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John Baez has been shared in 357 public circles

You can see here the 50 latest shared circles.
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AuthorFollowersDateUsers in CircleCommentsReshares+1Links
Rian Sigap475Get More Google+ Follower with  +TubeDEVILZ  January 15, 2015*****************************************************************HERE'S OF MY SHARED PUBLIC CIRCLE*****************************************************************Hope that you have been having a great week on Google+. Thank you for sharing and promoting this and for connecting up with all the great accounts I have included. Great With This Cilcle!!,Boost your visibility on Google+ - Share the circle!Boost your visibility on Google+ - Share the circle!To be added to the Circle you have to do these simple steps:1 Include me in your circles2 Click add people and create your circle3 +1 this circle4 Publicly share this circle to public, your circles and extended circles. ( dont forget share the circle and include yourself )5 If possible, leave a comment on this circle so I know you have done the three steps above (I say "if possible" as my circle comments more often than not hit the 500 comment limit).6  So I can easily find your share, always publicly share my original shared circle. You'll know if you're sharing the original one because you won't see "Jason Levy originally shared" above here. If you do see it, click on "originally shared" and it will bring you to this post.**************************************Follow Me Here : http://goo.gl/c18bpxAnd Subcribe : http://goo.gl/NT0MCkSpecial Invitation (Please +1 and Share) :+Alfina Dewi +Agus Septiann +Dini Ashanti +Amy Cesario +Sergii Daniloff +Danis Sanju +Lieven Damman +dini iftita +Lincoln Harrison +Riskhha Nur Hayati +Nanang Hendro +Hanste2015-01-16 20:15:35473000
Ryan Johnson23,295This 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  #sri_lanka #colombo #australia #adelaide #australia #cairns #australia #darwin #australia #hobart #new_zealand #auckland #new_zealand #wellington #papua_new_guinea #papua_new_guinea #awesome #AwesomePeople #AwesomeCircle #addmetoyourcircles #addcircle #addpeople #circlemeup #circlesdiscovery #circleshare #circlesharing #publiccircle #publicsharedcircles #SharedCircles #weeklyreview #sharedcircle #topsharedcircle #circleoftheday 2015-01-16 13:00:35472236
Circles and Photography35,700Builders 1     1.12.15Add this circle to Build-up your G+ network! Please ReShare.#circleoftheday #circleshare  #circlesharing     #circlesharingforthepeopleplc #sharedcircles     #sharedpubliccircles #sharedcircleoftheday #sharedcircleday   #publiccirclesproject #publiccircles #publicsharedcircles       #sharedpublicircles  #circle              2015-01-13 03:30:25499010
Terry Dyke1,332The #CulturalCreatives circle -- a carefully-vetted group of 100 artists, writers, makers, and thinkers on G+.They all have 1000+ followers and post actively. Most tend toward the humanist/progressive/green end of things, and all have a creatively provocative take on this stuff that fills our waking hours.If you are interested in joining the circle and expanding it, please do the following:1. Add this to your circles2. Add yourself to the circle3. Share the expanded circle to Public4. Include comments and #CulturalCreatives tagThanks!Terry Dyke#CulturalCreatives  #circles  #circlesharing   #sharedcircles    #publiccircles2015-01-13 00:56:20100000
RuMuZ NeYiMe1,336good morningadd friends list..#addcircle #addcircles #addpeople #awesomecircle #awesomecircles #awesomepeople #besharable #besocial #bestcircle #bestcircles #bestengagers #circleadd #circleall #circleme #circlemenow #circlemeup #circlenetwork #circleplus #circlesdiscovery #circleshare #circleshares #circlesharing #circleup #circleyoushare #coolpeople #engagerscircle #engagerscircles #findcircle #findcircles #follow4follow #followback #followme #fullcircleshare #influencermarketing #internetmarketing #morefollowers #networkcircle 2015-01-12 08:56:27466002
John Sean10,506This circle contains people who really are interesting and active people on Google Plus.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 circles2 - Share the circle (Publicly)3 - Add +1 to the post.4 - Leave a comment if you like.I will thankful if you plus and share this circle!#publiccircle #circleshare #circlesharing #philadelphia #phoenix #san_antonio #san_diego #san_francisco #san_jose #seattle #tampa #washington #american_samoa #american_samoa #pago_pago #fiji #fiji #nadi #fiji #suva #argentina #argentina #buenos_aires #argentina #cordoba #argentina #iguaza #argentina #mendoza #argentina #rosaio #argentina #san_carlos_de_bariloche #bolivia #bolivia #cochabamba 2015-01-12 06:41:19465012
Frank Gainsford53,086A circle of people who either post or share stuff that is about science 2015-01-09 10:11:40339000
