golay code.  You can learn more about it here:

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

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

Puzzle: I said the symmetry group of this code is M24.  But what do I mean, exactly, by a 'symmetry' of this code?

The extended binary Golay code is not only good for outer space.  In 1993, the US government issued standards for high frequency radio systems.  They require using this code for "forwards error correction" in "automatic link establishment"!  See page 51 here:

http://hflink.com/standards/FED_STD_1045A.pdf___

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2014-11-19 15:42:15 (57 comments, 18 reshares, 77 +1s)Open 

Severe inequality

In 2005, white US households were about 10 times more wealthy than black ones, as measured by median net worth.  By 2011, they were 14 times as wealthy!  

From 2005 to 2011, the Great Recession knocked American white households' median net worth down by 21%.   But for blacks, it dropped by 42%.  

The numbers are from the US Census:

https://www.census.gov/people/wealth/data/disttables.html

Severe inequality

In 2005, white US households were about 10 times more wealthy than black ones, as measured by median net worth.  By 2011, they were 14 times as wealthy!  

From 2005 to 2011, the Great Recession knocked American white households' median net worth down by 21%.   But for blacks, it dropped by 42%.  

The numbers are from the US Census:

https://www.census.gov/people/wealth/data/disttables.html___

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2014-11-18 15:57:33 (24 comments, 8 reshares, 78 +1s)Open 

Lockheed Martin tries nuclear fusion

I'm skeptical about this, but it would be a big deal if it works.  The company Lockheed Martin is famous for building airplanes.  But they have a research group that's working on nuclear fusion, and they're making very bold claims:

A reactor small enough to fit on a truck could provide enough power for a small city of up to 100,000 people.  Building on more than 60 years of fusion research, the Lockheed Martin Skunk Works approach to compact fusion is a high beta concept. This concept uses a high fraction of the magnetic field pressure, or all of its potential, so we can make our devices 10 times smaller than previous concepts. That means we can replace a device that must be housed in a large building with one that can fit on the back of a truck.

They've been publicizing their work on fusion since 2013. But... more »

Lockheed Martin tries nuclear fusion

I'm skeptical about this, but it would be a big deal if it works.  The company Lockheed Martin is famous for building airplanes.  But they have a research group that's working on nuclear fusion, and they're making very bold claims:

A reactor small enough to fit on a truck could provide enough power for a small city of up to 100,000 people.  Building on more than 60 years of fusion research, the Lockheed Martin Skunk Works approach to compact fusion is a high beta concept. This concept uses a high fraction of the magnetic field pressure, or all of its potential, so we can make our devices 10 times smaller than previous concepts. That means we can replace a device that must be housed in a large building with one that can fit on the back of a truck.

They've been publicizing their work on fusion since 2013.  But I haven't seen any serious technical details, and they seem to have a long way to go.   I'm curious why they want the publicity when it's quite likely they will fall flat on their face.  Are they trying to raise money from investors?   Here's something from Aviation Week:

This crucial difference means that for the same size, the CFR generates more power than a tokamak by a factor of 10. This in turn means, for the same power output, the CFR can be 10 times smaller. The change in scale is a game-changer in terms of producibility and cost, explains McGuire. “It’s one of the reasons we think it is feasible for development and future economics,” he says. “Ten times smaller is the key. But on the physics side, it still has to work, and one of the reasons we think our physics will work is that we’ve been able to make an inherently stable configuration.” One of the main reasons for this stability is the positioning of the superconductor coils and shape of the magnetic field lines. “In our case, it is always in balance. So if you have less pressure, the plasma will be smaller and will always sit in this magnetic well,” he notes.

Overall, McGuire says the Lockheed design “takes the good parts of a lot of designs.” It includes the high-beta configuration, the use of magnetic field lines arranged into linear ring “cusps” to confine the plasma and “the engineering simplicity of an axisymmetric mirror,” he says. The “axisymmetric mirror” is created by positioning zones of high magnetic field near each end of the vessel so that they reflect a significant fraction of plasma particles escaping along the axis of the CFR. “We also have a recirculation that is very similar to a Polywell concept,” he adds, referring to another promising avenue of fusion power research. A Polywell fusion reactor uses electromagnets to generate a magnetic field that traps electrons, creating a negative voltage, which then attracts positive ions. The resulting acceleration of the ions toward the negative center results in a collision and fusion.
 
The team acknowledges that the project is in its earliest stages, and many key challenges remain before a viable prototype can be built. However, McGuire expects swift progress. The Skunk Works mind-set and “the pace that people work at here is ridiculously fast,” he says. “We would like to get to a prototype in five generations. If we can meet our plan of doing a design-build-test generation every year, that will put us at about five years, and we’ve already shown we can do that in the lab.” The prototype would demonstrate ignition conditions and the ability to run for upward of 10 sec. in a steady state after the injectors, which will be used to ignite the plasma, are turned off. “So it wouldn’t be at full power, like a working concept reactor, but basically just showing that all the physics works,” McGuire says.

An initial production version could follow five years after that. “That will be a much bigger effort,” he says, suggesting that transition to full-scale manufacturing will necessarily involve materials and heat-transfer specialists as well as gas-turbine makers. The early reactors will be designed to generate around 100 MW and fit into transportable units measuring 23 × 43 ft. “That’s the size we are thinking of now. You could put it on a semi-trailer, similar to a small gas turbine, put it on a pad, hook it up and can be running in a few weeks,” McGuire says. The concept makes use of the existing power infrastructures to enable the CFR to be easily adapted into the current grid. The 100-MW unit would provide sufficient power for up to 80,000 homes in a power-hungry U.S. city and is also “enough to run a ship,” he notes.

For more details, see:

http://aviationweek.com/technology/skunk-works-reveals-compact-fusion-reactor-details

and other magazine articles at the bottom of the website I'm linking to.___

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2014-11-17 17:27:56 (39 comments, 37 reshares, 125 +1s)Open 

A new improved 2nd law of thermodynamics

The 2nd law of thermodynamics says entropy increases.  This is a powerful and useful idea, because you can compute the entropy of different things, like different mixtures of chemicals, and use this idea to see which things can or cannot turn into which other things.  But is the 2nd law always true?  And if so, why?

People have studied this for over a century.  It's clear by now that the 2nd law is only true under certain conditions... which happen to include the conditions we usually see around us.  That's no coincidence.  These are the situations that allow for life as we know it!  

But the laws of physics allow for situations where entropy is very high and occasionally decreases for a short time, then comes back up... and also situations where entropy starts out high and decreases. 

In the secondcase, we... more »

A new improved 2nd law of thermodynamics

The 2nd law of thermodynamics says entropy increases.  This is a powerful and useful idea, because you can compute the entropy of different things, like different mixtures of chemicals, and use this idea to see which things can or cannot turn into which other things.  But is the 2nd law always true?  And if so, why?

People have studied this for over a century.  It's clear by now that the 2nd law is only true under certain conditions... which happen to include the conditions we usually see around us.  That's no coincidence.  These are the situations that allow for life as we know it!  

But the laws of physics allow for situations where entropy is very high and occasionally decreases for a short time, then comes back up... and also situations where entropy starts out high and decreases. 

In the second case, we can just stick a minus sign in our definition of time and save the 2nd law that way: now entropy increases.  The first situation can't be saved that way.  And even in our universe, there will be tiny patches that work like this: if you've got a box of gas in equilibrium, there will be tiny patches where by random fluctuations, entropy decreases for a little while. 

Can we prove theorems saying that under certain precise conditions, the 2nd law of thermodynamics is true?   Yes!   Boltzmann did it a long time ago, and by now there are several theorems like this.   These theorems are limited in their power - they haven't put an end to the mysteries surrounding the 2nd law.  But they're useful because they let us see what's the logic behind the 2nd law, and where are the loopholes. 

Theorems say "if X then Y".   This doesn't mean Y is always true.  It means that if you want Y to be false, you need X to be false too!

One loophole is that the 2nd law only applies to 'closed systems' - systems that don't interact with the rest of the world.  You can lower the entropy of an 'open system' by transferring entropy to the rest of the world.  That's what you're doing whenever you clean your room! That's what the guy in this cartoon should do.

My student +Blake Pollard has a new paper where he generalizes the 2nd law to certain open systems called open Markov processes.   They're like random walks where the walkers can walk in and out of the room.  Very roughly speaking, he shows that for these systems, entropy can only decrease if it flows out of the system.    But he's written a great blog article that explains it more clearly than I just did:

http://johncarlosbaez.wordpress.com/2014/02/16/relative-entropy-part-4/

Check it out!___

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2014-11-16 20:06:09 (44 comments, 26 reshares, 132 +1s)Open 

Annie Oakley

Europeans have trouble understanding the USA's love affair with guns.  Me too.  But here is one of our folk heroes: Annie Oakley.

Born in 1860 in a log cabin in Ohio, at the age of 10 she was 'bound out' to a nearby family to help care for son, on the false promise of fifty cents a week and an education.  Being 'bound out' was pretty common for poor children: at best it was like becoming an apprentice, at worst it was pretty much like slavery.  Annie always called this family "the wolves" - and at the age of 12 she ran away back to her mother.

She'd started hunting at the age of 8 to help support her brothers and sisters, and she got good at shooting.  Really good.

A travelling show called Baughman & Butler came to town.  Butler was a marksman.  He placed a $100 bet that he could beat anylocal ... more »

Annie Oakley

Europeans have trouble understanding the USA's love affair with guns.  Me too.  But here is one of our folk heroes: Annie Oakley.

Born in 1860 in a log cabin in Ohio, at the age of 10 she was 'bound out' to a nearby family to help care for son, on the false promise of fifty cents a week and an education.  Being 'bound out' was pretty common for poor children: at best it was like becoming an apprentice, at worst it was pretty much like slavery.  Annie always called this family "the wolves" - and at the age of 12 she ran away back to her mother.

She'd started hunting at the age of 8 to help support her brothers and sisters, and she got good at shooting.  Really good.

A travelling show called Baughman & Butler came to town.  Butler was a marksman.  He placed a $100 bet that he could beat any local shooter.   The last thing he expected was a five-foot-tall, 15-year old challenger named Annie!  They took turns.  After missing on his 25th shot, Butler lost the match.

A year later, he married Annie.

In 1885, they joined Buffalo Bill Cody's travelling circus.  Buffalo Bill is another one of America's famous shooters.  He began his career by exterminating buffaloes and Indians.  But by this time, he was running a show called 'Buffalo Bill's Wild West'.  It featured notables such as Wild Bill Hickock and Sitting Bull - a chief of the Lakota tribe who had helped defeat Custer in a famous battle.

By this point, Annie Oakley had become almost superhuman.  Her most famous trick was to split a playing card, edge-on, and put several more holes in it before it could touch the ground - all using a rifle at 90 feet. 

According to the Encyclopedia Brittanica:

Oakley never failed to delight her audiences, and her feats of marksmanship were truly incredible. At 30 paces she could split a playing card held edge-on, she hit dimes tossed into the air, she shot cigarettes from her husband's lips, and, a playing card being thrown into the air, she riddled it before it touched the ground.

One day Chief Sitting Bull was watching when

Oakley playfully skipped on stage, lifted her rifle, and aimed the barrel at a burning candle. In one shot, she snuffed out the flame with a whizzing bullet.

Later the show went to Europe.  At his request, she used a bullet to knock the ashes off a cigarette held by Kaiser Wilhelm II.   Later some people later said that if Annie had shot the Kaiser, she could have prevented World War I. 

 In fact, after the war started, Oakley sent a letter to the Kaiser requesting a second shot. The Kaiser did not reply.

And so on.  Teddy Roosevelt turned down her request to lead a company of woman sharpshooters in the Spanish-American war.    She won 54 of 55 libel suits against various newspapers when the publisher William Randolph Hearst spread a false story that she'd been arrested for stealing to support a cocaine habit.  At the age of 62, she won a contest by hitting 100 clay targets in a row from a distance of 16 yards. 

At the age of 66, she died of pernicious anemia.  Her husband Butler was so grieved he stopped eating and died a couple of weeks later.

Apart from Hearst, Annie Oakland and her husband are the only Americans named in this story who never killed anyone with a gun.  Chief Sitting Bull was later shot and killed by the police.

I got interested in Annie Oakley last night while watching an episode of TV's best-kept secret, The Murdoch Mysteries.  You can see it on Netflix.  In this episode, Buffalo Bill's travelling show comes to Toronto, someone gets shot... and Annie Oakley is one of the suspects.   I was wondering how accurate it all was.  It seems pretty realistic, though I don't know if Buffalo Bill's Wild West ever went to Toronto.

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

#history  ___

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2014-11-14 16:55:46 (140 comments, 41 reshares, 146 +1s)Open 

In Alabama, 1/3 of black men cannot vote

Why not?  Because if you go to prison in that state, you may never be allowed to vote after that.  Since black men are imprisoned at an extremely high rate, many can't vote.   

There are 11 states in the US with laws like this. Others deny the vote for shorter periods of time - for example, while you're in prison or are on parole.  Here are some of the consequences:

31% of black men in Florida can't vote.

31% of black men in Alabama can't vote.

29% of black men in Mississippi can't vote.

28% of black men in Wyoming can't vote.

26% of black men in Iowa can't vote.

25% of black men in Virginia can't vote.

24% of black men in New Mexico can't vote.

24% of black men in Washington can't vote.

21% ofblack... more »

In Alabama, 1/3 of black men cannot vote

Why not?  Because if you go to prison in that state, you may never be allowed to vote after that.  Since black men are imprisoned at an extremely high rate, many can't vote.   

There are 11 states in the US with laws like this. Others deny the vote for shorter periods of time - for example, while you're in prison or are on parole.  Here are some of the consequences:

31% of black men in Florida can't vote.

31% of black men in Alabama can't vote.

29% of black men in Mississippi can't vote.

28% of black men in Wyoming can't vote.

26% of black men in Iowa can't vote.

25% of black men in Virginia can't vote.

24% of black men in New Mexico can't vote.

24% of black men in Washington can't vote.

21% of black men in Texas can't vote.

20% of black men in Delaware can't vote.

14% of black men in Tennessee can't vote.

12% of black men in Arizona can't vote.

10% of black men in Nevada can't vote.

These statistics are old, so they will have changed.  But I don't think they've gotten better - except that in the last 20 years, three states stopped doing this. 

An estimated 5.3 million Americans are denied the right to vote based on their felony convictions, 4 million of whom are out of prison. About a third of them are black, including 13% of all African-American men.

