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John Baez has been shared in 346 public circles

You can see here the 50 latest shared circles.
If this is your profile, you can check your dashboard to see all shared circles you have been included.

AuthorFollowersDateUsers in CircleCommentsReshares+1Links
Neil Bailey4,132If you received the notice you are in this circle, then well done.If you would like to be included in the next Circle Share, you only have to do these simple steps:1 - Include me in your circles (If you haven't already)2 - Share the circle (Publicly) - (cc) me in the comments on the share and I can add you to the next circle immediately.  Otherwise I may not notice your activity!3 - Add +1 to the post.4 - Leave a comment if you like.5 - Add the circle or just check it out.Follow your dreams, Share and Be Shared.More you share more you get! :)Thanks!#circles   #circleshare   #sharedcircle #circlesharing #followers #social #sharedcircles  #sharedpubliccircles #circleshared   #sharedcircleoftheday   #addmetoyourcircles #awesomepeople   #circlecount   #newfollowers #googleplus #meetingpeoplecircle2014-12-15 06:41:38487126
exceptional circles12,986A nice circle for today2014-12-13 20:57:4749889105118
Refurio Anachro4,796Engagers circle October + November. Hi there, you fantastic crowd! Not only have i been feverishly busy of late, and christmas upcoming, on top of that they had strangled my uplink for a week now. So i owe you all a pack of mathy posts, comments daft and curious, and many one-click salutes and appreciations. The people in this circle are friends, all of them curious and critical readers, and many writing original and genuinely interesting stuff. I'm sure everybody here is worthy of consideration to be a friend of yours. Have a look, add us now!Alexander Grothendieck has passed away. There have been some nice obituaries, but maybe not the ones in the press. I should go an collect some, maybe post them with a circle of people who appeared to care...https://plus.google.com/+RefurioAnachro/posts/K4xZTgT2Vf6Apparently, topological sorting can be done using a "normal" sorting algorithm. Do you know more? Or want to? Drop me a note!https://plus.google.com/+RefurioAnachro/posts/EkVfWkgCwik+David Roberts' call for participation, write maths in short words! Maybe just the right occupation for the months where the letters have fallen from the words, to rest beneath sentences in proofwood forest.https://plus.google.com/+RefurioAnachro/posts/UiaXnWuGcDQOn 1+2+3... = -1/12, following up Diagram 20 below. If you found other popular accounts lacking, maybe here's something differently too short for you. Thanks again +Stam Nicolis for prodding, me, who wouldn't see otherwise.https://plus.google.com/+RefurioAnachro/posts/jXAi8a7Gj52A poem by Marion D. Cohen, poet, mathematician.https://plus.google.com/+RefurioAnachro/posts/92PCBKoR7HwDiagram 20: X-Rays of the zeta functionhttps://plus.google.com/+RefurioAnachro/posts/Mux7WctktvoReshared a nice little illustration by +Owen Maresh https://plus.google.com/+RefurioAnachro/posts/DGSdqH5hEDmSeptember engagers circle:https://plus.google.com/+RefurioAnachro/posts/USZFSS95xfn+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  +Spherical Reflections), or got into a discussion with me. Thank you!#engage2014-12-09 10:38:3790000
Rogerio Manica29,965Engagers #10.I am sharing this circle of recent engagers to celebrate 30,000 followers, which will happen sometime later today. Even tough I was away for many weeks the numbers went only up and I thank circle sharers for keeping me inside their circles. I would also like to thank the people that have engaged with my wife's blog https://havefunwithkids.wordpress.com/ She has finally reached her first 100 followers. It is not an easy task to start a blog at this time.2014-12-09 09:02:213084414
Sharon Caroline3,867Hello my friends, good morning/evening for you all!Boost Your visibility On Google+! Add them all!Shared and be shared. :)Thank you for sharing and promoting this. :)#circleshare #sharedcircles #sharingcircles #sharedcircleoftheday2014-12-03 10:28:12501001
Becky Collins19,437Diet 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 ?2014-12-03 07:12:09426000
Rogerio Manica29,602Engagers #9Most of the people in this circle have engaged with posts from my wife's blog: https://havefunwithkids.wordpress.com/ I have also added engagers of recent photos and people that have included me in their recent shared circles. Engagers 10 will be a brand new circle that I am starting from scratch and will be circulated when I reach 30k followers.Thank you all for the continuous support.2014-12-02 11:32:54442231948
Richard Green88,785Engagers Showcase Circle, December 1, 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 the posts listed below, in the form of +1s, comments and reshares. I have not posted much in the last couple of months because I have been too busy, and so it has been a long time since the last reshare of the circle.Everyone mentioned below is also included in the circle.Millcreek Canyon Vista (reshared from +Tom Malloy)https://plus.google.com/101584889282878921052/posts/HuXLKw4GBwjAvoiding the unavoidablehttps://plus.google.com/101584889282878921052/posts/TnW3pTWt6d7Hydrangea flowershttps://plus.google.com/101584889282878921052/posts/3LDn2js6pWpMicroscopic Victorian arthttps://plus.google.com/101584889282878921052/posts/ieybEmL7tUCApproximating e using the digits 1–9https://plus.google.com/101584889282878921052/posts/W5E6HyihSuY“Vertebral 03 – Pendant Lampshade” by cordycepthttps://plus.google.com/101584889282878921052/posts/eSo9svRbLapCentred polygonal numbershttps://plus.google.com/101584889282878921052/posts/QowshFUnPZ2Reinventing the wheel: Reuleaux polygonshttps://plus.google.com/101584889282878921052/posts/gDxTM5Ko8hbSunrise at Maroon Lake (reshared from +Jason Hill)https://plus.google.com/101584889282878921052/posts/hvWMqo1HwvVSchmidt arrangementshttps://plus.google.com/101584889282878921052/posts/eM3adto6nsj“Dream Creatures” by Elido Turcohttps://plus.google.com/101584889282878921052/posts/ckjru8sN6AG“The Awakening III—Rebirth” by Luc Railhachttps://plus.google.com/101584889282878921052/posts/KyNg9DD4YnXPoincaré and topologyhttps://plus.google.com/101584889282878921052/posts/bmnd2URRAsfLytham St Annes (reshared from +Paul Haworth)https://plus.google.com/1015848892828789210522014-12-01 22:15:07443146111188
Rajani Vijaya0My Awesome CircleThis is my circle of the day :)Add people in this circle to increase your follower. Enjoy it!#CircleShare#CircleSharing#Circles#CircleOfTheDay#SharedCircles#Shared#SharedPublicCircles#SharedCircleOfTheDay#Engagers#ShareCircle#SocialMedia#EngagersCircle#Share#Google#SharingCircles#ADD#Friends#SEOtips#Website#Marketing#SEOmarketing#Google#WebDesign#SocialMedia#DigitalMarketing#Business#LocalSEO#OnlineMarketing#Search#SocialMediaMarketing#SEOservices#ContentMarketing#Blogging#SEM#WebDevelopment#SEOStrategy2014-11-28 01:58:10480012
Ryan Johnson19,323This circle contains people who are very active on Google+If you received a notification, please reshare to your circlesIf 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#circle #Sharedcircles #circleshare #publiccircle #followme #public #sharedpubliccircles #circleoftheday #circleall #circlecircle #circleday #Colombia  2014-11-27 11:17:05479849
Atanas Georgiev Atanasov2,751SCIENCE AND TECHNOLOGY FRIENDS : Circle V.1.11. Link to my own scientific research topic: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4212005/  To be included in future circle-editions, please ENGAGE: add me to your own circles/+1/re-share/comment on the original circle-post, or on some of the other science-related posts on my wall (this is needed since the number of people that could be circled is limited from Google – and therefore I am forced to keep included just the most active users). Currently featured science-related GOOGLE PLUS post:https://plus.google.com/115938908270684192009/posts/8LU6LVz75jxI 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   #nutrition    #ScienceSunday   #Sundayscience   #Science   #Research   #Tech #GameTech   #GameTechnology    #Microsoft #MicrosoftResearch    #innovation   #Inflammation   #Brain #mindcontrol   #photography   #tech   #socialmedia   #googleplus #naturalproduct   #artists   #foodies   #cars   2014-11-27 06:30:22451429
Rogerio Manica29,448Engagers #8Here comes my version 8 circle of engagers. Recently I had posted only a few entries related to our recent trip to the UK which were published in my wife's blog (http://havefunwithkids.wordpress.com/). I have added the people that have engaged with these posts. I have also posted a couple of videos of my kids in their school performance (http://youtu.be/brmuStXIe88) (http://youtu.be/IiL5Jh9Ncnk).2014-11-25 04:03:53302161029
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 12:58:0746211860145
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