Ruta a la Patagonia - Bariloche13,519┊ ☆ ┊☆ ┊ ☆ ┊Great Friends v17  CIRCLE  ┊ ☆ ┊☆ ┊☆┊_____________________________________________________*●❈●❈●❉●  Please Share From The Original Post! ●❈●❈●❉●▼▼▼▼▼▼▼ CLICK READ MORE FOR FULL CONTENT ▼▼▼▼▼▼▼This is the Great Friends v17 Google Plus CircleIf you want to participate please kindly frollow the following rules::-)1. Add me to your circles if you haven´t done it already2. Share this circle to Public3. Plus or coment this post so we know you wish to participate in upcoming circlesPlease note:● You must be an active Google+ user and shares useful content.● Your posts must be family-friendly. No adult, gambling, controversial, politics, religion blogs.Have a nice day!Your blogging friends of: +Ruta a la Patagonia - Bariloche  De camino al Sur el mejor hotel  para alojarte sobre ruta 5, antes de Santa Rosa La Pampa, esta en Trenque Lauquen: +Hotel Howard Johnson Trenque Lauquen Sobre ruta, con amplio parque, pileta climatizada, estacionamiento, wifi, restobar y mucho mas. Ya sea que vayas de camino a Bariloche, San Martin de los Andes, Villa la Angostura o cualquier otro destino de la cordillera o de la costa de la patagonia (por ruta 33).Consultanos:  www.hjtrenquelauquen.com.ar #patagonia   #SantaRosa   #Bariloche   #LaPampa   #Ruta5   #TrenqueLauquen   #Alojamiento   #HowardJohnson  2015-01-05 21:11:10500454976
Sakari Maaranen3,848Here's a circle that's about Life on Earth. Add these people and organizations for everything about the #environment , #biodiversity , and the kind of #values  that can bring sustainable development.This is a broad range of people, many of whom are not necessarily activists, but scientists and experts with generally the right kind of mindset and deep knowledge of these and related issues. Some are thinkers, artists, or younger people with similar interests.Shared because we need more this kind of thinking! Feel free to re-share —  #sustainability  deserves all our attention and is needed right now.Let's make 2015 the year of positive change!Oh, and please let me know, if I'm missing some active people or important organizations. Remember that I don't care about status. It doesn't matter if you are someone new or young or already a superstar, or if your main field is something else. All it takes is some genuine drive to engage and/or follow these topics. So don't be shy! You are as welcome as anyone.2015-01-03 23:20:49115200
Ruta a la Patagonia - Bariloche12,536┊ ☆ ┊☆ ┊ ☆ ┊World Gems v8 CIRCLE  ┊ ☆ ┊☆ ┊☆┊_____________________________________________________*●❈●❈●❉●  Please Share From The Original Post! ●❈●❈●❉●▼▼▼▼▼▼▼ CLICK READ MORE FOR FULL CONTENT ▼▼▼▼▼▼▼This is the World Gems v8 Google Plus CircleIf you want to participate please kindly frollow the following rules::-)1. Add me to your circles if you haven´t done it already2. Share this circle to Public3. Plus or coment this post so we know you wish to participate in upcoming circlesPlease note:● You must be an active Google+ user and shares useful content.● Your posts must be family-friendly. No adult, gambling, controversial, politics, religion blogs.Have a nice day!Your blogging friends of:  +Ruta a la Patagonia - Bariloche   De camino al Sur el mejor hotel  para alojarte sobre ruta 5, antes de Santa Rosa La Pampa, esta en Trenque Lauquen:  +Howard Johnson Hotel Trenque Lauquen Sobre ruta, con amplio parque, pileta climatizada, estacionamiento, wifi, restobar y mucho mas. Ya sea que vayas de camino a Bariloche, San Martin de los Andes, Villa la Angostura o cualquier otro destino de la cordillera o de la costa Patagonica, consultanos:www.hjtrenquelauquen.com.ar #patagonia   #Bariloche   #laAngostura   #SanMartin   #SantaRosa   #LaPampa   #ruta5   #TrenqueLauquen   #Hotel   #HowardJohnson   #Alojamiento  2014-12-27 17:25:54499394963
Rogerio Manica31,197Engagers #11Happy holidays everybody!! This is my last circle of recent engagers for this year and I would like to thank you all for your support and friendship. Next year I will create new circles of engagers that will be slightly more selective by keeping only the main profile of people based on engagement and quality of posted material.2014-12-22 23:42:13394211426
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 09:02:213086966129
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 11:32:544427561140
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 01:58:1048011912
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 11:17:05479454473
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 06:30:22451433271
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 04:03:53302603994
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 Atanasov1,550SCIENCE 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 06:22:16330123
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 Caroline1,691Hello 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 08:31:58463203
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 07:02:42362515
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 Atanasov1,054Science 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 08:37:19453526
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 12:58:0746211659144
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