For up-to-date information, go to the Sentencing Project website:

http://www.sentencingproject.org/detail/news.cfm?news_id=1877

They list percentages of African American citizens who are denied the right to vote, not African American men, so the figures look slightly less shocking.  More details are in this 2010 report:

http://sentencingproject.org/doc/publications/fd_State_Level_Estimates_of_Felon_Disen_2010.pdf

but for African American men the most up-to-date comprehensive statistics I've found are from 1996:

http://www.hrw.org/legacy/reports/reports98/vote/usvot98o-01.htm

so that's what I used in my list.___

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2014-11-14 16:25:49 (7 comments, 11 reshares, 63 +1s)Open 

One of humanity's deepest thinkers died yesterday. RIP Alexander Grothendieck.

In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. “You mean an actual number?” Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, “All right, take 57.”

That's from this article by Allyn Jackson about the man, his maths, and life:
http://www.ams.org/notices/200410/fea-grothendieck-part2.pdf

Here's the news of the death (French):
http://www.liberation.fr/sciences/2014/11/13/alexandre-grothendieck-ou-la-mort-d-un-genie-qui-voulait-se-faire-oublier_1142614

One of humanity's deepest thinkers died yesterday. RIP Alexander Grothendieck.

In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. “You mean an actual number?” Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, “All right, take 57.”

That's from this article by Allyn Jackson about the man, his maths, and life:
http://www.ams.org/notices/200410/fea-grothendieck-part2.pdf

Here's the news of the death (French):
http://www.liberation.fr/sciences/2014/11/13/alexandre-grothendieck-ou-la-mort-d-un-genie-qui-voulait-se-faire-oublier_1142614___

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2014-11-13 16:38:39 (38 comments, 5 reshares, 54 +1s)Open 

Tiger on the loose - near Paris!

A tiger has been seen in the town of Montévrain just outside Paris!  The weird part: nobody knows how it got there.

The wife of a supermarket owner in Montévrain was the first to see the tiger from the supermarket parking lot at 8:30 am local time today. She called her husband to say she thought she had seen a lynx, and took a photograph, which the couple then showed to municipal authorities.  Local and national police together with firefighters, immediately launched a search with helicopters.  Their goal was to capture it without hurting it, with a dart gun.

A young tiger was spotted two hours later, with the latest reports saying that it's hiding in some bushes behind some tennis courts. "We urge you to be careful and not to walk within the perimeter, which has been secured since this morning," Montévrainautho... more »

Tiger on the loose - near Paris!

A tiger has been seen in the town of Montévrain just outside Paris!  The weird part: nobody knows how it got there.

The wife of a supermarket owner in Montévrain was the first to see the tiger from the supermarket parking lot at 8:30 am local time today. She called her husband to say she thought she had seen a lynx, and took a photograph, which the couple then showed to municipal authorities.  Local and national police together with firefighters, immediately launched a search with helicopters.  Their goal was to capture it without hurting it, with a dart gun.

A young tiger was spotted two hours later, with the latest reports saying that it's hiding in some bushes behind some tennis courts. "We urge you to be careful and not to walk within the perimeter, which has been secured since this morning," Montévrain authorities wrote on their Facebook page.

It’s not yet clear where the animal came from!  The Mayor’s Office said they ruled out the theory that it came from a circus that was based in Montévrain until last Saturday. Police said that no tiger was present during the circus health inspection.

Let me know when they find it!  More details here, in French:

http://www.leparisien.fr/seine-et-marne-77/montevrain-un-tigre-en-liberte-retrouve-13-11-2014-4288567.php___

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2014-11-11 16:11:54 (19 comments, 26 reshares, 118 +1s)Open 

Arabesque

+Craig Kaplan has been taking ideas from Islamic wall tilings and adapting them to spheres.  It's a great way to bring new life to an glorious old tradition.  

See that star with 10 points and 5 nearest neighbors?  That's 5-fold symmetry.  You can't get perfect 5-fold symmetry in a tiling of the plane.  The best you can do is fake it in various ways - and by 1200 AD the great tile masters of Afghanistan, Iran, Morocco and Turkey had figured out most of these ways. 

Patterns with decagons and pentagons that fool the eye into thinking there's 5-fold symmetry!   Quasiperiodic tilings - later rediscovered by Penrose - that never quite repeat but have 5-fold symmetry on average.  Their discoveries were remarkable.

But when you tile a sphere, the dodecahedron comes to your aid: it has 5-fold symmetry, and things the old tileexperts ... more »

Arabesque

+Craig Kaplan has been taking ideas from Islamic wall tilings and adapting them to spheres.  It's a great way to bring new life to an glorious old tradition.  

See that star with 10 points and 5 nearest neighbors?  That's 5-fold symmetry.  You can't get perfect 5-fold symmetry in a tiling of the plane.  The best you can do is fake it in various ways - and by 1200 AD the great tile masters of Afghanistan, Iran, Morocco and Turkey had figured out most of these ways. 

Patterns with decagons and pentagons that fool the eye into thinking there's 5-fold symmetry!   Quasiperiodic tilings - later rediscovered by Penrose - that never quite repeat but have 5-fold symmetry on average.  Their discoveries were remarkable.

But when you tile a sphere, the dodecahedron comes to your aid: it has 5-fold symmetry, and things the old tile experts did with hexagons, you can now do with pentagons!  It's like a whole new world.

And the world expands even more when you use the hyperbolic plane: then you can get 7-fold symmetry, 8-fold symmetry and so on.  Kaplan has also studied that.

If you look carefully at this pattern, you'll see every star with 10 points is surrounded by 5 stars with 9 points... and every star with 9 points is surrounded by 6 stars, which alternate between having 9 and 10 points.  The stars with 10 points are the centers of the faces of a dodecahedron, so there are 12 of them.  The stars with 9 points are at the vertices of this dodecahedron, so there are 20 of them. 

The whole pattern is made of little things that look almost like triangles, but have bent edges.

Puzzle: how many of these little things are there?

I thank +Layra Idarani for making me pay attention to these details.

This image was created by TaffGoch based on a design by Craig Kaplan.   For more beautiful stuff, check out this page:

• Craig S. Kaplan, Computer graphics and geometric ornamental design, http://www.cgl.uwaterloo.ca/~csk/phd/

TaffGoch has a lot of great stuff here:

http://taffgoch.deviantart.com/

#geometry  

 ___

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2014-11-10 22:11:22 (20 comments, 37 reshares, 98 +1s)Open 

France prefers to pay twice for its researchers' work

Instead of giving everyone access to the work of its scientists - work it has financed - France prefers to pay 172 million euros to a Dutch publisher.

That's a headline in the newspaper Rue89.  In their latest round of negotiaitons, after threatening to quit taking Elsevier journals, the French Ministry of Research caved in and agreed on a secret contract with this publisher.  But now Rue89 has published a copy of this contract!   Here it is in French:

http://rue89.nouvelobs.com/sites/news/files/assets/document/2014/11/marche_elsevier.pdf

Here it is in English:

http://tinyurl.com/qjps3mv

The whole article is below, or here in English:

http://tinyurl.com/qfxtwz2

These English translations are pretty crude, made by Google Translate.  The Frenchmath... more »

France prefers to pay twice for its researchers' work

Instead of giving everyone access to the work of its scientists - work it has financed - France prefers to pay 172 million euros to a Dutch publisher.

That's a headline in the newspaper Rue89.  In their latest round of negotiaitons, after threatening to quit taking Elsevier journals, the French Ministry of Research caved in and agreed on a secret contract with this publisher.  But now Rue89 has published a copy of this contract!   Here it is in French:

http://rue89.nouvelobs.com/sites/news/files/assets/document/2014/11/marche_elsevier.pdf

Here it is in English:

http://tinyurl.com/qjps3mv

The whole article is below, or here in English:

http://tinyurl.com/qfxtwz2

These English translations are pretty crude, made by Google Translate.  The French mathematician Marie Farge, director of the science institute CNRS, writes:

It would be nice to have an English translation of both the paper and the contract. Could you please forward this news since the paper and the contract might not be accessible for long time?

This will certainly bring more transparency into the system and hopefully more consciousness among our colleagues. I am looking for lively discussions, thanks to this courageous paper. All the best!

Compare the situation in the Netherlands:

Negotiations between the Dutch universities and publishing company Elsevier on subscription fees and Open Access have ground to a halt. In line with the policy pursued by the Ministry of Education, Culture and Science, the universities want academic publications to be freely accessible. To that end, agreements will have to be made with the publishers. The proposal presented by Elsevier last week totally fails to address this inevitable change. The universities hope that Elsevier will submit an amended proposal. ‘From now on we will inform our researchers about the consequences of this deadlock’, says Gerard Meijer, president of Radboud University Nijmegen and chief negotiator on behalf of the VSNU.

This is a quote from here:

http://vsnu.nl/news/newsitem/11-negotiations-between-elsevier-and-universities-failed.html___

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2014-11-08 17:59:44 (44 comments, 37 reshares, 205 +1s)Open 

Cosmological billiards and spacetime crystals

In 1970, Belinksii, Khalatnikov and Lifschitz discovered that when you run time backwards toward the Big Bang, a homogeneous universe behaves like a billiard ball.  As you run time back, the universe shrinks, but also its shape changes.  Its shape moves around in some region of allowed shapes... and it 'bounces' off the 'walls' of this region!

These guys considered the simplest case: a universe with 3 dimensions of space and 1 dimension of time, containing gravity but nothing else.   In this case the region of allowed shapes is a triangle in the hyperbolic plane.  I showed it to you last time. 

So, running time backwards in this kind of universe is mathematically very much like watching a frictionless billiard ball bounce around on a strangely curved triangular pool table.

But you can playthe s... more »

Cosmological billiards and spacetime crystals

In 1970, Belinksii, Khalatnikov and Lifschitz discovered that when you run time backwards toward the Big Bang, a homogeneous universe behaves like a billiard ball.  As you run time back, the universe shrinks, but also its shape changes.  Its shape moves around in some region of allowed shapes... and it 'bounces' off the 'walls' of this region!

These guys considered the simplest case: a universe with 3 dimensions of space and 1 dimension of time, containing gravity but nothing else.   In this case the region of allowed shapes is a triangle in the hyperbolic plane.  I showed it to you last time. 

So, running time backwards in this kind of universe is mathematically very much like watching a frictionless billiard ball bounce around on a strangely curved triangular pool table.

But you can play the same game for other theories: gravity together with various kinds of matter, in universes with various numbers of dimensions.  And when people did this, they discovered something really cool.   Different possibilities gave different kinds of pool tables!

When space has some number of dimensions, the pool table has dimension one less.   As far as I know, it's always sitting inside 'hyperbolic space', a generalization of the hyperbolic plane.  And it's always a piece of a hyperbolic honeycomb - a very symmetrical way of chopping hyperbolic space into pieces.  

The picture here, drawn by +Roice Nelson, shows a hyperbolic honeycomb in 3-dimensional hyperbolic space.   So, one tetrahedron in this honeycomb could be the 'pool table' for a theory of gravity where space has 4 dimensions.  (In fact it doesn't quite work like this: we have to subdivide each tetrahedron shown here into 24 smaller tetrahedra to get the 'pool tables'.  But never mind.)

Even better, these stunningly symmetrical patterns arise from what I called spacetime crystals.   The technical term is 'hyperbolic Dynkin diagrams', and I told you about them earlier.   The picture here, in 3 dimensions, arises from a spacetime crystal in 4 dimensions.  That's how it always works: the crystal has one more dimension than the pool table.

And here's the really amazing thing: mathematicians have proved that the highest possible dimension for a spacetime crystal is 10.   This gives you a 9-dimensional pool table, which is the sort of thing that could show up in a theory of gravity where space has 10 dimensions.

And there is a theory of gravity in where space has 10 dimensions:  it's called 11-dimensional supergravity, because there's also 1 dimension of time in this theory.   String theorists like this theory of gravity a lot, because it seems to connect all the other stuff they're interested in. 

It turns out this particular theory of gravity gives a spacetime crystal called E10.  There are several other 10-dimensional spacetime crystals, but this is the best.

For a while I've been thinking that we should be able to describe E10 using the octonions, an 8-dimensional number system that shows up a lot in string theory.  I had a guess about how this should work.   And last week, my friend the science fiction writer Greg Egan proved this guess is right!

For the details, go here:

https://golem.ph.utexas.edu/category/2014/11/integral_octonions_part_7.html

This result probably came as no surprise to the real experts on cosmological billiards - I'm no expert, I just play a game now and then.   Here is a nice introduction by a real expert:

• Thibault Damour, Poincaré, relativity, billiards and symmetry, http://arxiv.org/abs/hep-th/0501168.

And here are some more detailed papers:

• Thibault Damour, Sophie de Buyl, Marc Henneaux and Christiane Schomblond, Einstein billiards and overextensions of finite-dimensional simple Lie algebras, http://arxiv.org/abs/hep-th/0206125.

• Axel Kleinschmidt, Hermann Nicolai, Jakob Palmkvist, Hyperbolic Weyl groups and the four normed division algebras, http://arxiv.org/abs/0805.3018.

#spnetwork arXiv:0805.3018 arXiv:hep-th/0206125 arXiv:hep-th/0501168 #gravity #geometry  ___

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2014-11-08 04:08:44 (14 comments, 11 reshares, 64 +1s)Open 

Man was winged hopefully. He had in him to go further than this short flight, now ending. He proposed even that he should become the Flower of All Things, and that he should learn to be the All-Knowing, the All-Admiring. Instead, he is to be destroyed. He is only a fledgling caught in a bush-fire. He is very small, very simple, very little capable of insight. His knowledge of the great orb of things is but a fledgling's knowledge. His admiration is a nestling's admiration for the things kindly to his own small nature. He delights only in food and the food-announcing call. The music of the spheres passes over him, through him, and is not heard.

Yet it has used him. And now it uses his destruction. Great, and terrible, and very beautiful is the Whole; and for man the best is that the Whole should use him.

But does it really use him? Is the beauty of the Whole... more »

Man was winged hopefully. He had in him to go further than this short flight, now ending. He proposed even that he should become the Flower of All Things, and that he should learn to be the All-Knowing, the All-Admiring. Instead, he is to be destroyed. He is only a fledgling caught in a bush-fire. He is very small, very simple, very little capable of insight. His knowledge of the great orb of things is but a fledgling's knowledge. His admiration is a nestling's admiration for the things kindly to his own small nature. He delights only in food and the food-announcing call. The music of the spheres passes over him, through him, and is not heard.