Activity

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

35
comments per post
29
reshares per post
121
+1's per post

2,412
characters per posting

Top posts in the last 50 posts

Most comments: 171

posted image

2014-11-03 18:18:00 (171 comments, 17 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: 137

posted image

2014-11-22 17:43:26 (52 comments, 137 reshares, 260 +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 »

Most plusones: 303

posted image

2014-10-09 20:07:55 (145 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

posted image

2014-12-18 16:38:45 (24 comments, 13 reshares, 113 +1s)Open 

Sky scraper

+Yonas Kidane wanted to see how big the Empire State Building would look if it were sitting on the asteroid 67P/Churyumov–Gerasimenko.  So, he figured it out and made this image, based on a photo taken by the Rosetta space probe.  The hard part was getting the shadow right.

His original post is here:

https://plus.google.com/u/0/117534618780996601977/posts/5rgkKpL6BqZ

This office building has a great view... but the commute is terrible.

Sky scraper

+Yonas Kidane wanted to see how big the Empire State Building would look if it were sitting on the asteroid 67P/Churyumov–Gerasimenko.  So, he figured it out and made this image, based on a photo taken by the Rosetta space probe.  The hard part was getting the shadow right.

His original post is here:

https://plus.google.com/u/0/117534618780996601977/posts/5rgkKpL6BqZ

This office building has a great view... but the commute is terrible.___

posted image

2014-12-17 16:47:20 (28 comments, 15 reshares, 62 +1s)Open 

The California Drought

Down here in Southern California, we've had three good rains since the summer.  Up north, they've gotten even more!   In the first storm, ending December 3rd, San Francisco got more rain than they did all last year!   They got 9.4 centimeters of rain  in four days, compared to just 8.6 in 2013. 

But we'd need a lot more rain to break the drought.   It will take about 11 trillion gallons of water - 42 cubic kilometers! - to fully recover from the drought.  That's what researchers at NASA say, based on satellite data including measurements of the Earth's gravitational field, which depends on how much groundwater there is.

They say that since 2011, the Sacramento and San Joaquin river basins have decreased in volume by four trillion gallons of water each year - 15 cubic kilometers.   About two-thirds of the loss is dueto deplet... more »

The California Drought

Down here in Southern California, we've had three good rains since the summer.  Up north, they've gotten even more!   In the first storm, ending December 3rd, San Francisco got more rain than they did all last year!   They got 9.4 centimeters of rain  in four days, compared to just 8.6 in 2013. 

But we'd need a lot more rain to break the drought.   It will take about 11 trillion gallons of water - 42 cubic kilometers! - to fully recover from the drought.  That's what researchers at NASA say, based on satellite data including measurements of the Earth's gravitational field, which depends on how much groundwater there is.

They say that since 2011, the Sacramento and San Joaquin river basins have decreased in volume by four trillion gallons of water each year - 15 cubic kilometers.   About two-thirds of the loss is due to depletion of groundwater beneath California's Central Valley.

http://www.jpl.nasa.gov/news/news.php?feature=4412

Some scientists studying tree rings have claimed that as measured by the Palmer Drought Severity Index - a measure of precipitation and evaporation - this is the worst drought California has seen in 1200 years:

http://www.livescience.com/49029-california-drought-worst-ever.html

I would like to see the evidence, and the definitions involved - but I haven't seen them yet.

#climate  ___

posted image

2014-12-16 18:37:18 (23 comments, 23 reshares, 100 +1s)Open 

Soon the famous mathematical physicist Barry Simon will be publishing a "comprehensive course in analysis" - 3000 pages of material, in 5 volumes:

Real Analysis
Basic Complex Analysis
Advanced Complex Analysis
Harmonic Analysis
Operator Theory

You can see a table of contents here. 

If you know Reed and Simon's series Methods of Modern Mathematical Physics you'll know why I'm excited.  The style of those books is simultaneously elegant yet friendly.  They put in just enough detail to make the proofs easy to follow... but include lots of side-remarks that explain the point of what's going on, and the history of the subject.

When I was a junior at Princeton I had to choose an advisor for my senior thesis.  I wanted to work on logic and quantum mechanics.  So I asked Simon Kochen if he would be my advisor.  He'sa log... more »

Soon the famous mathematical physicist Barry Simon will be publishing a "comprehensive course in analysis" - 3000 pages of material, in 5 volumes:

Real Analysis
Basic Complex Analysis
Advanced Complex Analysis
Harmonic Analysis
Operator Theory

You can see a table of contents here. 

If you know Reed and Simon's series Methods of Modern Mathematical Physics you'll know why I'm excited.  The style of those books is simultaneously elegant yet friendly.  They put in just enough detail to make the proofs easy to follow... but include lots of side-remarks that explain the point of what's going on, and the history of the subject.