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Most comments: 244

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2014-12-26 16:00:53 (244 comments, 203 reshares, 1443 +1s)Open 

Pals

This photo is almost unbearably cute!

It was taken by Barry Bland at TIGERS - The Institute for Greatly Endangered and Rare Species, in Myrtle Beach, Florida.

It's interesting to think about why this photo is so cute.

First of all, obviously, the young wolf and tiger seem like pals, walking in step - and the wolf is even smiling!  But more deeply, I think we like the idea that animals of different species, even fierce ones, could be friends.  The lamb may not lie down with the lion, but at least the tiger can play with the wolf!  It gives us hope.

Finally, these are young animals, and thus more friendly, playful and inquisitive than their adult versions... and more cute.  We seem to be innately fond of baby animals, perhaps thanks to our instinct to care for human babies. 

Dogs are neotenized wolves - adult dogs,espe... more »

Most reshares: 241

posted image

2015-01-25 19:08:51 (111 comments, 241 reshares, 1946 +1s)Open 

Sunrise - from an ice cave in Siberia

We can thank photographer Andrey Grachev for this view!  He walked across Lake Baikal, a huge lake in Siberia that freezes over in winter... and found this ice cave on Olkhon Island. 

You can see more of his photos here:

http://www.dailymail.co.uk/travel/travel_news/article-2918073/Photographer-braves-unstable-frozen-lake-capture-breathtaking-images-magical-ice-cavern-sunrise-Siberia.html

For an amazing picture of cracks in the ice on Lake Baikal see:

https://plus.google.com/u/0/117663015413546257905/posts/ABo1UqrHPVL

For almost five months a year, Lake Baikal is covered with ice.  Perhaps because it's so deep, it starts freezing only in January, long after the Siberian frosts become intense.  It usually thaws in May.   At its peak, the ice is between 1 and 2 meters thick.  Big cra... more »

Most plusones: 1946

posted image

2015-01-25 19:08:51 (111 comments, 241 reshares, 1946 +1s)Open 

Sunrise - from an ice cave in Siberia

We can thank photographer Andrey Grachev for this view!  He walked across Lake Baikal, a huge lake in Siberia that freezes over in winter... and found this ice cave on Olkhon Island. 

You can see more of his photos here:

http://www.dailymail.co.uk/travel/travel_news/article-2918073/Photographer-braves-unstable-frozen-lake-capture-breathtaking-images-magical-ice-cavern-sunrise-Siberia.html

For an amazing picture of cracks in the ice on Lake Baikal see:

https://plus.google.com/u/0/117663015413546257905/posts/ABo1UqrHPVL

For almost five months a year, Lake Baikal is covered with ice.  Perhaps because it's so deep, it starts freezing only in January, long after the Siberian frosts become intense.  It usually thaws in May.   At its peak, the ice is between 1 and 2 meters thick.  Big cra... more »

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2015-01-29 19:09:16 (17 comments, 10 reshares, 52 +1s)Open 

The mysteries of geometry

This shape is called the small cubicuboctahedron.   It looks pretty, but it conceals some mysteries.

For starters, the yellow pieces are actually 8-sided things - regular octagons - which go through the whole shape and are mostly hidden from view.  So we have:

6 red squares,
8 blue equilateral triangles,
6 yellow octagons.

Don't be fooled by how the octagons cross each other.  That creates 'false edges' that are not part of the game today!

As you traverse all the faces that meet at a vertex,  crossing edges and ignoring false edges (which is hard),  you'll go across:

... a square, an octagon, a triangle, an octagon...

and then you're back where you started! 