Yet it has used him. And now it uses his destruction. Great, and terrible, and very beautiful is the Whole; and for man the best is that the Whole should use him.

But does it really use him? Is the beauty of the Whole really enhanced by our agony? And is the Whole really beautiful? And what is beauty? Throughout all his existence man has been striving to hear the music of the spheres, and has seemed to himself once and again to catch some phrase of it, or even a hint of the whole form of it. Yet he can never be sure that he has truly heard it, nor even that there is any such perfect music at all to be heard. Inevitably so, for if it exists, it is not for him in his littleness.

But one thing is certain. Man himself, at the very least, is music, a brave theme that makes music also of its vast accompaniment, its matrix of storms and stars. Man himself in his degree is eternally a beauty in the eternal form of things. It is very good to have been man. And so we may go forward together with laughter in our hearts, and peace, thankful for the past, and for our own courage. For we shall make after all a fair conclusion to this brief music that is man.

This is the end of Olaf Stapledon's Last and First Men.   An early SF classic, it describes the history of humanity for the next two billion years, embodied 18 different species and living on several planets.   It was written in 1930, so try to forgive the sexist language and various kinds of naïveté.  You can read the whole book here:

http://gutenberg.net.au/ebooks06/0601101h.html___

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2014-11-06 16:31:17 (45 comments, 31 reshares, 118 +1s)Open 

Cosmological billiards

What happened before the Big Bang?  That's what I call a boring question.  

It might not make sense.  But don't believe anyone who confidently asserts that it doesn't make sense.  It might make sense.  We have no idea!  We just don't know enough about physics to make much progress on this question right now!  Maybe later.

What happened right after the Big Bang?  That's much more interesting, because we don't know the complete answer, but we know a lot of stuff, and we have at least a chance of making progress.

Here's something easy you can do: take a solution of Einstein's equation for gravity, run it back in time, and see what it says about the shape of the universe as you get closer and closer to the Big Bang. 

You might not think this is easy if you haven't taken a course onthis stuf... more »

Cosmological billiards

What happened before the Big Bang?  That's what I call a boring question.  

It might not make sense.  But don't believe anyone who confidently asserts that it doesn't make sense.  It might make sense.  We have no idea!  We just don't know enough about physics to make much progress on this question right now!  Maybe later.

What happened right after the Big Bang?  That's much more interesting, because we don't know the complete answer, but we know a lot of stuff, and we have at least a chance of making progress.

Here's something easy you can do: take a solution of Einstein's equation for gravity, run it back in time, and see what it says about the shape of the universe as you get closer and closer to the Big Bang. 

You might not think this is easy if you haven't taken a course on this stuff.   But it's really easy compared to, say, building a telescope and sending it into orbit.  You can do it with just a pencil and paper.  So you might as well try it and see what you get.

 In the simplest solutions, space is homogeneous and isotropic: for example completely flat, or completely round.   Then it stays that way as you go back in time.  That's what you usually read about in a basic course on this stuff.

But in some more interesting solutions, space is homogeneous but not isotropic.  That means it looks the same at every location, but not the same in every direction. 

In 1970, three Russian physicists named Belinskii, Khalatnikov and Lifshitz took these solutions, ran them back in time, and noticed something interesting.  The universe oscillated in shape ever more wildly as time went back towards the Big Bang!   And sometimes - depending on the particular solution - it would do so in a chaotic way.

Even better, they noticed that this problem was isomorphic to the problem of a ball rolling around in a 2-dimensional region. 

"Isomorphic" means that the math works the same way after you change the names of things.  For example, here, instead of working with time, you need to use minus the logarithm of time.  As time goes to zero (back to the Big Bang), minus the logarithm of time keeps increasing forever.   From this viewpoint there's time for a huge amount of happen as we get closer and closer to the Big Bang, but never quite get there!

And in these homogeneous but not isotropic solutions of the equations for gravity, as we get closer and closer to the Big Bang, the math works more and more like a billiard ball bouncing around in one of the triangles in this picture!

This picture shows the hyperbolic plane chopped up into triangles in a very symmetrical way.   Pick any triangle; then a point in that triangle describes a possible shape of the universe in the solutions that Belinskii, Khalatnikov and Lifshitz were studying.

Of course their work is oversimplified, because it left out all the forces besides gravity, it ignored quantum mechanics, and it assumed the universe was homogeneous.  So, don't take it too seriously!   But still, it pointed out a new possibility: the universe could wiggle around more and more wildly as we run time back toward the Big Bang.

Even more importantly,  from my perspective, it led to a huge amount of cool math connecting the equations of gravity to symmetrical ways of chopping up the hyperbolic plane into triangles, and higher-dimensional versions of that game.  And that's what I really wanted to talk about today, but I see this post is getting too long, so I'll stop for now.

This is their paper:

• V. A. Belinskii, I. M. Khalatnikov and E. M. Lifshitz, Oscillatory approach to a singular point in the relativistic cosmology, Adv. Phys. 19 (1970), 525.

Lifschitz, by the way, is one member of the famous physics textbook writing team Landau and Lifschitz.  You can also learn more about the Belinksii-Khalatnikov-Lifschitz singularity here:

• Wikipedia, BKL singularity, https://en.wikipedia.org/wiki/BKL_singularity

#spnetwork doi:10.1080/00018737000101171 #gravity #geometry___

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2014-11-03 18:18:00 (169 comments, 16 reshares, 151 +1s)Open 

Suspiciously cute

Last week many of my old posts started getting +1ed by beautiful women.

One theory is that at finally, at the age of 53 and happily married, I've become an irresistible babe magnet.  Maybe it's my skill at explaining advanced concepts in math and science.  Maybe it's my devilish good looks.

It's time to look for another theory.

Obviously, the other theory is that these are entities posing as beautiful women.  I wasn't born yesterday!   This type of thing has been happening for a long time on G+. 

But here's the interesting part.   Each old post is getting +1ed just once - each by a different entity.  Each of these entities does not merely post ads for some company, the way a typical fake does.  They post about a variety of things - but with a lot of photos of themselves riding horses, dressed upin Hall... more »

Suspiciously cute

Last week many of my old posts started getting +1ed by beautiful women.

One theory is that at finally, at the age of 53 and happily married, I've become an irresistible babe magnet.  Maybe it's my skill at explaining advanced concepts in math and science.  Maybe it's my devilish good looks.

It's time to look for another theory.

Obviously, the other theory is that these are entities posing as beautiful women.  I wasn't born yesterday!   This type of thing has been happening for a long time on G+. 

But here's the interesting part.   Each old post is getting +1ed just once - each by a different entity.  Each of these entities does not merely post ads for some company, the way a typical fake does.  They post about a variety of things - but with a lot of photos of themselves riding horses, dressed up in Halloween costumes, hanging out with friends, etc.. This suggests a fairly elaborate scheme.

Check them out:

+***, shown here, +1ed a post about Greg Egan's work on catacaustics.   She is "single" and "looking for a relationship".  

+*** +1ed a post about dance and gamelan in the town of Ubud.

+*** +1ed a post on how Shannon entropy is just part of a 1-parameter family of entropy concepts.

+*** +1ed a post on Islamic tile patterns.

+*** +1ed a post on the geometry of the hydronium ion.

And so on.

While writing this post, I found what all these entities' posts have in common - the thing that probably explains what is going on.

Puzzle: what is it?

EDIT: alas, their accounts have now been deleted, so you can't check them out anymore.___

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2014-11-03 16:13:05 (70 comments, 41 reshares, 101 +1s)Open 

The action-perception loop

A lot is happening when you walk down the street!  You're constantly processing new information from your senses and stored information from the past and using this to decide on your motions, which then affects your position and the environment around you.   It's a feedback loop: if you lean a bit too far to the left your muscles will automatically push you back to the right.  You don't need to consciously think about it very much.   It's called the action-perception loop.

Natural selection has pushed organisms to be very good at this loop.  Try catching a cockroach and you'll see what I mean!  

Among other things, we need to balance the cost of storing information about the past against the payoff of achieving our desired goals in the future.   

If information processing were free, we couldremember e... more »

The action-perception loop

A lot is happening when you walk down the street!  You're constantly processing new information from your senses and stored information from the past and using this to decide on your motions, which then affects your position and the environment around you.   It's a feedback loop: if you lean a bit too far to the left your muscles will automatically push you back to the right.  You don't need to consciously think about it very much.   It's called the action-perception loop.

Natural selection has pushed organisms to be very good at this loop.  Try catching a cockroach and you'll see what I mean!  

Among other things, we need to balance the cost of storing information about the past against the payoff of achieving our desired goals in the future.   

If information processing were free, we could remember everything and use it to make the best possible decision given all this information.  But it's not free.  Your brain uses about 40% of all your metabolic energy!  

So, we've got a fascinating problem here: optimizing decision-making when there's a cost for using information.  

Information is  something we can measure, thanks to the work of Claude Shannon.  So biologists and information theorists are starting to tackle this problem using math.  I'm learning about it here:

• Naftali Tishby and Daniel Polani, Information theory of decisions and actions, http://www.cs.huji.ac.il/labs/learning/Papers/IT-PAC.pdf

It combines lots of cool ideas.  They formulate this problem as something like a partially observed Markov decision process, use rate-distortion theory to analyze how much information needs to be stored to achieve a given reward, include that into the costs, and then use Bellman’s equation to solve the resulting optimization problem!

That's quite a mouthful.  I explain all the jargon here:

http://johncarlosbaez.wordpress.com/2014/10/30/sensing-and-acting-under-information-constraints/

As time passes I hope to write more about this.  I think these ideas will combine with my thoughts on network theory in a really cool way.   You can see a tiny network in the picture here... but the brain is a much more complicated network.

The picture here is from the University of Bielefeld’s Department of Cognitive Neuroscience:

http://www.uni-bielefeld.de/biologie/cns/

#informationtheory  ___

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2014-10-30 14:27:47 (28 comments, 2 reshares, 68 +1s)Open 

Shy bird

This puffin looks afraid to come out of its burrow.  It's probably perfectly happy.  But the lines around its eye make it look like a cartoon of nervousness.

Atlantic Puffins spend most of their time at sea - they're good at  swimming, using their wings to ‘fly’ underwater as they search for fish to eat.  They come to land each spring to breed in colonies on north Atlantic seacoasts and rocky islands.   They live in burrows - I just learned this yesterday!  Each pair lays one egg, with the male and female sharing incubation duties for about 40 days. After the chick hatches both parents feed it fish for approximately 45 days. After that the young puffling is ready to leave the nest.

You can see life in a puffin burrow here:

http://explore.org/live-cams/player/puffin-burrow-cam

I got my info from thatwebsite.  ... more »

Shy bird

This puffin looks afraid to come out of its burrow.  It's probably perfectly happy.  But the lines around its eye make it look like a cartoon of nervousness.

Atlantic Puffins spend most of their time at sea - they're good at  swimming, using their wings to ‘fly’ underwater as they search for fish to eat.  They come to land each spring to breed in colonies on north Atlantic seacoasts and rocky islands.   They live in burrows - I just learned this yesterday!  Each pair lays one egg, with the male and female sharing incubation duties for about 40 days. After the chick hatches both parents feed it fish for approximately 45 days. After that the young puffling is ready to leave the nest.

You can see life in a puffin burrow here:

http://explore.org/live-cams/player/puffin-burrow-cam

I got my info from that website.  And the picture here, taken on Skomer Island off southwestern Wales, was posted to Flickr by someone who goes by the name 'Clear Inner Vision':

https://www.flickr.com/photos/n031/7316927966/___

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2014-10-29 18:11:18 (63 comments, 89 reshares, 226 +1s)Open 

I'm listening to a talk on the origin of life at a workshop on Biological and Bio-Inspired Information Theory.  The speaker said something like this... and I was amazed, again, at how wonderful living organisms are.

You can see videos of the talks here:

http://www.birs.ca/videos/2014

I gave a talk on "Biodiversity, entropy and thermodynamics":

http://www.birs.ca/events/2014/5-day-workshops/14w5170/videos/watch/201410291038-Baez.mp4

but what really blew my mind was Naftali Tishby's talk on "Sensing and acting under information constraints - a principled approach to biology and intelligence":

http://www.birs.ca/events/2014/5-day-workshops/14w5170/videos/watch/201410281032-Tishby.mp4

It wasn't easy for me to follow - you should already know rate-distortion theory and the Bellman equation, and I... more »

I'm listening to a talk on the origin of life at a workshop on Biological and Bio-Inspired Information Theory.  The speaker said something like this... and I was amazed, again, at how wonderful living organisms are.

You can see videos of the talks here:

http://www.birs.ca/videos/2014

I gave a talk on "Biodiversity, entropy and thermodynamics":

http://www.birs.ca/events/2014/5-day-workshops/14w5170/videos/watch/201410291038-Baez.mp4

but what really blew my mind was Naftali Tishby's talk on "Sensing and acting under information constraints - a principled approach to biology and intelligence":

http://www.birs.ca/events/2014/5-day-workshops/14w5170/videos/watch/201410281032-Tishby.mp4

It wasn't easy for me to follow - you should already know rate-distortion theory and the Bellman equation, and I didn't - but it's great!  It's all about how living organisms balance the cost of storing information about the past against the payoff of achieving their desired goals in the future.  It's not fluff: it's a detailed mathematical model!  And it ends by testing the model on experiments with cats listening to music and rats swimming to land.

Here's a good paper about this stuff:

• Naftali Tishby and Daniel Polani, Information theory of decisions and actions, http://www.cs.huji.ac.il/labs/learning/Papers/IT-PAC.pdf

A conversation with Susanne Still convinced me even more that this is stuff I need to learn!  I hope to blog about it as I understand more.

In case you're wondering, rate-distortion theory is the branch of information theory that helps you find the minimum number of bits per second that must be communicated over a noisy channel so that the signal can be approximately reconstructed at the other end without exceeding a given distortion:

 https://en.wikipedia.org/wiki/Rate-distortion_theory

The Bellman equation lets you find an optimal course of action by optimizing what you do at each step:

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

#spnetwork doi:10.1007/978-1-4419-1452-1_19 #informationTheory #controlTheory #biology  ___

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2014-10-28 20:31:04 (32 comments, 35 reshares, 132 +1s)Open 

The mitochondrial genetic code

Your cells contain mitochondria, little factories that help convert food into useful chemical energy, using oxygen.   These guys were once bacteria in their own right!   About 1.5 billion years ago they joined forces with cells that couldn't handle oxygen.  Now you need them and they need you.