When I was a junior at Princeton I had to choose an advisor for my senior thesis.  I wanted to work on logic and quantum mechanics.  So I asked Simon Kochen if he would be my advisor.  He's a logician who is famous for the Kochen-Specker theorem - a result ruling out large classes of 'hidden variable theories', which seek to explain quantum mechanics in terms of classical mechanics.

He immediately asked me: "Do you know the spectral theorem?" 

I was taken aback.  I said no.   He said he wouldn't work with me.

This pissed me off.  What was so great about this theorem that Kochen wouldn't work with me if I didn't already know it?   So over the summer I got ahold of Reed and Simon's Methods of Modern Mathematical Physics - the first volume to be precise - and learned the spectral theorem.

It's a fundamental theorem that says how everything you can observe in quantum mechanics has a 'spectrum' of allowed values - like the spectrum of lines of light from the Sun, which are the allowed values of energies emitted as electrons hop from one energy level to another.  It's a great theorem.

I asked Edward Nelson if he would be my advisor.   He was another guy famous for his work on logic and quantum mechanics.  I told him what I wanted to work on: a blend of quantum mechanics and computability theory.  I wanted to see what quantum systems are able to compute.  He asked me a few questions about how I planed to do this.  I explained my approach.  He said no, he would not be interested.  

Only later did I learn how difficult it is to advise students who have strong views on what they want to work on. Usually they bite off more than they can chew.  So now I sympathize with him.

I then asked John Burgess in the philosophy department if he would be my advisor, and he said yes.   We never talked very much, but he was very helpful.   He knew about 'descriptive set theory', the study of different kinds of sets of real numbers, and how logically complicated they are.  It turns out that I'd been trying to reinvent this subject... so I was able to simplify my work a lot.  I proved a nice result.  Edward Nelson read my thesis, and he caught some small mistakes.  I corrected them and published it.  So everything worked out fine.

However, I got a bit tired of logic.   Reading Reed and Simon's books got me more interested in analysis and mathematical physics, so I took Elliot Lieb's grad course on 'functional analysis' in my senior year.   That got me even more interested, so I did my PhD thesis on analysis and quantum field theory - with Irving Segal, who was Edward Nelson's advisor.

I got hired at U.C. Riverside thanks to my work on analysis, and though I don't do much analysis anymore, I'm still considered an 'analyst' by my department... which means that they make me teach intro grad courses on real analysis.  And today I'm having a review session for my students in that course!

This might not have happened if I hadn't read Reed and Simon's books.

Learn some theorems!

https://en.wikipedia.org/wiki/Kochen-Specker_theorem

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

Or read my thesis:

http://math.ucr.edu/home/baez/recursivity.pdf

#analysis  ___

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2014-12-15 18:14:08 (41 comments, 16 reshares, 86 +1s)Open 

Delightful demicubes

If you take every other corner of a cube, you get the corners of a demicube

In 3 dimensions a demicube is just a regular tetrahedron!    So you get two tetrahedra in a cube, as shown here.  Together they form a stellated octahedron.  In other words, you can also get this shape by taking a regular octahedron and sticking a tetrahedron on each of its faces - getting a kind of 3-dimensional star!    It makes a great Christmas tree ornament.

What's a demicube in 4 dimensions?   A 4-dimensional cube has 2^4 = 16 corners, so the demicube has 8.  The cool part, special to 4 dimensions, is that all these corners point at right angles to each other, as viewed from the center of the cube!

If the corners of the 4-dimensional cube are

(±1, ±1, ±1, ±1)

then we can get a demicube by takingthose with an e... more »

Delightful demicubes

If you take every other corner of a cube, you get the corners of a demicube

In 3 dimensions a demicube is just a regular tetrahedron!    So you get two tetrahedra in a cube, as shown here.  Together they form a stellated octahedron.  In other words, you can also get this shape by taking a regular octahedron and sticking a tetrahedron on each of its faces - getting a kind of 3-dimensional star!    It makes a great Christmas tree ornament.

What's a demicube in 4 dimensions?   A 4-dimensional cube has 2^4 = 16 corners, so the demicube has 8.  The cool part, special to 4 dimensions, is that all these corners point at right angles to each other, as viewed from the center of the cube!

If the corners of the 4-dimensional cube are

(±1, ±1, ±1, ±1)

then we can get a demicube by taking those with an even number of minus signs.  That gives these four:

(1,  1,  1,  1)
(1,  1, -1, -1)
(1, -1,  1, -1)
(-1, 1,  1, -1)

and their negatives.  And if you know your math, you can check that the 'dot product' of any two of the vectors I listed is zero!   That means they all point at right angles to each other.

In fact, it means that the demicube in 4 dimensions is just the 4-orthoplex: the 4d analogue of an octahedron!  We usually make a 4-orthoplex by taking these four vectors:

(1, 0, 0, 0)
(0, 1, 0, 0)
(0, 0, 1, 0)
(0, 0, 0, 1)

and their negatives, and using those as corners.  Each pair of the vectors listed is at right angles to each other.  But the corners of a demicube work just as well, giving a 4-orthoplex that's twice as big in every direction, and rotated.

I don't know anything exciting to say about demicubes in 5 or 6 dimensions.  But in 7 dimensions something very nice happens! 

A 7-cube has 2^7 = 128 corners, so a 7-demicube has 64.  The 7-dimensional analogue of a tetrahedron, called a 7-simplex, has 8 corners.   Notice: 64 is 8 times 8.

Can we take a 7-demicube and partition its corners into 8 sets of 8, each set being the corners of a 7-simplex?  Yes we can! 

Greg Egan figured out how here:

https://golem.ph.utexas.edu/category/2014/12/integral_octonions_part_11.html#c047895

The trick involves the Fano plane. This is a little gadget with 7 points and 7 lines, where any two points lie on a single line and any two lines intersect in a single point.  If you haven't ever seen the Fano plane, what I just said is enough to draw it, so that might be fun to try... but beware: some of the lines will need to look curved if you draw it on an ordinary sheet of paper! 

So, in 7 dimensions there's a picture like the one here, but with 16 different simplexes stuck inside a cube, instead of just two.  That would be fun to see!

The next opportunity to partition the corners of a cube into simplices occurs in 15 dimensions. 

Puzzle 1: can you take the set of 15-bit strings and find 16 of them, each pair of which agrees in exactly 7 places?  

I don't know the answer.  But if you succeed, you'll have a 15-simplex inside the 15-cube, since you've taken the corners of a 15-cube:

(±1, ..., ±1)
 
and found 16 of them, any pair of which has a dot product of -1.  This is exactly what you need for the corners of a 15-simplex! 

Moreover, since any pair of your bit strings disagrees in an even number of places (namely 8), your simplex will actually lie in a demicube!