So this is a shape made of regular polygons, where every corner looks like everyother.... more »

The mysteries of geometry

This shape is called the small cubicuboctahedron.   It looks pretty, but it conceals some mysteries.

For starters, the yellow pieces are actually 8-sided things - regular octagons - which go through the whole shape and are mostly hidden from view.  So we have:

6 red squares,
8 blue equilateral triangles,
6 yellow octagons.

Don't be fooled by how the octagons cross each other.  That creates 'false edges' that are not part of the game today!

As you traverse all the faces that meet at a vertex,  crossing edges and ignoring false edges (which is hard),  you'll go across:

... a square, an octagon, a triangle, an octagon...

and then you're back where you started! 

So this is a shape made of regular polygons, where every corner looks like every other.   We call such a thing a uniform polyhedron.  But it's a weird kind, because it has faces that cross other faces forming false edges!  We call it a nonconvex uniform polyhedron.

With Greg Egan I've been digging into the math of such things, and it's deeper than I first expected.  For example:

You can study the small cubicuboctahedron intrinsically - ignoring how it's forced to cross itself when we stuff it inside 3-dimensional space, which is really not enough room for such a wonderful shape.  Then it's actually a 3-holed torus!  

If we draw a sphere around the small cubicuboctahedron, we can project each point on that shape radially outwards to the sphere.  This gives a map from the 3-holed torus to the sphere.  And as +Matt McIrvin helped me guess, this map is a branched cover, at least after you smooth it out a bit. 

What's a branched cover?  Well, if you ever thought about square roots, you'll know an example.  Most numbers have two square roots.  For example, 4 has 2 and -2 as its square roots.  What about -4?  Well, it has 2i and -2i as its square roots, if you use complex numbers.  But 0 has just one square root, namely itself! 

The complex numbers form a plane.  If you draw a picture of the square roots of complex numbers, you'll get two 'sheets' sitting over this plane, which come together and meet at one point, 0, which is called a branch point.  A picture would help:

http://tinyurl.com/square-root-branched-cover

This is the simplest branched cover.  It's a branched cover of the plane, with 1 branch point and 2 sheets.

But the small cubicuboctahedron gives a branched cover of the sphere with 8 branch points and 2 sheets!  And as you might guess from my example of the square root function, complex numbers play a big role in this game!

I'd like to say much more, but this is probably too much for most of you already.  I'll end with some puzzles and references:

Puzzle 1: How many faces does the small cubicuboctahedron have? Call this number F.

Puzzle 2: How many edges does it have, not counting false edges? Call this number E.

Puzzle 3: How many vertices does it have, not counting 'false vertices' where yellow octagons cross each other?  Call this number V.

Puzzle 4: Calculate F - E + V.    By a theorem of Euler, this equals

2 - 2g

where g is the number of holes in the 'g-holed torus' that is the small cubicuboctahedron's secret true self.

For more, go here:

http://en.wikipedia.org/wiki/Small_cubicuboctahedron#Related_tilings

You'll see that this 3-holed torus can be seen as a quotient space of the hyperbolic plane by a discrete group.  This group preserves a tiling of the hyperbolic plane by triangles, squares and octahedra! 

But there's much more, both on Wikipedia and here:

• Zvi Har'El, Uniform solution for uniform polyhedra, Geometriae Dedicata 47 (1993), 57-110, http://www.math.technion.ac.il/S/rl/docs/uniform.pdf.

#geometry  ___

posted image

2015-01-28 16:30:00 (20 comments, 55 reshares, 190 +1s)Open 

We'll arrive at the same time

The tautochrone is a curve with a remarkable property: if you let some beads slide down it, they all reach the bottom at the same time!  Ignoring friction, that is.

Even better, this curve is just an upside-down cycloid!  The cycloid is the curve you get by rolling a wheel on a flat road and tracing the motion of a point on the rim.

This was proved by Huyghens in 1659.   He also showed that the time of descent equals the time it takes for a rock to fall a distance of π/2 times the diameter of the wheel you used to make the cycloid!

Amazingly, Huyghens did all this without calculus.  Later, in 1690, Jakob Bernoulli solved this problem using calculus.  This was the first published paper that contains the word "integral" in its modern calculus meaning!

In the picture, each bead slidesdown t... more »

We'll arrive at the same time

The tautochrone is a curve with a remarkable property: if you let some beads slide down it, they all reach the bottom at the same time!  Ignoring friction, that is.