But they still have their own DNA, separate from the rest of the DNA in your cells.  Mitochondria are passed down only from mother to child, via the egg cell.  So, your mitochondrial DNA gives information about you, your mother, your grandmother, and so on.

Why does mitochondrial DNA come only from the mother?  For starters, an egg cell contains 100,000 to 1,000,000 molecules of mitochondrial DNA, while a sperm has only 100 to 1000.   On top of that, most mitochondria in a sperm cell stay in the tail, and sometimes thetail is ... more »

The mitochondrial genetic code

Your cells contain mitochondria, little factories that help convert food into useful chemical energy, using oxygen.   These guys were once bacteria in their own right!   About 1.5 billion years ago they joined forces with cells that couldn't handle oxygen.  Now you need them and they need you.

But they still have their own DNA, separate from the rest of the DNA in your cells.  Mitochondria are passed down only from mother to child, via the egg cell.  So, your mitochondrial DNA gives information about you, your mother, your grandmother, and so on.

Why does mitochondrial DNA come only from the mother?  For starters, an egg cell contains 100,000 to 1,000,000 molecules of mitochondrial DNA, while a sperm has only 100 to 1000.   On top of that, most mitochondria in a sperm cell stay in the tail, and sometimes the tail is lost during fertilization.  But on top of that, in mammals it seems the egg actively destroys any mitochondria that happen to get in from the sperm.

Puzzle 1: Why?

But in biology everything is complicated.  Biologists argue about how likely it is for people to inherit mitochondrial DNA from their father.   In a test of 172 sheep, three were found to inherit mitochondrial DNA from their father!   But in humans, there is so far just one recorded case of it happening.

Your mitochondrial DNA has just 37 genes.  It's made of about 16,600 base pairs: molecules called A, T, C and G, just like your ordinary DNA.  The information gets copied to RNA when the genes are used to make proteins, and the T gets copied to U, while the rest stay the same.

This chart shows the mitochondrion's genetic code.  More precisely: each codon, or triple of base pairs U, G, A, and C, is translated into an amino acid. Amino acids, the building blocks of proteins, have fun names like phenylalanine, leucine, isoleucine and so on - but in this chart they're abbreviated to Phe, Leu, Ile, etc.

The mitochondrial genetic code is a bit different than the genetic code used elsewhere in your cells!  The differences are marked in red:

• AUA codes for Met instead of Ile as it does elsewhere. 

• UGA codes for Trp rather than being a stop codon (meaning, a codon that says the protein is done).

• AGA and AGG are stop codons instead of coding for Arg.

The same code is used for mitochondria in all vertebrates, as far as I know.   Some invertebrates have slightly different mitochondrial genetic codes.

Puzzle 2: Why is the mitochondrial genetic code different in the above ways? 

This is an extremely hard puzzle, and I doubt anyone knows the answer for sure, since it could simply be due to random events that happened billions of years ago.  But I bet people have thought about it, and I'd love to know any good ideas they've had.

Here's a clue: when the mitochondrial genetic code differs from the 'usual' one, it tends to be simpler!   All the mitochondrial genetic code consists of blocks of 2 or 4 codons that do the same thing.  Most of the usual code is this way - but AUA and UGA break that rule.

For more, try:

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

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

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

#genetics  ___

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2014-10-26 14:51:07 (11 comments, 101 reshares, 241 +1s)Open 

Looping Lorentzian lattice

A rapidly moving observer will see time (the vertical axis) and space (the horizontal axis) in a different way than you do at rest.  As their speed increases  the warping increases. 

Each black dot is a point in spacetime.  As viewed by faster and faster observers, it moves along a hyperbola.  But after a while, the whole lattice of black dots gets back to the same pattern it started with!

The warpings of spacetime shown here are called Lorentz transformations.  Greg Egan made this movie to illustrate how we can do a Lorentz transformation to a lattice in spacetime and get back the same lattice.  This is the one of the symmetries that you get in what I was calling a 'spacetime crystal' - technically, a lattice coming from a hyperbolic Dynkin diagram.

For many beautiful pictures related to looping Lorentzianlattic... more »

Looping Lorentzian lattice

A rapidly moving observer will see time (the vertical axis) and space (the horizontal axis) in a different way than you do at rest.  As their speed increases  the warping increases. 

Each black dot is a point in spacetime.  As viewed by faster and faster observers, it moves along a hyperbola.  But after a while, the whole lattice of black dots gets back to the same pattern it started with!

The warpings of spacetime shown here are called Lorentz transformations.  Greg Egan made this movie to illustrate how we can do a Lorentz transformation to a lattice in spacetime and get back the same lattice.  This is the one of the symmetries that you get in what I was calling a 'spacetime crystal' - technically, a lattice coming from a hyperbolic Dynkin diagram.

For many beautiful pictures related to looping Lorentzian lattices, try:

• Jos Leys, Lorenz and modular flows: a visual introduction, http://www.josleys.com/articles/ams_article/Lorenz3.htm

The set of all Lorentzian lattices where each parallelogram has area 1 forms a 3d space with a trefoil knot removed!   As we keep applying Lorentz transforms to a lattice, it traces out a curve in this space.

For more on spacetime crystals, see my earlier post:

https://plus.google.com/u/0/117663015413546257905/posts/6MEi2TAvPaS

#geometry  ___

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2014-10-24 16:33:55 (56 comments, 12 reshares, 88 +1s)Open 

Toward a post-carbon world

Big news!   European leaders have signed a deal obliging the EU to cut greenhouse gases by at least 40% by 2030, relative to the 1990 level of emissions.  They'd already agreed to cut them 20% by 2020, and they're close to meeting that goal already. 

The president of the EU Council said:

"It was not easy, not at all, but we managed to reach a fair decision. It sets Europe on an ambitious yet cost effective climate and energy path. Climate change is one of the biggest challenges of mankind. Ultimately this is about survival."

This is great.  And there's some self-interest here, too.  The EU plans to make money in the long term by getting ahead of other countries on implementing technologies that reduce carbon emissions.  If they lead by example, it increases the chance that others will follow.
more »

Toward a post-carbon world

Big news!   European leaders have signed a deal obliging the EU to cut greenhouse gases by at least 40% by 2030, relative to the 1990 level of emissions.  They'd already agreed to cut them 20% by 2020, and they're close to meeting that goal already. 

The president of the EU Council said:

"It was not easy, not at all, but we managed to reach a fair decision. It sets Europe on an ambitious yet cost effective climate and energy path. Climate change is one of the biggest challenges of mankind. Ultimately this is about survival."

This is great.  And there's some self-interest here, too.  The EU plans to make money in the long term by getting ahead of other countries on implementing technologies that reduce carbon emissions.  If they lead by example, it increases the chance that others will follow.

As well as the greenhouse gas pact, two 27% targets were agreed - for renewable energy market share and increase in energy efficiency. But the renewable energy agreement is binding only on the EU as a whole, not on particular nations, raising tricky issues. And the energy efficiency target is optional.

The reason?  Its seems some of the poorer nations raised objections: Portugal, Poland, and... umm... Britain?

Apparently the British Prime Minister David Cameron is "keen to minimise any perceived loss of UK sovereignty over energy policy." He tried to cut the energy efficiency target to 25%, but he accepted 27% as long as it was not binding on Britain.  So, now it's not binding on any nation, though somehow it's supposedly binding on the EU as a whole.

Poland relies a lot on coal, and it managed to get concessions that will allow it to get hundreds of millions of euros to modernise its coal-fired power plants. Of eight EU nations eligible for these subsidies, Poland will get 60% of the total up until 2019.

“It’s scandalous,” said Julia Michalak, a spokeswoman for Climate Action Network Europe. “A continuation of free emission permits for Poland’s coal-reliant energy system would be a grave mistake. Leaders who came to Brussels to agree new historic climate goals, are actually discussing whether to hand out money to Europe’s dirtiest power plants.”

But still: we're seeing progress. 

I got a lot of this material from here:

http://www.theguardian.com/world/2014/oct/24/eu-leaders-agree-to-cut-greenhouse-gas-emissions-by-40-by-2030

#carbonfootprint  ___

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2014-10-23 16:08:43 (60 comments, 55 reshares, 145 +1s)Open 

Spacetime crystals

You know about crystals in space.  What's a crystal in spacetime?    It's a repetitive pattern that has a lot of symmetries including reflections, translations, rotations and Lorentz transformations.   Rotations mix up directions in space.  Lorentz transformations mix up space and time directions.

We can study spacetime crystals mathematically - and the nicest ones are described by gadgets called hyperbolic Dynkin diagrams, which play a fascinating role in string theory.

How do these diagrams work?

Each dot stands for a reflection symmetry of our spacetime crystal.  Dots not connected by an edge are reflections along axes that are at right angles to each other.  Dots connected by various differently labelled edges are reflections at various other angles to each other.  To get a spacetime crystal, the diagramneeds to... more »

Spacetime crystals

You know about crystals in space.  What's a crystal in spacetime?    It's a repetitive pattern that has a lot of symmetries including reflections, translations, rotations and Lorentz transformations.   Rotations mix up directions in space.  Lorentz transformations mix up space and time directions.

We can study spacetime crystals mathematically - and the nicest ones are described by gadgets called hyperbolic Dynkin diagrams, which play a fascinating role in string theory.

How do these diagrams work?

Each dot stands for a reflection symmetry of our spacetime crystal.  Dots not connected by an edge are reflections along axes that are at right angles to each other.  Dots connected by various differently labelled edges are reflections at various other angles to each other.  To get a spacetime crystal, the diagram needs to obey some rules.

The number of dots in the diagram, called its rank, is the dimension of the spacetime the crystal lives in.  So, the picture here shows a bunch of crystals in 5-dimensional spacetime.

Victor Kac, the famous mathematician who helped invent these spacetime crystals, showed they can only exist in dimensions 10 or below.  He showed that:

there are 4 in dimension 10
there are 5 in dimension 9
there are 5 in dimension 8
there are 4 in dimension 7

In 1979, two well-known mathematicians named Lepowsky and Moody showed there were infinitely many spacetime crystals in 2 dimensions... but they classified all of them.

In 1989, Saclioglu tried to classify the spacetime crystals in dimensions 3 through 6.  He got a list of 118.

But he left a bunch out!  A more recent list, compiled very carefully by a big team of mathematicians, gives 220:

there are 22 in dimension 6
there are 22 in dimension 5
there are 53 in dimension 4
there are 123 in dimension 3

If they're right, there's a total of 238 spacetime crystals with dimensions between 3 and 10.  

I think it's really cool how 10 is the maximum allowed dimension, and the number of spacetime crystals explodes as we go to lower dimensions... becoming infinite in dimension 2.

String theory lives in 10d spacetime, so it's perhaps not very shocking that some 10-dimensional spacetime crystals are important in string theory - and also supergravity, the theory of gravity that pops out of superstring theory.    The lower-dimensional ones seem to appear when you take 10d supergravity and 'curl up' some of the space dimensions to get theories of gravity in lower dimensions.

Greg Egan and I have been playing around with these spacetime crystals.  I've spent years studying crystal-like patterns in space, so it's fun to start looking at them in spacetime.  I'd like to say a lot more about them - but my wife is waiting for me to cook breakfast, so not now!

Nobody calls them 'spacetime crystals', by the way - to sound smart, you gotta say 'hyperbolic Dynkin diagrams'.  Here's the paper by that big team:

• Lisa Carbone, Sjuvon Chung, Leigh Cobbs, Robert McRae, Debajyoti Nandi, Yusra Naqvi and Diego Penta, Classification of hyperbolic Dynkin diagrams, root lengths and Weyl group orbits, http://arxiv.org/abs/1003.0564.

+J Gregory Moxness created nice pictures of all 238 hyperbolic Dynkin diagrams and put them on Wikicommons:

https://en.wikipedia.org/wiki/User:Jgmoxness

and that's where I got my picture here!

#spnetwork arXiv:1003.0564 #symmetry #KacMoody #Dynkin #geometry  ___

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2014-10-20 05:38:33 (8 comments, 4 reshares, 66 +1s)Open 

In case you started thinking the whole world is just that tiny part of it that makes sense...

Lotus Flower

I will shape myself into your pocket
Invisible, do what you want, do what you want
I will shrink and I will disappear
I will slip into the groove and cut me up, and cut me up

There's an empty space inside my heart
Where the weeds take root
And now I'll set you free
I'll set you free
There's an empty space inside my heart
Where the weeds take root
Tonight I'll set you free
I'll set you free

Slowly we unfurl
As lotus flowers
'Cause all I want is the moon upon a stick
Just to see what if
Just to see what is
I can't kick your habit
Just to feed your fast ballooning head
Listen to your heart

We will shrink and be quiet as mice... more »

In case you started thinking the whole world is just that tiny part of it that makes sense...

Lotus Flower

I will shape myself into your pocket
Invisible, do what you want, do what you want
I will shrink and I will disappear
I will slip into the groove and cut me up, and cut me up

There's an empty space inside my heart
Where the weeds take root
And now I'll set you free
I'll set you free
There's an empty space inside my heart
Where the weeds take root
Tonight I'll set you free
I'll set you free

Slowly we unfurl
As lotus flowers
'Cause all I want is the moon upon a stick
Just to see what if
Just to see what is
I can't kick your habit
Just to feed your fast ballooning head
Listen to your heart

We will shrink and be quiet as mice
While the cat is away; do what we want
Do what we want

There's an empty space inside my heart
Where the weeds take root
So now I set you free
I set you free

'Cause all I want is the moon upon a stick
Just to see what if
Just to see what is

A bird has flown into my room

Slowly we unfurl
As lotus flowers
'Cause all I want is the moon upon a stick
I dance around a pit
The darkness is beneath
I can't kick your habit
Just to feed your fast ballooning head
Listen to your heart___

posted image

2014-10-18 15:55:11 (19 comments, 28 reshares, 88 +1s)Open 

Inspirograph

http://nathanfriend.com/inspirograph/

If you ever had a spirograph, or even better if you never had one: now there's a good one on your web browser! 

It's easy to use.  You just move one gear around the other using your mouse (or finger).  I still prefer the actual spirograph: working with actual physical tools is a much more immersive experience than twiddling a computer.  People are getting starved for contact with interesting matter.  But not everyone has access to a spirograph!