If this puzzle was too easy, move on to:

Puzzle 2: can you take the set of 15-bit strings and partition it into sets of 16, such that any two strings in a given subset agree in exactly 7 places? 

If so, you'll have found a way to partition the vertices of a 15-dimensional demicube into 15-dimensional simplices!   2048 of them, in fact.

Some people actually get paid to work on this stuff.  They're called mathematicians... or more precisely, experts on incidence geometry and codes.

#geometry  ___

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2014-12-15 05:38:22 (16 comments, 1 reshares, 31 +1s)Open 

Late night music

Richie Hawtin's 1998 album Artifakts starts with three linked tunes, all hypnotic - but this, the third, is the real star. 

Richie Hawtin is making largely beautiful music, and sometimes he's making it eat into your brain, too. - Musical Express

There are three big events in this piece:

1) the start, when an icy-cold, three note melody wafts into view like a wistful signal from a long-dead alien civilization;

2) the moment one minute later when some slinky, crisp percussion enters;

3) the moment around 2:30 when the more lively Roland TB-303 synthesizer line appears.

The 303 is a famous instrument.  It was made for only 18 months in the 1980s, but later it became popular in acid house music, because you can get it to play a repeating pattern and then turn knobs to adjust the sound in a veryf... more »

Late night music

Richie Hawtin's 1998 album Artifakts starts with three linked tunes, all hypnotic - but this, the third, is the real star. 

Richie Hawtin is making largely beautiful music, and sometimes he's making it eat into your brain, too. - Musical Express

There are three big events in this piece:

1) the start, when an icy-cold, three note melody wafts into view like a wistful signal from a long-dead alien civilization;

2) the moment one minute later when some slinky, crisp percussion enters;

3) the moment around 2:30 when the more lively Roland TB-303 synthesizer line appears.

The 303 is a famous instrument.  It was made for only 18 months in the 1980s, but later it became popular in acid house music, because you can get it to play a repeating pattern and then turn knobs to adjust the sound in a very fluid and lively sort of way.   Richie Hawtin made great use of it in his early work, and I love how he uses it. 

There are a lot of other smaller excitements besides these three big events, but the piece is mainly about a cool, enigmatic mood and a dreamy suspension of time.  

There are two kinds of time here: one in which the melody spirals around and around, seeming to go nowhere - and the other, where it slowly changes, but mostly in timbre.  So, this is very good late night music for me, when I want something shimmering in the background while I work, gently energizing me.

If you like this piece, you may like the whole triptych of which it's the climax:

https://www.youtube.com/watch?v=mfZdTk5D9ew

For a picture of the TB-303, and more, see:

http://www.vintagesynth.com/roland/303.php

For a documentary on this instrument, see:

https://www.youtube.com/watch?v=omHUR3R0Qqw

#favoritemusic___

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2014-12-13 16:03:14 (71 comments, 33 reshares, 123 +1s)Open 

Context matters!

The two disks are exactly the same in every way. Only their surroundings differ. 

This is a metaphor for many things in life.  What's the best example where the context of an event changed your perception of it?

For more great visual effects by the psychologist Akiyoshi Kitaoka, go here:

http://www.ritsumei.ac.jp/~akitaoka/index-e.html#allpages

Context matters!

The two disks are exactly the same in every way. Only their surroundings differ. 

This is a metaphor for many things in life.  What's the best example where the context of an event changed your perception of it?

For more great visual effects by the psychologist Akiyoshi Kitaoka, go here:

http://www.ritsumei.ac.jp/~akitaoka/index-e.html#allpages___

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2014-12-12 18:16:50 (53 comments, 47 reshares, 143 +1s)Open 

See the small squares divided into 2 black squares and 2 white ones?  These are confusing your poor eyes.

This is not a spiral

It’s a series of concentric circles.

More: http://makezine.com/2010/04/04/this-is-not-a-spiral/___See the small squares divided into 2 black squares and 2 white ones?  These are confusing your poor eyes.

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2014-12-11 18:54:31 (67 comments, 30 reshares, 109 +1s)Open 

The Salt Pit

On Dec. 31, 2003, I took a bus from Germany to Macedonia. When we arrived, my nightmare began. Macedonian agents confiscated my passport and detained me for 23 days. I was not allowed to contact anyone, including my wife.

At the end of that time, I was forced to record a video saying I had been treated well. Then I was handcuffed, blindfolded and taken to a building where I was severely beaten. My clothes were sliced from my body with a knife or scissors, and my underwear was forcibly removed. I was thrown to the floor, my hands pulled behind me, a boot placed on my back. I was humiliated.

Eventually my blindfold was removed, and I saw men dressed in black, wearing black ski masks. I did not know their nationality. I was put in a diaper, a belt with chains to my wrists and ankles, earmuffs, eye pads, a blindfold and a hood. I was thrown... more »

The Salt Pit

On Dec. 31, 2003, I took a bus from Germany to Macedonia. When we arrived, my nightmare began. Macedonian agents confiscated my passport and detained me for 23 days. I was not allowed to contact anyone, including my wife.

At the end of that time, I was forced to record a video saying I had been treated well. Then I was handcuffed, blindfolded and taken to a building where I was severely beaten. My clothes were sliced from my body with a knife or scissors, and my underwear was forcibly removed. I was thrown to the floor, my hands pulled behind me, a boot placed on my back. I was humiliated.

Eventually my blindfold was removed, and I saw men dressed in black, wearing black ski masks. I did not know their nationality. I was put in a diaper, a belt with chains to my wrists and ankles, earmuffs, eye pads, a blindfold and a hood. I was thrown into a plane, and my legs and arms were spread-eagled and secured to the floor. I felt two injections and became nearly unconscious. I felt the plane take off, land and take off. I learned later that I had been taken to Afghanistan.

Khaled El-Masri wrote this back in 2005, and I added it to my collection of posts about the US-run torture program:

http://www.math.ucr.edu/home/baez/torture/

In Afghanistan, he was interrogated in the Salt Pit, a CIA-run 'black site'.  We are now learning more about this place.

There, I was beaten again and left in a small, dirty, cold concrete cell. I was extremely thirsty, but there was only a bottle of putrid water in the cell. I was refused fresh water.

That first night I was taken to an interrogation room where I saw men dressed in the same black clothing and ski masks as before. They stripped and photographed me, and took blood and urine samples. I was returned to the cell, where I would remain in solitary confinement for more than four months.

He was interrogated, force-fed, lost 60 pounds.  His requests to see a lawyer were ignored.  Eventually he was blindfolded, handcuffed, chained to an airplane seat, and taken to Albania, where he was left in the mountains.  Eventually he made it back to his home in Germany. 

His crime?  His name resembled that of the terror suspect Khalid al-Masri.