Even better, this curve is just an upside-down cycloid!  The cycloid is the curve you get by rolling a wheel on a flat road and tracing the motion of a point on the rim.

This was proved by Huyghens in 1659.   He also showed that the time of descent equals the time it takes for a rock to fall a distance of π/2 times the diameter of the wheel you used to make the cycloid!

Amazingly, Huyghens did all this without calculus.  Later, in 1690, Jakob Bernoulli solved this problem using calculus.  This was the first published paper that contains the word "integral" in its modern calculus meaning!

In the picture, each bead slides down the tautochrone, with a little arrow showing the component of its acceleration vector tangent to the curve.  At right we see a graph of the distance each bead travels as a function of time.

It's easy to show that the tautochrone is an upside-down cycloid if you use calculus and Lagrangian mechanics - the approach to classical mechanics that says roughly this: a system will move in a way that minimizes its total action.  Its action is its kinetic energy minus its potential energy, integrated over time.

For the details, see:

http://en.wikipedia.org/wiki/Tautochrone_curve

Puzzle 1: what's the most elegant way to see that the tautochrone is an upside-down cycloid? 

Puzzle 2: the tautochrone is also the brachistochrone: the curve where it takes as little time as possible for a bead starting at rest to slide under the force of gravity from the start to the end.  What's the most elegant way to see this?___

posted image

2015-01-27 17:53:00 (50 comments, 37 reshares, 131 +1s)Open 

Morse code

There are different kinds of Morse code - this is International Morse Code

Each letter or number is represented by a sequence of dots and dashes.  When you type these out on a telegraph, a dash should be 3 times as long as a dot.  Each dot or dash is followed by a short silence, as long as a dot. The letters of a word should be separated by a silence that's 3 dots long, and words should be separated by a silence that's 7 dots long.

How long is a dot?  That depends on your skills!

The codes for numbers make a pattern.  The codes for letters look chaotic.   But they're not: they're chosen so that commonly used letters have short codes!   The system is nicely explained using a tree:

http://commons.wikimedia.org/wiki/File:Morse_code_tree3.png

E and T are on top: they have theshortest... more »

Morse code

There are different kinds of Morse code - this is International Morse Code

Each letter or number is represented by a sequence of dots and dashes.  When you type these out on a telegraph, a dash should be 3 times as long as a dot.  Each dot or dash is followed by a short silence, as long as a dot. The letters of a word should be separated by a silence that's 3 dots long, and words should be separated by a silence that's 7 dots long.

How long is a dot?  That depends on your skills!

The codes for numbers make a pattern.  The codes for letters look chaotic.   But they're not: they're chosen so that commonly used letters have short codes!   The system is nicely explained using a tree:

http://commons.wikimedia.org/wiki/File:Morse_code_tree3.png

E and T are on top: they have the shortest codes, because they are very commonly used letters.  The E is a single dot and the T is a single dash.   Then come I, A, M and N.  And so on.

How good is International Morse Code?  For that you should compare the tree it uses to the tree it would use if it were as good as possible.   The best possible way is called a Huffman coding.  You can see it on page 16 here:

• Ingrid Daubechies, The mathematics of communication, https://web.math.princeton.edu/~ingrid/VUB/VUB_Spring_2010.pdf

http://en.wikipedia.org/wiki/Morse_code

#coding  ___

posted image

2015-01-26 21:45:51 (27 comments, 17 reshares, 63 +1s)Open 

Ancient ice

If the Greenland ice sheet completely melts, the sea will rise 7.2 meters.  This will drown most of the world’s coastal cities - unless we move them or build dikes.  So ice on Greenland is important. 

It's also a fascinating record of the past!  Scientists just made this wonderful cross-section of Greenland, showing 4 kinds of ice:

• Green is ice from snow that fell on Greenland after the last ice age.  That's after 12,000 years ago.

• Blue is ice from the last ice age.  That's between 12,000 and 115,000 years ago.

• Red is ice from the warm period before the last ice age - the Eemian interglacial.  That's between 115,000 and 130,000 years ago.

• Gray is ice we don't understand yet.

This cross-section is just part of a detailed 3d map of Greenland, built using ice coresamples and rada... more »

Ancient ice

If the Greenland ice sheet completely melts, the sea will rise 7.2 meters.  This will drown most of the world’s coastal cities - unless we move them or build dikes.  So ice on Greenland is important. 