It's written using TypeScript - a typed superset of JavaScript that compiles to plain JavaScript:

http://www.typescriptlang.org/

#geometry  

Inspirograph

http://nathanfriend.com/inspirograph/

If you ever had a spirograph, or even better if you never had one: now there's a good one on your web browser! 

It's easy to use.  You just move one gear around the other using your mouse (or finger).  I still prefer the actual spirograph: working with actual physical tools is a much more immersive experience than twiddling a computer.  People are getting starved for contact with interesting matter.  But not everyone has access to a spirograph!

It's written using TypeScript - a typed superset of JavaScript that compiles to plain JavaScript:

http://www.typescriptlang.org/

#geometry  ___

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2014-10-17 16:06:27 (123 comments, 111 reshares, 203 +1s)Open 

The moving sofa problem

You've probably tried to move a sofa around a bend in a hallway.  It's annoying.   But it leads to some fun math puzzles.  Let's keep things simple and work in 2 dimensions.  Then the moving sofa problem asks:

What is the shape of largest area that can be maneuvered through an L-shaped hallway of width 1? 

This movie shows one attempt to solve this problem.  It's called the Hammersley sofa, since it was discovered by John Hammersley.  It has an area of

π/2 + 2/pπ = 2.20741609916...

But it's not the best known solution!  In 1992, Joseph Gerver found a shape of area 2.2195 that works.

On the other hand, Hammersley showed that any solution has area at most

2 sqrt(2) = 2.82842712475...

So, the moving sofa problem remains unsolved.  Anothereasily stat... more »

The moving sofa problem

You've probably tried to move a sofa around a bend in a hallway.  It's annoying.   But it leads to some fun math puzzles.  Let's keep things simple and work in 2 dimensions.  Then the moving sofa problem asks:

What is the shape of largest area that can be maneuvered through an L-shaped hallway of width 1? 

This movie shows one attempt to solve this problem.  It's called the Hammersley sofa, since it was discovered by John Hammersley.  It has an area of

π/2 + 2/pπ = 2.20741609916...

But it's not the best known solution!  In 1992, Joseph Gerver found a shape of area 2.2195 that works.

On the other hand, Hammersley showed that any solution has area at most

2 sqrt(2) = 2.82842712475...

So, the moving sofa problem remains unsolved.  Another easily stated but very hard geometry problem!

You can see Joseph Gerver's sofa on this page by my friend Steve Finch:

http://web.archive.org/web/20080107101427/http://mathcad.com/library/constants/sofa.htm

Basically he rounded off some of the corners of Hammersley's sofa!

The movie here was made by Claudio Rocchini, and appears on the Wikipedia article:

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

#geometry  ___

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2014-10-16 19:59:25 (14 comments, 21 reshares, 99 +1s)Open 

The densest way to pack octagons

This image made by +Graeme McRae shows the densest way to pack equal-sized regular octagons in the plane.  The cool part: the density is slightly less than the best you can do for circles!

You can pack equal-sized circles with a density of at most

90.68996%

or so.  For equal-sized regular octagons, the best you can do is

90.61636%

That's just about 0.07% worse, but it's enough to prove that a circle isn't the pessimal plane packer - that is, the shape whose densest packing is the lowest of all. 

And one reason that is interesting is that Stanislaw Ulam conjectured that in 3 dimensions, the sphere is the pessimal packer!  That conjecture is still open.

For more about this story, visit my blog Visual Insight:
h... more »

The densest way to pack octagons

This image made by +Graeme McRae shows the densest way to pack equal-sized regular octagons in the plane.  The cool part: the density is slightly less than the best you can do for circles!

You can pack equal-sized circles with a density of at most

90.68996%

or so.  For equal-sized regular octagons, the best you can do is

90.61636%

That's just about 0.07% worse, but it's enough to prove that a circle isn't the pessimal plane packer - that is, the shape whose densest packing is the lowest of all. 

And one reason that is interesting is that Stanislaw Ulam conjectured that in 3 dimensions, the sphere is the pessimal packer!  That conjecture is still open.

For more about this story, visit my blog Visual Insight:

http://blogs.ams.org/visualinsight/2014/10/15/packing-regular-octagons/

#geometry  ___

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2014-10-13 16:21:49 (18 comments, 35 reshares, 80 +1s)Open 

Network Theory

The world needs some new kinds of math, which deal with networks.  So, check out this video! I start with a quick overview of network theory, and then begin building a category where the morphisms are electrical circuits - a warmup for more complicated kinds of networks.  You can also read lecture notes:

• Network theory (part 30), http://johncarlosbaez.wordpress.com/2014/10/03/network-theory-part-30/

With luck, this video will be the first of a series. I’m giving a seminar on network theory at U.C. Riverside this fall. I’ll start by sketching the results in this new paper:

• John Baez and Brendan Fong, A compositional framework for passive linear networks, http://math.ucr.edu/home/baez/circuits.pdf

A couple of weeks ago I talked about having a Hangout on Air, but I didn't get my act together in time to set one upwith a ca... more »

Network Theory

The world needs some new kinds of math, which deal with networks.  So, check out this video! I start with a quick overview of network theory, and then begin building a category where the morphisms are electrical circuits - a warmup for more complicated kinds of networks.  You can also read lecture notes:

• Network theory (part 30), http://johncarlosbaez.wordpress.com/2014/10/03/network-theory-part-30/

With luck, this video will be the first of a series. I’m giving a seminar on network theory at U.C. Riverside this fall. I’ll start by sketching the results in this new paper:

• John Baez and Brendan Fong, A compositional framework for passive linear networks, http://math.ucr.edu/home/baez/circuits.pdf

A couple of weeks ago I talked about having a Hangout on Air, but I didn't get my act together in time to set one up with a camera of high enough quality to make my writing readable on the long whiteboard in this room.  So, I decided to use a videocam provided by U.C. Riverside.  It can't hook up directly to my laptop, since I don't have FireWire, but it creates mp3 files that I can upload later.  These have a maximum length of about 30 minutes, but using Windows Live Movie Maker it was easy to stitch them together into a single file.  Then I uploaded this to YouTube.

So, that's the best I've been able to do so far.  If you have questions about the seminar, you can ask them here - or even better, on the Azimuth blog, where the conversation has already started, and you can write in TeX:

• Network theory seminar (part 1), http://johncarlosbaez.wordpress.com/2014/10/11/network-theory-seminar-part-1/

In the future, I'll announce seminar videos on that blog.

I thank Blake Pollard for filming this seminar, and Muhammad “Siddiq” Siddiqui-Ali for providing the videocamera and technical support.___

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2014-10-10 22:35:04 (7 comments, 34 reshares, 146 +1s)Open 

Hyperbubbles

This great image by +Paul Nylander is based on a tiling of the hyperbolic plane.  You can tile it with regular heptagons, three meeting at each corner.  It's called the {7,3} tiling - and I've shown you several versions already!  But in this one, each heptagon is artistically drawn as a bubble, thanks to a suggestion by Robert Bagula.

I think I learned about this picture from +Steve Esterly - he liked how bubbles of foam on a cup of coffee get smaller near the edges, sort of like how things get scrunched up near the edges of the hyperbolic plane when we squash it down to the Poincare disk.

You can see more of Paul Nylander's hyperbolic art here:

http://bugman123.com/Hyperbolic/index.html

If you aren't friends with the {7,3} tiling yet, get to know it here:
... more »

Hyperbubbles

This great image by +Paul Nylander is based on a tiling of the hyperbolic plane.  You can tile it with regular heptagons, three meeting at each corner.  It's called the {7,3} tiling - and I've shown you several versions already!  But in this one, each heptagon is artistically drawn as a bubble, thanks to a suggestion by Robert Bagula.

I think I learned about this picture from +Steve Esterly - he liked how bubbles of foam on a cup of coffee get smaller near the edges, sort of like how things get scrunched up near the edges of the hyperbolic plane when we squash it down to the Poincare disk.

You can see more of Paul Nylander's hyperbolic art here:

http://bugman123.com/Hyperbolic/index.html

If you aren't friends with the {7,3} tiling yet, get to know it here:

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

#geometry  ___

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2014-10-09 20:07:55 (146 comments, 58 reshares, 303 +1s)Open 

Hollow Earth

There have been lots of theories saying the Earth is hollow, but I know only one that could be true.

Edmond Halley, the guy who discovered the famous comet, had a theory where Earth consists of a hollow shell about 800 kilometers thick, two smaller shells nested inside, and a ball in the middle - all separated by atmospheres and rotating at different speeds!  It sounds nutty, but Halley was trying to explain the Earth's rather complicated magnetic fields: each of his shells was magnetic.

People sometimes accuse Leonhard Euler, the famous mathematician and physicists, of believing the Earth was hollow.  But that's not true.  In fact, all Euler did was propose a famous thought experiment:

Puzzle: If you could drill a hole all the way through the Earth, and drop a stone in, what would happen?

With the passage of time,sci... more »

Hollow Earth

There have been lots of theories saying the Earth is hollow, but I know only one that could be true.

Edmond Halley, the guy who discovered the famous comet, had a theory where Earth consists of a hollow shell about 800 kilometers thick, two smaller shells nested inside, and a ball in the middle - all separated by atmospheres and rotating at different speeds!  It sounds nutty, but Halley was trying to explain the Earth's rather complicated magnetic fields: each of his shells was magnetic.

People sometimes accuse Leonhard Euler, the famous mathematician and physicists, of believing the Earth was hollow.  But that's not true.  In fact, all Euler did was propose a famous thought experiment:

Puzzle: If you could drill a hole all the way through the Earth, and drop a stone in, what would happen?

With the passage of time, scientists have been learning more about geology, so the remaining people who believe in hollow Earth theories get crazier and crazier.  For example, the neo-Nazi Ernst Zündel wrote a book entitled UFOs - Nazi Secret Weapons? claiming that Hitler and his men had boarded submarines at the end of the war, escaped to Argentina, and then established a base for flying saucers in a hole leading to the inside of the Earth at the South Pole. Zündel also suggested that the Nazis had originated as a separate race that had come from inside the Earth.

My favorite hollow Earth theories are the ones that say we are actually inside!   That's right: all of outer space fits inside an Earth-sized ball, while rocks fill the infinite outside of the universe!    Or maybe the outside is outer space too... but we're stuck on the inside.

I'm not sure what this theory is supposed to accomplish, but it's fun to think about.   What you think are stars may actually be lights from cities on the other side of the Earth!  When the Sun sets.... well, there are different ideas about where it goes.  Perhaps it goes into a hole?  Or perhaps its light bends in such a way that only half the Earth gets sunlight at any one time.

In fact, if you change enough laws of physics, the inside-out Earth theory becomes hard to refute.   It's becomes our usual theory of the universe described in "inside-out coordinates", so it becomes equivalent to our usual ideas - except perhaps for one tiny point, which I'll mention later. 

There's an Egyptian mathematician Mostafa Abdelkader who actually took this line of thought seriously.  According to Martin Gardner, this guy claimed that light rays travel in circular paths, and slow as they approach the center of the spherical star-filled hollow Earth.  Everything shrinks and slows down as it gets close to the center, so nothing can ever get there.  On the other hand, a drill would get longer as you used it to dig outward into the rock.  

This theory gets a bit freaky if you drill all the way through the Earth.  At some point your drill tip has to suddenly appear on the opposite side of the Universe!

But if you allow this, there's not much difference between the inside-out Universe and ours.  True, it's missing one point compared to our usual Universe - namely, the point right at the center of the Earth!   But to make up for this, it has an extra point, namely the "point at infinity", in the center of this inside-out Universe. 

In fact, if we include both these points, the Universe itself becomes a "3-sphere",  which has two "hemispheres", namely the inside and the outside of the Earth.  One is solid rock, the other empty space.  And in fact Dante describes this cosmology in his Divine Comedy.   He was a smart dude.

It's a fun exercise in the philosophy of science to figure out why a theory that makes almost identical predictions to our usual theories is nonetheless considered much worse.

For more on hollow Earth theories, try:

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

For more on the myth that Euler believed in a hollow Earth:

http://eulerarchive.maa.org/hedi/HEDI-2007-04.pdf

The nice mathematical way to turn space inside out is called conformal inversion.  It sends straight lines and circles to straight lines and circles, and it preserves angles.  You can read more about it here:

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

And here's the webpage where I got this marvelous picture:

http://aetherforce.com/the-hollow-earth-in-science-by-duane-griffin/

Here is seems that the 'point at infinity' is contained inside a glowing octahedral Heaven, the Moon is bowl-shaped, the Sun's light bends, and there's a hole in the North Pole, leading... where?

Zündel, by the way, was jailed several times for publishing books like The Hitler We Loved and Why and Did Six Million Really Die?:

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

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2014-10-05 16:36:09 (38 comments, 56 reshares, 190 +1s)Open 

The mystery of neutrinos

Neutrinos come in 3 kinds, but the 3 kinds with definite masses are not the same as the 3 kinds with definite flavors.   They're related by a 3×3 matrix.

Every particle is also a wave, and a particle at rest is a standing wave: it wiggles without going anywhere, like a vibrating piano string.   The mass, or rest mass, of a particle is simply the frequency at which this standing wave vibrates. 

But sometimes a particle at rest can wiggle in several different ways with different frequencies.  A neutrino can wiggle in 3 ways, and these are called its mass eigenstates.  They have boring names: 1, 2 and 3.

The flavor of a neutrino says how it interacts with other particles via the weak force. The flavors have cool names: e (electron), μ (muon), and τ (tau particle).
 
But a neutrino with adefinite f... more »

The mystery of neutrinos

Neutrinos come in 3 kinds, but the 3 kinds with definite masses are not the same as the 3 kinds with definite flavors.   They're related by a 3×3 matrix.

Every particle is also a wave, and a particle at rest is a standing wave: it wiggles without going anywhere, like a vibrating piano string.   The mass, or rest mass, of a particle is simply the frequency at which this standing wave vibrates. 

But sometimes a particle at rest can wiggle in several different ways with different frequencies.  A neutrino can wiggle in 3 ways, and these are called its mass eigenstates.  They have boring names: 1, 2 and 3.

The flavor of a neutrino says how it interacts with other particles via the weak force. The flavors have cool names: e (electron), μ (muon), and τ (tau particle).
 