In 2006 as U.S. Federal District Judge dismissed a lawsuit he filed against the CIA, stating that a public trial would "present a grave risk of injury to national security."  A Court of Appeals also dismissed the case, and in 2008 so did the U.S. Supreme Court.

In the newly released U.S. Senate report,  a supervisor is quoted as saying the Salt Pit was "good for interrogations because it is the closest thing … to a dungeon."

Guards and interrogators tiptoed through the darkness, carrying headlamps to count detainees packed into two dozen cells. Their lights illuminated prisoners hanging from overhead bars, next to buckets on the floor to catch their waste. One hung there for 17 days.

Another detainee "looked like a dog that had been kenneled," wrote an interrogator. "When the doors to their cells were opened, they cowered," according to CIA documents quoted in the report.

Indeed, reports of sleep and sensory deprivation; of nudity and unhealthful, unsanitary food; of cold showers and ice buckets; and of rough takedowns and mock executions never were reported to supervisors.

http://www.latimes.com/world/afghanistan-pakistan/la-fg-torture-salt-pit-20141210-story.html

The moral?  I don't have a moral.  But it's curious: anyone in the US who cared has known the rough outlines of what we've been doing for at least 12 years.  Read my torture blog!   Yet now some people are acting surprised.   Where were they back then?___

2014-12-10 13:06:42 (17 comments, 3 reshares, 80 +1s)Open 

My talk on climate networks is coming up in one hour! There are 2100 people at this conference - six hotels are completely sold out - and I'm giving the only talk from 9 to 10 am, in a HUGE room, so everyone can attend.

The pressure is on!  Luckily, while I'm shy and quiet in ordinary life, a big audience transforms me into a showman - I like being the center of attention.  I only learned this after teaching for several years.  At first I hated teaching - standing in front of a crowd of bored teenagers talking about calculus.  Then I realized this was my big chance to entertain people, get them interested in the things I love, and - let's be frank - show off. 

Then came the internet, and I realized I could entertain the world from the privacy of my own home.  But there's nothing like a crowd of hundreds to make me excited.  One man's stage fright is another'sadrena... more »

John Baez on Networks in Climate Science So we've all been in the position: it's the middle of the conference, you've spent so much time talking late into the evenings with colleagues (possibly well-lubricated talking) and you're not sure whether to wake up and go to the morning's opening talk. Today at NIPS 2014 John Baez (who I count as a friend, so I do have an interest) will be giving the first talk of the morning on Networks in Climate Science, and I reckon if you're at the conference there's probably three good reasons to attend: (i) he always tends to have lots of solid, thought provoking material in is talks; (ii) he's a very engaging speaker and (iii) being outside the established ML community he has an interesting perspective on some of the issues.

So don't touch that snooze button!___My talk on climate networks is coming up in one hour! There are 2100 people at this conference - six hotels are completely sold out - and I'm giving the only talk from 9 to 10 am, in a HUGE room, so everyone can attend.

The pressure is on!  Luckily, while I'm shy and quiet in ordinary life, a big audience transforms me into a showman - I like being the center of attention.  I only learned this after teaching for several years.  At first I hated teaching - standing in front of a crowd of bored teenagers talking about calculus.  Then I realized this was my big chance to entertain people, get them interested in the things I love, and - let's be frank - show off. 

Then came the internet, and I realized I could entertain the world from the privacy of my own home.  But there's nothing like a crowd of hundreds to make me excited.  One man's stage fright is another's adrenaline rush.

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2014-12-07 16:02:22 (34 comments, 40 reshares, 141 +1s)Open 

We're all in the gutter, but some of us are looking at the stars

This is the ceiling of the tomb of the famous Persian poet Hafez, who was born in the city of Shiraz in 1315, and died there in 1390. The current version of the tomb dates back only to 1935 and was designed by the French architect and archaeologist André Godard.  But the design is beautiful! 

There's a lot of fun stuff to see if you zoom in, but let's think about the star.

Puzzle 1: How many points do the stars here have?

Puzzle 2: How many different kinds of stars are there with this many points?

For example, there's just one kind of 5-pointed star, but two kinds of 7-pointed star.   There's a 7-pointed star with blunt points where you draw a line from each dot to the dot 2 after it, and one with sharp points where you draw a line fromeach... more »

We're all in the gutter, but some of us are looking at the stars

This is the ceiling of the tomb of the famous Persian poet Hafez, who was born in the city of Shiraz in 1315, and died there in 1390. The current version of the tomb dates back only to 1935 and was designed by the French architect and archaeologist André Godard.  But the design is beautiful! 

There's a lot of fun stuff to see if you zoom in, but let's think about the star.

Puzzle 1: How many points do the stars here have?

Puzzle 2: How many different kinds of stars are there with this many points?

For example, there's just one kind of 5-pointed star, but two kinds of 7-pointed star.   There's a 7-pointed star with blunt points where you draw a line from each dot to the dot 2 after it, and one with sharp points where you draw a line from each dot to the dot 3 after it.

Puzzle 3: How many different kinds of n-pointed stars are there?

Puzzle 4: How many of these are connected?

For example, there is no connected 6-pointed star.   If you take a regular hexagon and draw a line from each dot to the dot 2 after it, you get the traditional Star of David, which consists of two separate triangles.  If you draw a line from each dot to the dot 1 after it, you get the hexagon... you can decide if that counts as a star.  If you draw a line from each dot to the dot 3 after it, you get three straight lines meeting at the center... you can decide if that counts as a star.  And that's all you can get, if you're following the rules I have in mind!

You can get a higher-quality version of this image starting here:

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

It was put on Wikicommons by 'Pentocelo'.

Hafez was a Sufi, and his poems show that:

Change rooms in your mind for a day.
All the hemispheres in existence
Lie beside an equator
In your heart.

For more in translation, try:

http://peacefulrivers.homestead.com/hafiz.html___

posted image

2014-12-05 22:27:34 (39 comments, 19 reshares, 118 +1s)Open 

The beauty of the 24th dimension

My main hobby these days is working with Greg Egan on lattices.  Roughly, these are repeating patterns of points, like the centers of atoms in a crystal.  But you can study lattices in different dimensions - and a lot of fun happens in 24 dimensions!

If you look for the densest ways to pack spheres in different dimensions, you'll be led to some interesting lattices.  In 3 dimensions, the usual way of stacking oranges gives the D3 lattice: when you center your spheres at points of this lattice, each sphere touches 12 others.   This is known to be the densest packing of spheres in 3 dimensions.

In 4 dimensions the densest known sphere packing comes from the D4 lattice, where each sphere touches 24 others.