It's also a fascinating record of the past!  Scientists just made this wonderful cross-section of Greenland, showing 4 kinds of ice:

• Green is ice from snow that fell on Greenland after the last ice age.  That's after 12,000 years ago.

• Blue is ice from the last ice age.  That's between 12,000 and 115,000 years ago.

• Red is ice from the warm period before the last ice age - the Eemian interglacial.  That's between 115,000 and 130,000 years ago.

• Gray is ice we don't understand yet.

This cross-section is just part of a detailed 3d map of Greenland, built using ice core samples and radar from planes.  Here's a great video that shows the whole 3d map and how it was made:

http://youtu.be/u0VbPE0TOtQ

The Greenland ice sheet is melting at a rate of about 200 cubic kilometers per year.  The rate is increasing at about 17±8 cubic kilometers per year each year.  This sounds bad.  Indeed, Greenland is contributing about as much to sea level rise as Antarctica.  But the Greenland ice sheet won't go away soon.  It has about 2,850,000 cubic kilometers of ice! 

Ice from the last interglacial - the Eemian - was only recently found in Greenland.   For more, read this story by Eric Steig:

http://www.realclimate.org/index.php/archives/2013/01/the-greenland-melt/

Puzzle 1: if you extrapolate the constantly accelerating rate of melting that I described, when would the Greenland ice sheet be completely melted?  Of course this is naive, but the calculation is easy and fun.

Puzzle 2: about how many gigatonnes of water are in a cubic kilometer?

Puzzle 3: if it were spread equally over the whole ocean, how much would a cubic kilometer of water raise the sea level?

Puzzle 4: what's the absurd mistake in this web page:

http://www.mpimet.mpg.de/en/kommunikation/fragen-zu-klima-faq/how-much-will-the-sea-level-rise.html

#greenland #ice  ___

posted image

2015-01-25 19:08:51 (111 comments, 241 reshares, 1946 +1s)Open 

Sunrise - from an ice cave in Siberia

We can thank photographer Andrey Grachev for this view!  He walked across Lake Baikal, a huge lake in Siberia that freezes over in winter... and found this ice cave on Olkhon Island. 

You can see more of his photos here:

http://www.dailymail.co.uk/travel/travel_news/article-2918073/Photographer-braves-unstable-frozen-lake-capture-breathtaking-images-magical-ice-cavern-sunrise-Siberia.html

For an amazing picture of cracks in the ice on Lake Baikal see:

https://plus.google.com/u/0/117663015413546257905/posts/ABo1UqrHPVL

For almost five months a year, Lake Baikal is covered with ice.  Perhaps because it's so deep, it starts freezing only in January, long after the Siberian frosts become intense.  It usually thaws in May.   At its peak, the ice is between 1 and 2 meters thick.  Big cra... more »

Sunrise - from an ice cave in Siberia

We can thank photographer Andrey Grachev for this view!  He walked across Lake Baikal, a huge lake in Siberia that freezes over in winter... and found this ice cave on Olkhon Island. 

You can see more of his photos here:

http://www.dailymail.co.uk/travel/travel_news/article-2918073/Photographer-braves-unstable-frozen-lake-capture-breathtaking-images-magical-ice-cavern-sunrise-Siberia.html

For an amazing picture of cracks in the ice on Lake Baikal see:

https://plus.google.com/u/0/117663015413546257905/posts/ABo1UqrHPVL

For almost five months a year, Lake Baikal is covered with ice.  Perhaps because it's so deep, it starts freezing only in January, long after the Siberian frosts become intense.  It usually thaws in May.   At its peak, the ice is between 1 and 2 meters thick.   Big cracks can be 10 to 30 kilometers long!

Lake Baikal is the world's largest freshwater lake.   It contains roughly 20% of the world's unfrozen surface fresh water!  It's 1600 meters deep - that's almost a mile! - and it's 640 kilometers long.  It's over 30,000 square kilometers in area.  It holds over 24,000 cubic kilometers of water. 

Lake Baikal is also the oldest freshwater lake - about 25 million years old.  Most lakes don't last very long.  For example, the Great Lakes between Canada and the US started forming only 10,000 years ago, with the retreat of ice at the end of the last glacial period.  I bet they've come and gone many times!  But 25 million years goes back into the Oligocene, before the glacial cycles we're used to.  