But a neutrino with a definite flavor does not have a definite mass!   A neutrino with a definite flavor is a superposition, or linear combination, of mass eigenstates.   The first equation here says how that works.   You can understand this if you know about matrix multiplication.  We use a  3×3 matrix, the neutrino mixing matrix, to write the neutrinos with definite flavor as linear combinations of the mass eigenstates.

What are the numbers in this matrix?  Experimentalists have worked very hard over the last few decades trying to measure them.  We know some better than others.

They are complex numbers, but there's an interesting guess about their absolute values, shown in the second equation.  This guess is called the tribimaximal matrix.

That's a goofy-sounding name!   Where did it come from?

With this matrix, the 2nd column is all 1/3's.  This means that the 2nd mass eigenstate consists of equal parts of e, μ and τ, so we say it's trimaximally mixed.  The 3rd column has two 1/2's.  This means that the 3rd mass eigenstate consists of equal parts of μ and τ, so we say it's bimaximally mixed

Is the tribimaximal matrix right?  It is consistent with all known experiments...

... or at least it was until 2012, when this paper seems to have ruled it out:

• The Daya Bay Collaboration, Observation of electron-antineutrino disappearance at Daya Bay, Physics Review Letters 108 (2012): 171803, http://arxiv.org/abs/1203.1669.

Great title, eh?

I'm still hoping this experiment made a mistake.  If the tribimaximal matrix is wrong, neutrinos remain profoundly mysterious: they're described by a bunch of numbers and we have no idea why these numbers are what they are - since we don't even know exactly what they are.

I wish I knew more about this stuff.  For more, try:

https://en.wikipedia.org/wiki/Pontecorvo-Maki-Nakagawa-Sakata_matrix
https://en.wikipedia.org/wiki/Tribimaximal_mixing

By the way, the neutrino mixing matrix is 'unitary': its inverse is the complex conjugate of its transpose.  So, the rows say how flavors are superpositions of mass eigenstates - and the columns say how mass eigenstates are superpositions of flavors!  I used this when I explained the tribimaximal idea.
 
Also, because this matrix is unitary, when you take the absolute value squared of the entries, you get a matrix of nonnegative numbers where each row sums to 1 and each column sums to 1.  So, as soon as we decide it looks like this:

?     1/3     ?
?     1/3     1/2
?     1/3     1/2

we know what it must be:

2/3    1/3    0
1/6    1/3    1/2
1/6    1/3    1/2

#spnetwork arxiv:1203.1669 #neutrinos___

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2014-10-03 18:31:31 (63 comments, 29 reshares, 87 +1s)Open 

3-adic integers

Math is full of wonderful number systems.  Here are the 3-adic integers.  A 3-adic integer is like an integer written in base 3, except that its digits can go on forever to the left.  So, something like this:

   .......1012010212

The usual positive integers sit inside the 3-adic integers.  For example, the number 21 (in base 3) gives the 3-adic integer

   .......0000000021

But there are lots of other 3-adic integers: the ones that go on forever.

We can add and multiply these in the usual way, carrying the digits as usual.  It just takes longer.  For example:

    ...........1012010212
+  ...........0120210101
------------------------------------
     ...........1202221020

The picture here, created by Christopher Cutler, uses a clever trick to draw 3-adicintegers as point... more »

3-adic integers

Math is full of wonderful number systems.  Here are the 3-adic integers.  A 3-adic integer is like an integer written in base 3, except that its digits can go on forever to the left.  So, something like this:

   .......1012010212

The usual positive integers sit inside the 3-adic integers.  For example, the number 21 (in base 3) gives the 3-adic integer

   .......0000000021

But there are lots of other 3-adic integers: the ones that go on forever.

We can add and multiply these in the usual way, carrying the digits as usual.  It just takes longer.  For example:

    ...........1012010212
+  ...........0120210101
------------------------------------
     ...........1202221020

The picture here, created by Christopher Cutler, uses a clever trick to draw 3-adic integers as points on the plane.  The 3-adic integers are the black stuff.

See how the black stuff is contained in 3 gray circles?  Those are the 3-adic integers whose last digit is 0, 1, and 2.  

Each gray circle contains 3 smaller gray circles.  those are the 3-adic integers whose second to last digit is 0, 1, and 2. 

And so on, forever... so we get a fractal in the plane!

In this picture,  a few selected 3-adic integers have lines pointing to them from pretty color pictures.  There's a clever way to draw a 'color portrait' of each 3-adic integer, which is explained in my blog post on this stuff:

http://blogs.ams.org/visualinsight/2014/10/01/2-adic-integers/

My blog post actually talks about the 2-adic integers, but they are similar.  We can define p-adic integers for any p = 2, 3, 4, ..., but they work best when p is prime.  In that case we can divide two p-adic integers and get a more general p-adic number.

A p-adic number has digits going on forever before the decimal point, but only a finite way after the decimal point.  Here's a typical 3-adic number:

   .......1012010212.1022

Puzzle 1:  I didn't say anything about negative 3-adic integers.  The reason is that we never need to stick a minus sign in front of a 3-adic integer!   To see what I mean, find a 3-adic integer that when added to

   ......0000000001

gives
 
    ......000000000

Your answer will be the 3-adic version of -1, even though there's no minus sign in front of it!

Puzzle 2:  Find two 4-adic integers that aren't zero but give zero when you multiply them.  This means that dividing 4-adic integers doesn't work well.  Only when p is prime can we divide by any nonzero p-adic number.

p-adic numbers are very important in the branch of math called... surprise... number theory!   You can read more about them here:

http://en.wikipedia.org/wiki/P-adic_number___

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2014-10-02 15:40:38 (35 comments, 19 reshares, 98 +1s)Open 

Infinity-gons

Here's a picture of the hyperbolic plane tiled with apeirogons - polygons with infinitely many sides!  

Regular polygons with infinitely many sides are one of those things that make hyperbolic geometry more fun than ordinary Euclidean geometry. 

"Apeiro-" is a Greek prefix meaning "infinite" or "unlimited".  You know the word "perimeter", meaning boundary or limit?  Well, "apeiro-" means having no limit.  Greek mathematicians were pretty nervous about the infinite, because they thought its unlimitedness made it inherently vague, even chaotic.

The apeirogons are white, and there are 3 meeting at each corner. 

In blue we see the dual tiling:  there's one blue vertex in the middle of each white polygon, and one blue edge crossing each white edge, and oneblue po... more »

Infinity-gons

Here's a picture of the hyperbolic plane tiled with apeirogons - polygons with infinitely many sides!  

Regular polygons with infinitely many sides are one of those things that make hyperbolic geometry more fun than ordinary Euclidean geometry. 

"Apeiro-" is a Greek prefix meaning "infinite" or "unlimited".  You know the word "perimeter", meaning boundary or limit?  Well, "apeiro-" means having no limit.  Greek mathematicians were pretty nervous about the infinite, because they thought its unlimitedness made it inherently vague, even chaotic.

The apeirogons are white, and there are 3 meeting at each corner. 

In blue we see the dual tiling:  there's one blue vertex in the middle of each white polygon, and one blue edge crossing each white edge, and one blue polygon containing each white vertex.

Here's are two funs puzzle:

Puzzle 1: What would a regular apeirogon look like in Euclidean geometry?

Puzzle 2: Can you tile the Euclidean plane with regular apeirogons?

And here are two puzzles that are easy if you know what the Schläfli symbol of a tiling is, impossible otherwise:

Puzzle 3: what's the Schläfli symbol of the white tiling?

Puzzle 4: what's the Schläfli symbol of the blue tiling?

The picture was drawn by Don Hatch:

http://www.plunk.org/~hatch/HyperbolicTesselations/

but I'm afraid peeking at his page may give away the answer to the puzzles!

#geometry___

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2014-10-02 00:24:15 (64 comments, 65 reshares, 189 +1s)Open 

M-theory and octonions

My former student +John Huerta (at left) has just finished an amazing paper that I've got to tell you about!

You've probably heard rumors that superstring theory lives in 10 dimensions and something more mysterious called M-theory lives in 11.  You may have wondered why.  

In fact, there's a nice way to define superstrings in dimensions 3, 4, 6, and 10 - at least before you take quantum mechanics into account.  Of these theories, you can only consistently quantize the 10-dimensional version.   But never mind that.  What's so great about the numbers 3, 4, 6 and 10?

What's so great is that they're 2 more than 1, 2, 4, and 8.

If you try to set up a nice number system where you can add, multiply, subtract and divide, it only works in dimensions 1, 2, 4, and 8.  The real numbers form a line,and th... more »

M-theory and octonions

My former student +John Huerta (at left) has just finished an amazing paper that I've got to tell you about!

You've probably heard rumors that superstring theory lives in 10 dimensions and something more mysterious called M-theory lives in 11.  You may have wondered why.  

In fact, there's a nice way to define superstrings in dimensions 3, 4, 6, and 10 - at least before you take quantum mechanics into account.  Of these theories, you can only consistently quantize the 10-dimensional version.   But never mind that.  What's so great about the numbers 3, 4, 6 and 10?

What's so great is that they're 2 more than 1, 2, 4, and 8.

If you try to set up a nice number system where you can add, multiply, subtract and divide, it only works in dimensions 1, 2, 4, and 8.  The real numbers form a line, and that's 1-dimensional.  The complex numbers form a plane, and that's 2-dimensional.  There are also more esoteric options: the quaternions are 4-dimensional, and the octonions are 8-dimensional.  When you try to go beyond these, you lose the law that

|xy| = |x| |y|

and things aren't so nice.

I've spent decades studying the quaternions and octonions, just because they're weird and interesting.  Why do the dimensions double each time in this game?  I've learned the answer, and I could tell you - but it might shatter your brain.  What happens if you go further, to dimension 16?  I've learned a bit about that too, though I bet there are big mysteries still lurking here. 

I also learned that being an expert on this stuff does not make you popular at parties.

One cool thing is this.  A string is a curve, so it's 1-dimensional, but as time passes it traces out a 2-dimensional surface.  So, if we have a string floating around in some spacetime, we've got a 2-dimensional surface together with some extra dimensions of spacetime.   

But for the string to be 'super' - for it to have supersymmetry, a symmetry between bosons and fermions - we need a certain special equation to be true.  And it's true precisely when we can take the extra dimensions and think of them as one of our nice number systems.  

So, we need 1, 2, 4 or 8 extra dimensions.  So the total dimension of spacetime needs to be 3, 4, 6, or 10.

(That's a very rough sketch of a complicated argument, of course.  I'm leaving out the details, but later I'll show you where to find them.)

We can also look at theories of 'branes', which are like strings but higher-dimensional.  Instead of a curve, a 2-brane is a 2-dimensional surface.  As time passes, it traces out a 3-dimensional surface.  So, if we have a 2-brane floating around in some spacetime, we've got a 3-dimensional surface together with some extra dimensions of spacetime.   And it turns out that 2-branes can also have supersymmetry when the extra  dimensions can be seen as one of our nice numbers systems!

So now the total dimension of spacetime needs to be 3 more than 1, 2, 4, and 8.  It needs to be 4, 5, 7 or 11.

When we take quantum mechanics into account it seems that the 11-dimensional theory works best... but the quantum aspects are still mysterious, murky and messy compared to superstring theory, so it's called M-theory.

There's stuff we don't understand, and stuff we do.  In his new paper, John Huerta has pushed forward the line separating the two.  He's shown that using the octonions we can build a 'super-3-group', an algebraic structure that seems just right for understanding the symmetries of supersymmetric 2-branes in 11 dimensions.  

I could say a lot more, but it's better if you read this:

• John Baez and John Huerta, The strangest numbers in string theory, http://math.ucr.edu/home/baez/octonions/strangest.html

This is a fun and easy article about this stuff, which we wrote for Scientific American.  

Then, if that's too easy, try this:

• John Baez and John Huerta, Division algebras and supersymmetry I, http://arxiv.org/abs/0909.0551.

Here we get into the details, and explain the special equation that makes superstrings work nicely in 3, 4, 6, and 10 dimensions - and how it follows from having a nice number system in dimensions 1, 2, 4 and 8.  This stuff was known before, but not explained all in one place.

Next, try this:

• John Baez and John Huerta, Division algebras and supersymmetry II, http://arxiv.org/abs/1003.3436.

Here we explain the special equation that makes supersymmetric 2-branes work in dimension 4, 5, 7 and 11.  More importantly, we start studying how the symmetries of superstrings and super-2-branes come out of the nice number systems.  Physicists use gadgets called 'Lie algebras' to study symmetry... so they should like these generalizations, called 'Lie 2-superalgebras' and 'Lie 3-superalgebras'.

Next, try this:

• John Huerta, Division algebras and supersymmetry III, http://arxiv.org/abs/1109.3574.

At this point John Huerta sailed off on his own!  

Physicists like Lie algebras, but what they really love are 'Lie groups'.  Lie algebras are just a trick for studying Lie groups: it's the groups that directly describe symmetry.  In this paper John cooked up the 'Lie 2-supergroups' that govern classical superstrings in dimensions 3, 4, 6 and 10.  Just as a group is a special sort of category, a 2-group is a special sort of 2-category.  So at this point John got into 'higher category theory' - one of my favorite subjects.

And here's his new paper, the last of the series:

• John Huerta, Division algebras and supersymmetry IV, http://arxiv.org/abs/1409.4361.

Here John built the 'Lie 3-supergroups' that govern classical super-2-branes in dimensions 3, 4, 6 and 10.  A 3-group is a special sort of 3-category.  

I really love how the math of superstrings and M-theory emerge nicely from combining the octonions with higher category theory. In case you're wondering: I have no strong opinion about whether these ideas apply to our physical universe.  I see no convincing experimental evidence in favor of string theory or M-theory.  All I know is that they're beautiful.  

Maybe they apply to some other universes that are less messed-up than ours.  Maybe we're in some sort of purgatory for species who still need to learn basic math.  If so, John Huerta just placed out.  :-)

#spnetwork arxiv:1409.4361 #octonions #superstrings #Mtheory___

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2014-09-30 18:42:45 (39 comments, 5 reshares, 30 +1s)Open 

Video camera for Hangout on Air: how to do it?

I want a video camera that can do a better job for Hangout on Air than the built-in webcam in my laptop.  I want something cheap and easy.  What's best?

I'm completely clueless.  I've never done anything like this.  One website mentions 5 kinds of cameras:

1. Built in webcams
2. USB connected cameras
3. HDMI cameras
4. SDI cameras
5. Component or composite (analog) output cameras
6. Cameras with wi-fi capability

I don't want a camera that produces output that needs to be converted before I can feed it into a Hangout on Air.  So I think I want a USB connected camera.  Right?   If so, what's a good one?