These D lattices are easy to build: you draw a higher-dimensional checkerboard withalte... more »

The beauty of the 24th dimension

My main hobby these days is working with Greg Egan on lattices.  Roughly, these are repeating patterns of points, like the centers of atoms in a crystal.  But you can study lattices in different dimensions - and a lot of fun happens in 24 dimensions!

If you look for the densest ways to pack spheres in different dimensions, you'll be led to some interesting lattices.  In 3 dimensions, the usual way of stacking oranges gives the D3 lattice: when you center your spheres at points of this lattice, each sphere touches 12 others.   This is known to be the densest packing of spheres in 3 dimensions.

In 4 dimensions the densest known sphere packing comes from the D4 lattice, where each sphere touches 24 others.

These D lattices are easy to build: you draw a higher-dimensional checkerboard with alternating red and black hypercubes, and put a dot in the middle of each red hypercube.

When you pack sphere using these D lattices in higher and higher dimensions, there's more and more room left over between your spheres.  And when you get to 8 dimensions, something funny happens!  There's so much room left that you can slip in another whole set of spheres packed the same way! 

So, you can double the density with this improved lattice.  It's called the E8 lattice and you see it as a peak in the graph here.  With this lattice, each sphere touches 240 others.   Nobody has proved that this is the densest sphere packing possible in 8 dimensions.  But in 2009, Henry Cohn and Abhinav Kumar proved that no other packing can beat its density by a factor of more than

1.00000000000001

So, I'm willing to bet that it's the best.

What I really like about 8 dimensions is that there's an 8-dimensional number system where you can add, subtract, multiply and divide. 

I'm sure you know how a 1-dimensional ruler is labelled by ordinary real numbers.  You can add, subtract, multiply and divide those.  If you try to do this trick in higher dimensions, you'll notice something weird: you can only do it in dimensions 1, 2, 4, and 8. 

In 2 dimensions you can use the complex numbers, and in 4 you can use the quaternions.  In 8 dimensions you can use the octonions, and that's where the game ends!   So the octonions are special.  They play a role in string theory - so if string theory ever turns out to be right, maybe the octonions will actually count as useful.  Right now they're just amazingly beautiful and lots of fun.

But back to lattices!  The simplest lattice lives in 1 dimension: it's the evenly spaced numbers on your ruler, called integers:

... -2, -1, 0, 1, 2, ...
 
You can add, subtract and multiply integers and get integers... but not divide them: that takes you out of the integers.

There are versions of the integers for complex numbers, quaternions and octonions too!  The Hurwitz integral quaternions form the D4 lattice that I mentioned earlier.  And the Cayley integral octonions form the E8 lattice.  It's actually the arithmetic of these integral octonions that fascinates me, more than the sphere packing business.

But as you can see from the graph, there's a really interesting mountain peak called the Leech lattice.  This gives the densest known way to pack spheres in 24 dimensions.   Nobody has proved it's the best - but Cohn and Kumar proved that no other packing in this dimension can beat its density by more than a factor of

1.00000000000000000000000000000165

It's a lot harder to describe the Leech lattice than the others I mentioned so far.  Each sphere touches 196,560 others... and the pattern is rather tricky.

But 24 is 3 times 8, so you might hope to build the Leech lattice from 3 copies of the E8 lattice... and you can!  But you need a fairly clever trick.  Various people have described this trick in different ways, but I like Greg Egan's the best.  I explain it here:

https://golem.ph.utexas.edu/category/2014/11/integral_octonions_part_9.html

It relies on a great feature of the E8 lattice.  You can rotate it in a way that turns every point by the same specific angle, and expand it by factor of sqrt(2), and this transformation maps the E8 lattice into itself.  Any way of doing this gives a way to build the Leech lattice. 

The graph of densities here is taken from Conway and Sloane's Sphere Packings, Lattices and Groups.  They take spheres of radius 1, work out the density of sphere centers, take its logarithm in base 2... and then add n(24-n)/96.  This is a parabola peaked at 12.  I find this touch a bit distracting.

There's a huge amount more to say about this graph, and all this stuff... but just ask questions if you want.

Here's the paper I mentioned:

• Henry Cohn and Abhinav Kumar, Optimality and uniqueness of the Leech lattice among lattices, http://arxiv.org/abs/math.MG/0403263.

#spnetwork arXiv:math.MG/0403263 #lattices #geometry  ___

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2014-12-01 16:54:34 (11 comments, 4 reshares, 47 +1s)Open 

Climate networks

Complex network theory is the study of large networks.  Can we use this to get new insights about the Earth's climate?  That's what I'm talking about next week at a conference called the Neural Information Processing Seminar.  

This map was made by a team of scientists using complex network theory.   They claim that the center of the Pacific Ocean - the El Niño basin - plays a crucial role in the Earth's climate.

Take temperature data at a grid of points on the Earth.  Subtract out the seasonal variations in temperature.  Then, for each pair of points, work out the mutual information between the temperature at one point and the other point.  In other words: how many bits of information these two pieces of data have in common!  

Next, connect two points with an edge if their mutual informationis much h... more »

Climate networks

Complex network theory is the study of large networks.  Can we use this to get new insights about the Earth's climate?  That's what I'm talking about next week at a conference called the Neural Information Processing Seminar.  

This map was made by a team of scientists using complex network theory.   They claim that the center of the Pacific Ocean - the El Niño basin - plays a crucial role in the Earth's climate.

Take temperature data at a grid of points on the Earth.  Subtract out the seasonal variations in temperature.  Then, for each pair of points, work out the mutual information between the temperature at one point and the other point.  In other words: how many bits of information these two pieces of data have in common!  

Next, connect two points with an edge if their mutual information is much higher than average.  (How much?  You can adjust this.)  Now you have a climate network

The red region here consists of points that are connected to at least 10 times as many other points as average.  So,  if you know the temperature at these places, you know a lot of information about the temperature at a lot of other places! 

Why?  This is where the climate disruption called El Niño starts.  We may be due for another one soon.

Recently some people have been trying to use climate networks to predict El Niños.   The Azimuth Project - a team of scientists and programmers I'm working with - has been reviewing this work.  That's what my talk is about.  You can see a draft here:

http://johncarlosbaez.wordpress.com/2014/11/29/climate-networks/

I really hope you comment on it and help improve it! 

What I like about this talk is that a lot of people at the Azimuth Project have helped create it: Jan Galkowski, Graham Jones, Nadja Kutz, Daniel Mahler, Blake Pollard, Paul Pukite, Dara Shayda, David Tanzer, David Tweed, Steve Wenner, and others.  I'm no good at programming and software, but a lot of these people are!

This map was not created by us; it's in this paper:

• J. F. Donges, Y. Zou, N.Marwan and J. Kurths, Complex networks in climate dynamics, European Physics Journal Special 174 (2009), 157–179.  Available at https://www.pik-potsdam.de/members/kurths/publikationen/2009/complex-networks.pdf.