Lake Baikal is in a rift valley, created by the Baikal Rift Zone. So the lake itself is a kind of crack - that even now is expanding at 2 centimeters per year!

#ice  ___

posted image

2015-01-24 17:06:53 (38 comments, 9 reshares, 59 +1s)Open 

4 you

+Scott Carter has found a way to offend people while counting in base two.  He holds his thumb, index finger and middle finger up to stand for a 1 in the 1's place, the 2's place and the 4's place. 

This gives binary digits a whole new meaning... or maybe its original meaning.

Puzzle: what's the original reason the digits of a number are called 'digits'? 

He writes:

Some might find the gestures that are located at 101 or 100 to be obscene, but this is the preliminary method of using binary (up or down) digits (fingers) to count. On my right hand, I can count to 31=2*2*2*2*2-1=1+2+4+8+16. On both hands, I can count to 1023. With a little more dexterity in my toes, I could count quite a few bits more.

By the way, this illustration is one of a zillion that will be in my upcomingpa... more »

4 you

+Scott Carter has found a way to offend people while counting in base two.  He holds his thumb, index finger and middle finger up to stand for a 1 in the 1's place, the 2's place and the 4's place. 

This gives binary digits a whole new meaning... or maybe its original meaning.

Puzzle: what's the original reason the digits of a number are called 'digits'? 

He writes:

Some might find the gestures that are located at 101 or 100 to be obscene, but this is the preliminary method of using binary (up or down) digits (fingers) to count. On my right hand, I can count to 31=2*2*2*2*2-1=1+2+4+8+16. On both hands, I can count to 1023. With a little more dexterity in my toes, I could count quite a few bits more.

By the way, this illustration is one of a zillion that will be in my upcoming paper that re-writes the 4D proof of Heron's formula. We are trying to get some animations working for this as well.

He posted this after seeing my hypercube of bits:

https://plus.google.com/u/0/117663015413546257905/posts/VteWm45DCff

Using the fingers of both hands, you can make gestures that stand for all the corners of a 10-dimensional cube!  This could be useful for string theorists.  However, some of these gestures could get you into fights.

Here is the American politician Rahm Emmanuel proving that he can count to 31 using Scott's system:

http://www.mindonfire.com/images/rahmgestures.jpg___

posted image

2015-01-23 15:54:45 (39 comments, 47 reshares, 201 +1s)Open 

A Martian devil

Mars is a windy place!  This dust devil, roughly 20 kilometers high but just 70 meters wide, was seen whirling through northern Mars on March 14, 2007.  It was imaged by a high resolution camera on the Mars Reconnaissance Orbiter... and NASA made this animation based on what they saw.

Dust devils happen on Earth too - I often see them in the deserts around here!  They're spinning columns of air, made visible by the dust they pull off the ground.  Unlike tornadoes, they usually form on clear days when the ground is heated by the sun, warming the air just above the ground. 

As hot air rises, it can start to rotate, by chance... and as more hot air rushes in to replace the air that is rising, the rotation becomes stronger.  So the dust devil grows and sustains itself, becoming a quick way for hot air to rise... until it dies.
... more »

A Martian devil

Mars is a windy place!  This dust devil, roughly 20 kilometers high but just 70 meters wide, was seen whirling through northern Mars on March 14, 2007.  It was imaged by a high resolution camera on the Mars Reconnaissance Orbiter... and NASA made this animation based on what they saw.

Dust devils happen on Earth too - I often see them in the deserts around here!  They're spinning columns of air, made visible by the dust they pull off the ground.  Unlike tornadoes, they usually form on clear days when the ground is heated by the sun, warming the air just above the ground. 

As hot air rises, it can start to rotate, by chance... and as more hot air rushes in to replace the air that is rising, the rotation becomes stronger.  So the dust devil grows and sustains itself, becoming a quick way for hot air to rise... until it dies.

Puzzle: why does it die?

In short, a dust devil is a great example of how efficient increase in entropy can actually create ordered structures, which however have a finite lifetime.  You are an example of this.

This dust devil happened in Amazonis Planitia  during the late spring, two weeks short of the northern summer solstice, when the ground in the northern mid-latitudes is heated most strongly by the sun.

The Mars Reconnaissance Orbiter has been examining the Red Planet with six science instruments since 2006.   You can see thousands of images taken by HiRISE - the High Resolution Imaging Science Experiment - at this website:

http://hirise.lpl.arizona.edu

They're awesome!