Ideally I could just attach it to my laptop and it would automagically override the built-in webcam.   More likely I'll need to say some sortof magic ... more »

Video camera for Hangout on Air: how to do it?

I want a video camera that can do a better job for Hangout on Air than the built-in webcam in my laptop.  I want something cheap and easy.  What's best?

I'm completely clueless.  I've never done anything like this.  One website mentions 5 kinds of cameras:

1. Built in webcams
2. USB connected cameras
3. HDMI cameras
4. SDI cameras
5. Component or composite (analog) output cameras
6. Cameras with wi-fi capability

I don't want a camera that produces output that needs to be converted before I can feed it into a Hangout on Air.  So I think I want a USB connected camera.  Right?   If so, what's a good one?

Ideally I could just attach it to my laptop and it would automagically override the built-in webcam.   More likely I'll need to say some sort of magic spell.  I have no idea what.

I'm having trouble finding out about this online.  Hmm, I guess this helps a bit:

Any camera that is going to be used with a Hangout does, though, need a clean video output, without any overlays for camera controls or settings. Broadcast and most consumer camcorders have this, but not all DSLRs do.

Hangouts work using a Google browser plug in, and currently this seems to support in-built webcams, USB webcams or USB class devices, PCI video capture cards and Thunderbolt-based cards.

Both PCI and Thunderbolt cards can capture over HDMI or SDI and many also support capture from analogue sources, a cheap way to connect up a standard definition camera, especially if it has component video out.

(That last paragraph sailed right over my head so I'm going to ignore it.)

The plug in Google uses does not, sadly, support the Firewire connections present on all but the latest Macs, as well as some PCs, and on a wide range of camcorders.

(I'm using a Windows machine and I have no idea what Firewire is, so I'm hoping I can ignore this too.)

HDMI to Thunderbolt adapters, such as the Blackmagic Intensity Extreme used for our Hangout On Air recording, are not especially expensive, so this might not be a deal-breaker. But it does mean carrying an extra piece of kit, as well as using both a computer equipped with Thunderbolt, and a camera with HDMI or SDI.

(Ignore.)

Using an external camera allows Hangouts and Hangouts On Air to broadcast in HD. But hooking up a pro camera brings immediate advantages, beyond simply resolution. Cameras can be set to the best white balance for the room, focusing is usually more accurate, and the image less noisy.

(Yes, that's the point.  Also, I'll be walking back and forth in front of a long whiteboard.)

http://www.audiovideopro.net/using-google-plus-hangouts-air-broadcasting/2/___

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2014-09-30 05:52:46 (123 comments, 24 reshares, 113 +1s)Open 

"The radical activists are doomed"

That's what the government-run newspaper in Hong Kong says.  But is it true? 

Of course it's not "radical activists" who are protesting China's attempt to subvert Hong Kong democracy by hand-picking which candidates get to run.  It's tens of thousands of ordinary people, most of whom sprang into action when police began using pepper spray and tear gas. 

But are they doomed? 

I'm less optimistic about mass street protests than I was a few years ago. In Iran, the Green Movement was crushed by force.  In Egypt, the Tahrir Square movement was first coopted by the Muslim Brotherhood and then crushed by the military.  In the United States, Occupy Wall Street was beaten back with pepper spray but ultimately doomed by its lack of organization and the apathy of citizens at large.  Ineach ca... more »

"The radical activists are doomed"

That's what the government-run newspaper in Hong Kong says.  But is it true? 

Of course it's not "radical activists" who are protesting China's attempt to subvert Hong Kong democracy by hand-picking which candidates get to run.  It's tens of thousands of ordinary people, most of whom sprang into action when police began using pepper spray and tear gas. 

But are they doomed? 

I'm less optimistic about mass street protests than I was a few years ago. In Iran, the Green Movement was crushed by force.  In Egypt, the Tahrir Square movement was first coopted by the Muslim Brotherhood and then crushed by the military.  In the United States, Occupy Wall Street was beaten back with pepper spray but ultimately doomed by its lack of organization and the apathy of citizens at large.  In each case, the street protests failed to get support from enough well-placed people in the corridors of power.

Hong Kong is a bit different because the protests are ultimately against the encroachment of an outside power: China.  Yes, it's "one country", but it's also "two systems", and Hong Kongers have their own distinct patriotism.  Will they be able to organize well enough to keep what democracy and freedom they have?  I hope so.  I have a real fondness for this place; I'd hate to see it be crushed.

The US and Britain, self-proclaimed supporters of democracy worldwide, seem to be sitting this one out:

The muted response from Western capitals has dismayed Hong Kong’s protesters. Many feel that Britain, as the former colonial power, has failed in its legal and moral responsibility to speak out. The Foreign and Commonwealth Office issued an anodyne statement welcoming “the confirmation that China’s objective is for the election of Hong Kong’s chief executive through universal suffrage,” while noting that it would disappoint some people.

And on Monday, the U.S. Consulate in Hong Kong issued a statement saying it supported Hong Kongers’ fundamental freedoms, such as peaceful assembly and expression, yet it would not take sides in the discussion of the island’s political development.

Thus writes Louisa Lim here:

http://www.nytimes.com/2014/09/30/opinion/occupy-central-protests-hong-kong-people.html

#hongkong  ___

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2014-09-29 18:30:31 (44 comments, 13 reshares, 69 +1s)Open 

Soon I'll restart my seminar on Network Theory.    It'll happen Mondays 3:10-4:30 pm in room 268 of the Surge Building at U.C. Riverside, starting on October 6th.  

If you want to come, please do!  But one of my grad students will be watching by Skype.  So maybe it makes sense to run the seminar as a Google Hangout and let everyone watch?

Here's the idea:

Network theory uses the tools of modern math - categories, operads and more - to study complex systems made of interacting parts. The idea of this seminar is to start from scratch, explain what my grad students have been doing, and outline new projects.

A lot has happened since I left town in January:

1) +Franciscus Rebro has been working on 'cospans', a very general way to treat a physical system with inputs and outputs and treat it as a morphism in category. Thisunderl... more »

Soon I'll restart my seminar on Network Theory.    It'll happen Mondays 3:10-4:30 pm in room 268 of the Surge Building at U.C. Riverside, starting on October 6th.  

If you want to come, please do!  But one of my grad students will be watching by Skype.  So maybe it makes sense to run the seminar as a Google Hangout and let everyone watch?

Here's the idea:

Network theory uses the tools of modern math - categories, operads and more - to study complex systems made of interacting parts. The idea of this seminar is to start from scratch, explain what my grad students have been doing, and outline new projects.

A lot has happened since I left town in January:

1) +Franciscus Rebro has been working on 'cospans', a very general way to treat a physical system with inputs and outputs and treat it as a morphism in category. This underlies all the other projects.

2) Brendan Fong, a student at Oxford, is working on a category where the morphisms are electrical circuits, and composing morphisms is sticking together circuits:

• Brendan Fong, A compositional approach to control theory, http://math.ucr.edu/home/baez/Brendan_Fong_Transfer_Report.pdf.

3) +Blake Pollard, a student in the physics department, has been studying Markov processes. In a Markov process, things randomly hop from one vertex of a graph to another along edges. Blake has created a category where morphisms are ‘open’ Markov process, in which things flow in and out of certain special vertices called ‘terminals’. 

4) +Jason Erbele has been working on categories in control theory, the branch of math used to study physical systems that interact with the outside world via inputs and outputs. After finishing this paper:

• John Baez and Brendan Fong, Categories in control, http://arxiv.org/abs/1405.6881.

he’s been taking more concept from control theory and formalizing them using categories.

5) Oh yeah, and what about me? I gave a series of lectures on network theory at Oxford, and you can see videos of them here:

• John Baez, Network Theory, http://math.ucr.edu/home/baez/networks_oxford/

+Jacob Biamonte and I have also more or less finished a book on chemical reaction networks and Petri nets:

• John Baez and Jacob Biamonte, Quantum Techniques for Stochastic Mechanics, http://math.ucr.edu/home/baez/stoch_stable.pdf

But I won’t be talking about that - I want to talk about the new work my students are doing! 

A big picture is emerging, and I want to talk about it and flesh it out.
To warm up, you can see where we left off last fall:

• Electrical circuits, and getting differential equations from circuit diagrams: http://johncarlosbaez.wordpress.com/2013/04/03/network-theory-part-27/

• The analogy between electronics and mechanics: http://johncarlosbaez.wordpress.com/2013/04/10/network-theory-part-28/

• Analogies between mechanics, electronics, hydraulics, thermal physics, chemistry, heat flow and economics: http://johncarlosbaez.wordpress.com/2013/04/23/network-theory-part-29/

These analogies mean the network theory of electric circuits is really just a way of talking about networks in many subjects.  The simplest and most famous example: the mass on a spring with friction shown here is analogous to a so-called RLC circuit.  

#spnetwork arXiv:1405.6881 #networks  ___

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2014-09-28 16:38:31 (34 comments, 39 reshares, 157 +1s)Open 

The perfect pool break

This is what happens if the cue ball hits the other balls in a perfectly centered way.  This image, created by +Jim Belk, is part of the most popular discussion so far on Mathoverflow - a site where mathematicians ask and answer questions.

The assumptions behind this simulation:

◆ All balls are assumed to be perfectly elastic and almost perfectly rigid.

◆ Each ball has a mass of 1 and a radius of 1.  (The units don't really matter.)

◆ The cue ball has a initial speed of 10.

◆ The force between two balls is zero if their distance is more than 2, and 10^11 times (2 - d)^{3/2} if their distance d is less than 2. So, the force becomes huge if the balls actually press into each other.

The whole collision takes place in a time of 2 times 10^-7, with most of the momentummoving do... more »

The perfect pool break

This is what happens if the cue ball hits the other balls in a perfectly centered way.  This image, created by +Jim Belk, is part of the most popular discussion so far on Mathoverflow - a site where mathematicians ask and answer questions.

The assumptions behind this simulation:

◆ All balls are assumed to be perfectly elastic and almost perfectly rigid.

◆ Each ball has a mass of 1 and a radius of 1.  (The units don't really matter.)

◆ The cue ball has a initial speed of 10.

◆ The force between two balls is zero if their distance is more than 2, and 10^11 times (2 - d)^{3/2} if their distance d is less than 2. So, the force becomes huge if the balls actually press into each other.

The whole collision takes place in a time of 2 times 10^-7, with most of the momentum moving down the edges of the triangle.  You can see the collision happening here:

http://i.stack.imgur.com/WY37i.gif

Jim Belk's post also shows some variations with different laws governing the force between balls:

http://mathoverflow.net/a/156407/2893

If you haven't visited Mathoverflow, this is a great way to start!  Many mathematicians find it addictive.  In math you can make huge progress by talking to other people, and now all mathematicians in the world have a place to do this!   Also, Mathoverflow has a system where you get points for posting questions and answers that other people like.  Young hotshots love racking up points.

I don't spend much time there myself, because I'm more interested in developing new frameworks than proving difficult theorems, and I'm past the stage where I want to learn as much math as possible.  (For me that stage lasted from the ages of about 15 to 50; it's definitely over now.)  But when I feel someone must know the answer to a question and I can't find the answer, Mathoverflow is where I go.

#mathoverflow  ___

posted image

2014-09-26 16:02:50 (5 comments, 6 reshares, 73 +1s)Open 

Geysers on a moon of Saturn

Enceladus, the sixth-largest moon of Saturn, has geysers that shoot jets of water into space!  They look beautiful in these photographed by the Cassini probe.  They create an invisible ring of ice crystals around Saturn: the E ring.  And now they have been mapped!

There are at least 100 geysers near the south pole of Enceladus.  They come from four big cracks in its rocky surface, called tiger stripes.  Some of the water they shoot out falls back to the surface as snow, but about 200 kilograms per second shoots out into space.  The tiger stripes also emit about 5 gigawatts of heat.

The geysers have recently been mapped by Carolyn Porco and her collaborators at NASA:

http://photojournal.jpl.nasa.gov/catalog/PIA17188

There are still many interesting puzzles about what creates these geysers.Porco... more »

Geysers on a moon of Saturn

Enceladus, the sixth-largest moon of Saturn, has geysers that shoot jets of water into space!  They look beautiful in these photographed by the Cassini probe.  They create an invisible ring of ice crystals around Saturn: the E ring.  And now they have been mapped!

There are at least 100 geysers near the south pole of Enceladus.  They come from four big cracks in its rocky surface, called tiger stripes.  Some of the water they shoot out falls back to the surface as snow, but about 200 kilograms per second shoots out into space.  The tiger stripes also emit about 5 gigawatts of heat.

The geysers have recently been mapped by Carolyn Porco and her collaborators at NASA:

http://photojournal.jpl.nasa.gov/catalog/PIA17188

There are still many interesting puzzles about what creates these geysers. Porco believes Enceladus has a sea of liquid water under its icy shell!  But do tidal stresses along the walls of the tiger stripe cracks form liquid water near the surface? Or do tidal stresses create a deep system of cracks extending tens of kilometers to the sea below?

I think it's cool how volcanos on Earth involve molten rock, but similar phenomena on the cold outer moons of our Solar System can happen with liquid water, methane or ammonia!  This is called cryovolcanism:

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

The picture here is one of many in this paper, which you can read for free online:

•  Carolyn Porco, Daiana DiNino, and Francis Nimmo, How the geysers, tidal stresses, and thermal emission across the Southern polar terrain of Enceladus are related, The Astronomical Journal 148 (2014), http://www.ciclops.org/media/sp/2014/7927_18911_0.pdf.

Abstract: We present the first comprehensive examination of the geysering, tidal stresses, and anomalous thermal emission across the south pole of Enceladus and discuss the implications for the moon's thermal history and interior structure. A 6.5 yr survey of the moon's south polar terrain (SPT) by the Cassini imaging experiment has located ~100 jets or geysers erupting from four prominent fractures crossing the region. Comparing these results with predictions of diurnally varying tidal stresses and with Cassini low resolution thermal maps shows that all three phenomena are spatially correlated. The coincidence of individual jets with very small (~10 m) hot spots detected in high resolution Cassini VIMS data strongly suggests that the heat accompanying the geysers is not produced by shearing in the upper brittle layer but rather is transported, in the form of latent heat, from a sub-ice-shell sea of liquid water, with vapor condensing on the near-surface walls of the fractures. Normal stresses modulate the geysering activity, as shown in the accompanying paper; we demonstrate here they are capable of opening water-filled cracks all the way down to the sea. If Enceladus' eccentricity and heat production are in steady state today, the currently erupting material and anomalous heat must have been produced in an earlier epoch. If regional tidal heating is occurring today, it may be responsible for some of the erupting water and heat. Future Cassini observations may settle the question.