Alas, I just discovered something annoying about this map.  It's not based on actual temperature data: it's based on a climate model.  I don't hate models, but the actual data is easy to get, and they could have used that.  I bet the results would be similar, but why make us guess?___

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2014-11-29 16:44:51 (25 comments, 62 reshares, 237 +1s)Open 

Mr. Tower's spherical steam engine

In the 1800s there was an intense exploration of different designs for steam engines.  One of the most unusual is this spherical steam engine designed by a fellow named Beauchamp Tower.    It got a lot of publicity around 1885.   It was actually used for generating electricity to light carriages on the locomotives of the Great Eastern Railway in Britain!   It was also used on some ships.

 But it needed a lot of steam for the power it produced - perhaps due to leaks - so it never really caught on.

I got this picture, made by Bill Todd, from Douglas Self's wonderfu online museum of old technologies.  He writes:

The operation of the engine is not easy to comprehend, but goes something like this: The "cylinder" is spherical, and contains two quarter-spheres, with a thin circular disc between them.The two ... more »

Mr. Tower's spherical steam engine

In the 1800s there was an intense exploration of different designs for steam engines.  One of the most unusual is this spherical steam engine designed by a fellow named Beauchamp Tower.    It got a lot of publicity around 1885.   It was actually used for generating electricity to light carriages on the locomotives of the Great Eastern Railway in Britain!   It was also used on some ships.

 But it needed a lot of steam for the power it produced - perhaps due to leaks - so it never really caught on.

I got this picture, made by Bill Todd, from Douglas Self's wonderfu online museum of old technologies.  He writes:

The operation of the engine is not easy to comprehend, but goes something like this: The "cylinder" is spherical, and contains two quarter-spheres, with a thin circular disc between them. The two quarter-spheres rotate and engage rather like a universal joint, creating four cavities in the sphere, two of which are expanding and two contracting at any moment. By suitably timing admission and exhaust, rotational power is generated.

For much more, see:

http://www.douglas-self.com/MUSEUM/POWER/tower/tower.htm

Beauchamp Tower's main claim to fame was not this engine, but his discovery of full-film lubrication: with a suitable flow of oil, the surfaces of ball bearings will never actually touch, and they won't wear down.  He also invented a slide rule that uses metallic tapes that wind from one roller to another. 

A true steampunk!  The energy and crazy cleverness that goes into computer technology today, went into mechanical devices back then.

#steampunk  ___

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2014-11-28 18:00:53 (18 comments, 8 reshares, 75 +1s)Open 

Boron - not boring

This is boron carbide, an extremely hard ceramic material used in macho gear like tank armor, bulletproof vests, and engine sabotage powder. 

(Engine sabotage powder?  Yes, you can pour this into the oil supply, and it will make a car engine grind itself to death.)

If diamond has a hardness of 10, this comes in at 9.497.  But its crystal structure is even cooler than diamond!

A group of 12 boron atoms likes to form an icosahedron.   You can see 8 of these icosahedra here - the green things.  These form the corners of a rhombohedron - a kind of squashed cube.  These repeat over and over, forming a rhombohedral lattice.  

But that's not all!   The icosahedra are connected by struts!  These struts have carbon atoms at their ends and a boron in the middle.  Only one strut is shown in detail here. The carbon at... more »

Boron - not boring

This is boron carbide, an extremely hard ceramic material used in macho gear like tank armor, bulletproof vests, and engine sabotage powder. 

(Engine sabotage powder?  Yes, you can pour this into the oil supply, and it will make a car engine grind itself to death.)

If diamond has a hardness of 10, this comes in at 9.497.  But its crystal structure is even cooler than diamond!

A group of 12 boron atoms likes to form an icosahedron.   You can see 8 of these icosahedra here - the green things.  These form the corners of a rhombohedron - a kind of squashed cube.  These repeat over and over, forming a rhombohedral lattice.  

But that's not all!   The icosahedra are connected by struts!  These struts have carbon atoms at their ends and a boron in the middle.  Only one strut is shown in detail here.  The carbon atoms are the black balls and the boron is the little green ball.

Overall there are 4 boron atoms per carbon atom, so people call boron carbide B₄C. 

Puzzle 1: why are there 4 borons per carbon?  I haven't done the counting, so I don't understand this.

Puzzle 2: what's the difference between a rhombus and a parallelogram?

Puzzle 3: what's the difference between a rhombohedron and a paralleliped?

Puzzle 4: what's the difference between a rhombohedral crystal and an 'orthorhombic' crystal? 

Another macho application of boron carbide is to shielding and control rods for nuclear reactors!  The reason is that boron can absorb neutrons without forming long-lived radioactive isotopes.

The structure of boron carbide has even more subtle features, which I don't understand.  Maybe I'm not looking at the pictures carefully enough!

Puzzle 5: where are the octahedra made of boron atoms?

For clues, read this:

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

The picture here was made by 'Materialscientist' and placed on Wikicommons:

https://commons.wikimedia.org/wiki/File:Borfig11a.png

#chemistry  ___

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2014-11-27 16:47:13 (65 comments, 27 reshares, 187 +1s)Open 

The universal Turing machine

I'm looking forward to The Imitation Game, a movie about Alan Turing.  But right now I'm reading the book that inspired it: Alan Turing: the Enigma, by Andrew Hodges.

In 1936, Turing conceived of a universal computing device - a machine that could compute anything that's computable.   He used this idea to prove that some questions could not be systematically answered by any computing device. 

But by 1945, thanks to his job cracking German codes, he became more interested in actually building such a machine:

There will positively be no internal alterations made even if we wish suddenly to switch from calculating the energy levels of a xenon atom to the enumeration of the groups of order 720.

Or as he put it in 1948:

We do not need to have an infinity of different machines doingdif... more »

The universal Turing machine

I'm looking forward to The Imitation Game, a movie about Alan Turing.  But right now I'm reading the book that inspired it: Alan Turing: the Enigma, by Andrew Hodges.

In 1936, Turing conceived of a universal computing device - a machine that could compute anything that's computable.   He used this idea to prove that some questions could not be systematically answered by any computing device. 

But by 1945, thanks to his job cracking German codes, he became more interested in actually building such a machine:

There will positively be no internal alterations made even if we wish suddenly to switch from calculating the energy levels of a xenon atom to the enumeration of the groups of order 720.

Or as he put it in 1948:

We do not need to have an infinity of different machines doing different jobs. A single one will suffice.  The engineering job of producing various machines for various jobs is replaced by the office work of 'programming' different the universal machine to do these jobs.

Programming a single machine to do endless different jobs!   His awesome realization is still overturning our civilization today. 