#spnetwork doi:10.1088/0004-6256/148/3/45 #astronomy #enceladus___

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2014-09-25 17:48:03 (63 comments, 45 reshares, 111 +1s)Open 

Rise and Fall of the Roman Empire

A thousand years of history flash before your eyes in half a minute!

Lots of fun puzzles here:

Puzzle 1: In 320 and 300 BC, chunks of the Italy turn red.  Why?

Puzzle 2: In 270 BC, a lot of Italy turns red.  Why?

Puzzle 3: In 190 BC, a bunch of what's now Spain turns red.  Why? 

Puzzle 4: In 40 BC, the area that's now France turns red.  Why?

Puzzle 5: In 20 AD, the color changes from red to purple.  Why?

Puzzle 6: In 70 AD, a bunch of Britain turns purple.  Why?

... skipping a few centuries here....

Puzzle 7: In 405 AD, the Roman Empire splits into Eastern and Western halves, shown here in different colors.  Why? 

Puzzle 8: In 420 AD, the Western Roman Empire leaves Britain. Why?
more »

Rise and Fall of the Roman Empire

A thousand years of history flash before your eyes in half a minute!

Lots of fun puzzles here:

Puzzle 1: In 320 and 300 BC, chunks of the Italy turn red.  Why?

Puzzle 2: In 270 BC, a lot of Italy turns red.  Why?

Puzzle 3: In 190 BC, a bunch of what's now Spain turns red.  Why? 

Puzzle 4: In 40 BC, the area that's now France turns red.  Why?

Puzzle 5: In 20 AD, the color changes from red to purple.  Why?

Puzzle 6: In 70 AD, a bunch of Britain turns purple.  Why?

... skipping a few centuries here....

Puzzle 7: In 405 AD, the Roman Empire splits into Eastern and Western halves, shown here in different colors.  Why? 

Puzzle 8: In 420 AD, the Western Roman Empire leaves Britain.  Why?

Puzzle 9: In 480 AD, the Western Roman Empire almost completely disappears!  Why?

Puzzle 10: By 520 AD, the Western Roman Empire is gone.  Why?

When I ask "why", I'm just asking for some basic information on what happened, not the deep causes.  But history buffs should feel free to show off their expertise!  I'm hoping to learn a lot here. 

The image here was created by 'Nyq' and put on Wikicommons:

http://commons.wikimedia.org/wiki/File:Roman_Empire_map.gif

#history  ___

posted image

2014-09-24 15:52:13 (29 comments, 24 reshares, 172 +1s)Open 

This selfie was taken by Vermeer's maid Griet in 1665, moments after the painter completed his masterwork Girl With a Pearl Earring.

This selfie was taken by Vermeer's maid Griet in 1665, moments after the painter completed his masterwork Girl With a Pearl Earring.___

posted image

2014-09-23 14:59:35 (37 comments, 24 reshares, 155 +1s)Open 

Rolling smoothed octagons

This picture by Greg Egan is more profound than it looks.  The area of each white space doesn't change as the wheels roll.  But that's not all!

What shape is the worst at densely packing the plane?  To make this question interesting we should ask about convex shapes - remember, a shape is convex if whenever two points are in it, so is the line between them.  Then the answer seems to be a regular heptagon... though nobody has been able to prove this.  The densest packing of regular heptagons fills up about 89.3% of the plane.  For comparison, circles fill up 90.7%.

But suppose we also demand that our shape be centrally symmetric - meaning that if you reflect it vertically and then reflect it horizontally, it looks the same.  Packings of centrally symmetric convex shapes are a lot easier to understand. 
What&... more »

Rolling smoothed octagons

This picture by Greg Egan is more profound than it looks.  The area of each white space doesn't change as the wheels roll.  But that's not all!

What shape is the worst at densely packing the plane?  To make this question interesting we should ask about convex shapes - remember, a shape is convex if whenever two points are in it, so is the line between them.  Then the answer seems to be a regular heptagon... though nobody has been able to prove this.  The densest packing of regular heptagons fills up about 89.3% of the plane.  For comparison, circles fill up 90.7%.

But suppose we also demand that our shape be centrally symmetric - meaning that if you reflect it vertically and then reflect it horizontally, it looks the same.  Packings of centrally symmetric convex shapes are a lot easier to understand. 

What's the worst centrally symmetric convex shape for densely packing the plane?  It seems to be a regular octagon with its corners smoothed in a certain clever way.  Nobody has proved this... but Thomas Hales, the guy who proved Kepler's conjecture on densely packing spheres, wants to prove this conjecture too! 

The densest packing of these smoothed octagons fills up about 90.2% of the plane. 

But there's not just one densest packing!  You can turn the smoothed octagons while keeping the density the same!  And that's what you see here. 

Notice that the centers of the octagons move as we turn them.  This makes me a bit sea-sick.  But the math is beautiful.

There's a lot more to say about this, and I said it here:

https://golem.ph.utexas.edu/category/2014/09/a_packing_pessimization_proble.html

I recommend this paper, too:

• Yoav Kallus, Least efficient packing shapes, http://arxiv.org/abs/1305.0289.

#spnetwork arXiv:1305.0289 #geometry #packing  ___

posted image

2014-09-22 18:28:03 (86 comments, 34 reshares, 120 +1s)Open 

More 2nd laws of thermodynamics

The 2nd law of thermodynamics says that entropy increases.   Entropy is a way to measure how disordered or 'randomized' things are.  So very roughly, the 2nd law says that things get messed up.

Amazingly, we can make this quite precise and use it to figure out answers to practical problems.   It's a long story.   But one small step is to combine this law with conservation of energy.  If we do this, we can see that something called 'free energy' decreases.

In physics and chemistry, free energy doesn't mean energy that you can get for free.   It means energy that's not in the form of heat - energy that is not  randomized like the wiggling of atoms.

More precisely, free energy is

E - TS

where E is energy, T is temperature and S is entropy.  It turns out that TSis heat ene... more »

More 2nd laws of thermodynamics

The 2nd law of thermodynamics says that entropy increases.   Entropy is a way to measure how disordered or 'randomized' things are.  So very roughly, the 2nd law says that things get messed up.

Amazingly, we can make this quite precise and use it to figure out answers to practical problems.   It's a long story.   But one small step is to combine this law with conservation of energy.  If we do this, we can see that something called 'free energy' decreases.

In physics and chemistry, free energy doesn't mean energy that you can get for free.   It means energy that's not in the form of heat - energy that is not  randomized like the wiggling of atoms.

More precisely, free energy is

E - TS

where E is energy, T is temperature and S is entropy.  It turns out that TS is heat energy.  So, E - TS is the energy that's not in the form of heat. 

It's obvious that if the energy E doesn't change and the entropy S goes up, the free energy E - TS goes down... at least if the temperature is positive!  

So, assuming conservation of energy, this is another way to say the 2nd law of thermodynamics: free energy decreases.   Energy in the form of heat goes up, energy in other forms goes down.

There's a lot more to say about this old stuff - it's endlessly tricky and fascinating.  But here's something new. Together with some coauthors, +Stephanie Wehner (shown here) at the +Centre for Quantum Technologies has discovered a bunch more 2nd laws of thermodynamics, saying a bunch of other things, like free energy but different, also decrease!

For large physical systems, like a steam engine, these other 2nd laws are usually almost indistinguishable from the usual one.  But for small systems, like collections of a few atoms, they are really different.  So they could help us understand thermodynamics of microscopic objects.

Check out the paper - it's pretty readable, especially near the start:

•  Fernando G.S.L. Brandao, Michał Horodecki, Nelly Huei Ying Ng, Jonathan Oppenheim and Stephanie Wehner, The second laws of quantum thermodynamics, http://arxiv.org/abs/1305.5278.

Puzzle: I said that the free energy F = E - TS decreases as entropy increases if T > 0.  What about when T < 0?  What does this mean?

#spnetwork arXiv:1305.5278 #thermodynamics  ___

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2014-09-20 18:53:10 (45 comments, 23 reshares, 119 +1s)Open 

Packing octagons

The mathematician Stanislaw Ulam was a clever and versatile guy.  He worked in the Manhattan project and later helped invent the hydrogen bomb.  He invented the Monte Carlo method, where you use a computer to estimate the average value of a quantity by repeatedly sampling it in a random way.  After a conversation over spaghetti with the Italian physicist Fermi, they discovered chaos in a computer experiment and wrote the famous Fermi–Pasta–Ulam paper, in which they named the food as a coauthor.  But he also did amazing work in mathematical logic, combinatorics, and more.

He made a lot of great conjectures.  His friend the Italian mathematician Gian-Carlo Rota claimed this was because he was too lazy to prove theorems.

One conjecture was this: spheres are the worst convex shape for packing!

Let me explain.  Ashape in ... more »

Packing octagons

The mathematician Stanislaw Ulam was a clever and versatile guy.  He worked in the Manhattan project and later helped invent the hydrogen bomb.  He invented the Monte Carlo method, where you use a computer to estimate the average value of a quantity by repeatedly sampling it in a random way.  After a conversation over spaghetti with the Italian physicist Fermi, they discovered chaos in a computer experiment and wrote the famous Fermi–Pasta–Ulam paper, in which they named the food as a coauthor.  But he also did amazing work in mathematical logic, combinatorics, and more.

He made a lot of great conjectures.  His friend the Italian mathematician Gian-Carlo Rota claimed this was because he was too lazy to prove theorems.

One conjecture was this: spheres are the worst convex shape for packing!

Let me explain.  A shape in Euclidean space is convex if for any pair of points in this shape, the line segment connecting them also lies in this shape.  Take any 3-dimensional convex shape.  The Ulam conjecture says you can pack space with copies of this this shape with a density that's at least 74.048% - the density for spheres.

It hasn't been proved, but no counterexample is known.

What about the same question in 2 dimensions?  Is the circle the worst shape for packing the plane?  Here are some fun puzzles that Greg Egan and I came up with yesterday:

Puzzle 1:  The picture shows a famous way of packing regular octagons.  What is the density of this packing?

Puzzle 2: Does this density exceed the maximum density for circles, 90.68997%? 

Puzzle 3: If not, can you find a denser way to pack regular octagons?  What's the best you can do?  Can you get a density of more than 90.68997%?

Puzzle 4: In this post, what did I say that was false?  Correct it.

The picture here was created by Turmamataplicada:

https://commons.wikimedia.org/wiki/File:Truncated_Square_Tiling.JPG

#geometry  ___

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2014-09-19 00:20:07 (19 comments, 5 reshares, 106 +1s)Open 

I'm leaving Singapore today.  This wall painting in Chinatown, modeled after a classical Chinese painting, captures a bit of what I like about the place.   It's a mix of old and new, East and West. 

Last weekend, Lisa and I saw a Chinese opera - part of a free series in Hong Lim Park.  Chinese opera used to be really popular in Singapore, with the stars being the equivalent of pop idols today.  Now its appeal is dwindling, but there was still a big crowd - and some old guys were punching their fists in the air when the star-crossed lovers finally triumphed in the end.  It was set in the Ming Dynasty, and featured an emperor who snuck out of the palace and wound up marrying a peasant girl.  I enjoyed it a lot more than I expected.  Why?  Because big computer screens showed translations of the lyrics into English!  Without that, I might have roughly followed the plot, but Iwouldn'... more »

I'm leaving Singapore today.  This wall painting in Chinatown, modeled after a classical Chinese painting, captures a bit of what I like about the place.   It's a mix of old and new, East and West. 

Last weekend, Lisa and I saw a Chinese opera - part of a free series in Hong Lim Park.  Chinese opera used to be really popular in Singapore, with the stars being the equivalent of pop idols today.  Now its appeal is dwindling, but there was still a big crowd - and some old guys were punching their fists in the air when the star-crossed lovers finally triumphed in the end.  It was set in the Ming Dynasty, and featured an emperor who snuck out of the palace and wound up marrying a peasant girl.  I enjoyed it a lot more than I expected.  Why?  Because big computer screens showed translations of the lyrics into English!  Without that, I might have roughly followed the plot, but I wouldn't have gotten the jokes.

I'm going back to Riverside to teach.  I've got a light teaching load this fall, just grad-level real analysis (the first quarter of a 3-part course) and my seminar - where I'll take the work my grad students have been doing on network theory and put it together into a nice story.  I'll try to write lecture notes in the form of blog articles, but I find that fun and relaxing.  So, I'll be able to put some energy into the talk I'm giving this December at NIPS, the Neural Information Processing Seminar, a big annual conference on neural networks.  I want to talk about El Niño prediction, climate networks and machine learning.  But I've got a lot to learn, especially about machine learning.  The Azimuth Code Project team want to carry out some computer experiments in that direction.  It should be fun, as long as I resign myself to giving a talk that's just "work in progress", not completed and polished.

I'll do some heavier teaching in  the winter quarter, but the spring will be a non-teaching quarter.  This seems awfully cushy, but my department chair noticed I'd taught too much last year - more than I'm paid to do!   And in the spring, I'll be helping run two workshops.  One is on information and entropy in biology, at the National Institute for Mathematical and Biological Synthesis, in Knoxville Tennessee.   The other is on network theory, at the Institute for Scientific Interchange, in Torino Italy.

Then in June I'll come back to work at the CQT in Singapore!

Sorry to bore you with personal details, but when your life is boxed up and ready to move, you tend to think about the future. 

#singapore  ___

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2014-09-18 04:11:22 (43 comments, 35 reshares, 164 +1s)Open 

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  ___

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2014-09-17 02:58:44 (55 comments, 42 reshares, 140 +1s)Open 

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, 87 reshares, 302 +1s)Open 

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, 124 +1s)Open 

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  ___

posted image

2014-09-13 06:34:55 (166 comments, 27 reshares, 177 +1s)Open 

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!

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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.

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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...

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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.

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The picture is from here:

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

Buttons

A special service of CircleCount.com is the following button.

The button shows the number of followers you have directly in a small button. You can add this button to your website, like the +1-Button of Google or the Like-Button of Facebook.






You can add this button directly in your website. For more information about the CircleCount Buttons and the description how to add them to another page click here.

John BaezCircloscope