We have almost no idea where all this is leading.  The revolution has just begun.  I'm old enough to have used computers that stored their data on paper tape with holes in it.  They were huge.  Now my cell phone is vastly more powerful, and it fits in my pocket.  I can use it to find the nearest restaurant.  I can use it to process photographs.  I can use it to read about Alan Turing.   I can talk to it.  What I can do with it is limited only by the collective programming skills of humanity. 

And someday, it will be able to program.

As Andrew Hodges put it:

And thus it was that in this remote station... working with one assistant in a small hut, and thinking in his spare time, an English homosexual atheist mathematician had conceived of the computer.

Read this book!___

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2014-11-25 17:25:21 (35 comments, 31 reshares, 168 +1s)Open 

Diamonds are forever

Diamonds are one of hardest known substances.  They're made of carbon, with each atom connected to 4 others in a pattern called the diamond cubic

The same pattern appears in crystals of silicon, germanium, and tin.  These are other elements in the same column of the periodic table.  They all like to hook up with 4 neighbors.

The diamond cubic is elegant but a bit tricky.  Look at it carefully here!  We start by putting an atom at each corner of a cube.  Then we put an atom in the middle of each face of the cube.   So far, this is called a face-centered cubic.

But then: the tricky part!  We put 4 more atoms inside the cube.  Each of these has 4 nearest neighbors, which form the corners of a tetrahedron.

What are the coordinates of these points?  It's good to start with a 4×4×4 cube.  Itscorners are:more »

Diamonds are forever

Diamonds are one of hardest known substances.  They're made of carbon, with each atom connected to 4 others in a pattern called the diamond cubic

The same pattern appears in crystals of silicon, germanium, and tin.  These are other elements in the same column of the periodic table.  They all like to hook up with 4 neighbors.

The diamond cubic is elegant but a bit tricky.  Look at it carefully here!  We start by putting an atom at each corner of a cube.  Then we put an atom in the middle of each face of the cube.   So far, this is called a face-centered cubic.

But then: the tricky part!  We put 4 more atoms inside the cube.  Each of these has 4 nearest neighbors, which form the corners of a tetrahedron.

What are the coordinates of these points?  It's good to start with a 4×4×4 cube.  Its corners are:

(0,0,0)   (4,0,0)
(0,4,0)   (4,4,0)

(0,0,4)   (4,0,4)
(0,4,4)   (4,4,4)

The middles of its faces are
 
(2,2,0)  (2,0,2)  (0,2,2)
(2,2,4)  (2,4,2)  (4,2,2)

We can take the four extra points to be

(1,1,3)  (1,3,1)  (3,1,1) 
             (3,3,3)

So, here's a nice way to describe all the points in the diamond cubic.  They're points (x,y,z) where:

•  x, y, and z are all even or all odd

•  x+y+z is either a multiple of 4, or one more than a multiple of 4.

Tricky, eh?

Part of why it's tricky is that there was a choice.  We could also switch the 1's and 3's in the four extra points, using

(3,3,1)  (3,1,3)  (1,3,3) 
             (1,1,1)

instead.  Then we'd get a diamond cubic with points (x,y,z) where:

•  x, y, and z are all even or all odd

•  x+y+z is either a multiple of 4, or one less than a multiple of 4.

Puzzle 1: is the diamond cubic a 'lattice' in the mathematical sense?  A lattice is a discrete set of points that is closed under addition and subtraction.

Puzzle 2: take n-tuples of numbers where:

•  the numbers are all even or all odd

•  their sum is either a multiple of 4, or two more than a multiple of 4.

Does this give you a lattice?  The answer may depend on n.

Puzzle 3: For experts: when you get a lattice in Puzzle 2, what is this lattice called? 

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

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2014-11-24 17:56:05 (17 comments, 11 reshares, 90 +1s)Open 

Don't go there

The Taklamakan Desert in western China is one of the toughest, most forbidding deserts in the world: icy cold in winter, hot in summer, bone-dry with plenty of sand storms all year round.  Its name means "those who go there do not return".  Now there's a highway across it!  But before you plan a road trip, talk to the locals.

Annoyingly,  I can't remember the coordinates of this Google map!

Puzzle: where exactly is this place?

It's somewhere on the Tarim Desert Highway.  But this highway has several parts.   The longest part, the Lunmin Highway, links the city of Luntai on the northern edge of the desert with Minfeng on the southern edge.  This road is 550 kilometers long - with about 450 kilometers in uninhabited areas covered by shifting sand dunes.  It's the longest desert highway in theworld!more »

Don't go there

The Taklamakan Desert in western China is one of the toughest, most forbidding deserts in the world: icy cold in winter, hot in summer, bone-dry with plenty of sand storms all year round.  Its name means "those who go there do not return".  Now there's a highway across it!  But before you plan a road trip, talk to the locals.

Annoyingly,  I can't remember the coordinates of this Google map!

Puzzle: where exactly is this place?

It's somewhere on the Tarim Desert Highway.  But this highway has several parts.   The longest part, the Lunmin Highway, links the city of Luntai on the northern edge of the desert with Minfeng on the southern edge.  This road is 550 kilometers long - with about 450 kilometers in uninhabited areas covered by shifting sand dunes.  It's the longest desert highway in the world!

Halfway along this highway there are some restaurants and a gas station, painted bright blue with a red roof. 

To keep shifting sands from covering the highway, bushes were planted next to it, and a huge irrigation system was constructed to keep them alive.   I don't think they have that here.  Even with it, they probably need to clean up after sand storms.

Except for the people who work at the restaurant, gas station and irrigation system, the region is otherwise entirely uninhabited.  What a job!

The closest I ever came to this place was the Yadan Geological Park, over 1000 kilometers away - but separated by nothing but different kinds of desert.

You can see the gas station here:

http://www.amusingplanet.com/2013/05/the-green-belt-along-worlds-longest.html

For more:

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

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2014-11-22 17:43:26 (52 comments, 137 reshares, 260 +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/___

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2014-11-21 16:07:28 (26 comments, 6 reshares, 69 +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 (47 comments, 39 reshares, 127 +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 (62 comments, 19 reshares, 84 +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, 9 reshares, 81 +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 (48 comments, 38 reshares, 132 +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, 135 +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, 149 +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, 65 +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 (42 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 (46 comments, 36 reshares, 206 +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 (16 comments, 12 reshares, 63 +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, 119 +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 (171 comments, 17 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.___

posted image

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, 90 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, 102 reshares, 242 +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, 78 +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___

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2014-10-18 15:55:11 (19 comments, 29 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, 204 +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.___

posted image

2014-10-10 22:35:04 (7 comments, 35 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  ___

posted image

2014-10-09 20:07:55 (145 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, 88 +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___

posted image

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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