#astronomy  ___

posted image

2015-01-22 17:47:24 (41 comments, 29 reshares, 82 +1s)Open 

A hypercube of bits

This is the kind of thing mathematicians know about almost instinctively - but most ordinary folks don't.  It's a 4-dimensional cube drawn in a very nice way, with each corner labeled by a string of 4 bits. 

If you haven't ever thought about this stuff, try these puzzles!

Puzzle 1: Something nice happens when you start at any number and move west-northwest.  For example, when you go from 0110 to 1110.  What always happens when you move this way?

Puzzle 2: What happens when you move east-northeast?   For example, when you go from 0110 to 0111.

Puzzle 3: What happens when you move north-northwest?  For example, when you go from 1001 to 1101.

Puzzle 4: What happens when you move north-northeast?  For example, when you go from 1001 to 1011.

Puzzle 5:There a... more »

A hypercube of bits

This is the kind of thing mathematicians know about almost instinctively - but most ordinary folks don't.  It's a 4-dimensional cube drawn in a very nice way, with each corner labeled by a string of 4 bits. 

If you haven't ever thought about this stuff, try these puzzles!

Puzzle 1: Something nice happens when you start at any number and move west-northwest.  For example, when you go from 0110 to 1110.  What always happens when you move this way?

Puzzle 2: What happens when you move east-northeast?   For example, when you go from 0110 to 0111.

Puzzle 3: What happens when you move north-northwest?  For example, when you go from 1001 to 1101.

Puzzle 4: What happens when you move north-northeast?  For example, when you go from 1001 to 1011.

Puzzle 5: There are 8 bit strings on the outside of this picture.  What happens when you go from one of these to the one directly opposite?

Puzzle 6: There are also 8 bit strings on the inside of this picture.  What happens when you go from one of these to the one directly opposite?

Puzzle 7: How many pictures of cubes can you find in this picture?  The cubes will be a bit slanted.

This picture is a tiny part of a huge subject called coding theory, which about efficiently sending messages as strings of bits, while making it hard for one message to get mistaken for another when an error occurs.

http://en.wikipedia.org/wiki/Coding_theory

I got this image from here:

http://commons.wikimedia.org/wiki/File:Hypercubestar_binary.svg

but modified it.

#geometry  ___

posted image

2015-01-20 16:57:48 (28 comments, 19 reshares, 72 +1s)Open 

How symmetrical is a neutron?

A neutron is a spinning bag of charged particles, so we shouldn't be surprised that it acts like a little magnet.  We say it has a magnetic dipole moment.   Like the Earth, it has a north magnetic pole and a south pole.  The blue arrow called μ here points to the north pole.

A neutron might also have an electric dipole moment.  That would happen if there were more positive charge near one pole, and more negative charge near the other pole.  Then we could draw a red arrow called d pointing toward the positive charges.  

In the picture at left, the red arrow points the same way as the blue arrow.  But nobody knows if there is a red arrow!  So far nobody has seen an electric dipole moment for a neutron.  It's either zero, or very small.

A water molecule has an electric dipole moment:it's s... more »

How symmetrical is a neutron?

A neutron is a spinning bag of charged particles, so we shouldn't be surprised that it acts like a little magnet.  We say it has a magnetic dipole moment.   Like the Earth, it has a north magnetic pole and a south pole.  The blue arrow called μ here points to the north pole.

A neutron might also have an electric dipole moment.  That would happen if there were more positive charge near one pole, and more negative charge near the other pole.  Then we could draw a red arrow called d pointing toward the positive charges.  

In the picture at left, the red arrow points the same way as the blue arrow.  But nobody knows if there is a red arrow!  So far nobody has seen an electric dipole moment for a neutron.  It's either zero, or very small.

A water molecule has an electric dipole moment: it's shaped like a head with two big ears, and there's more positive charge near the ears.  You might argue that the electric dipole moment of the neutron should be zero because - unlike the water molecule - the neutron is round.  There's a kernel of truth to that.

Indeed, if the electric dipole moment wasn't zero, it would violate some symmetries that the neutron seems to have!

P symmetry, or parity, is the symmetry where you reverse all 3 spatial directions: send each point (x,y,z) to the opposite point (-x,-y,-z).  If you do this to a spinning sphere, it still spins the same way, so the arrow μ is unchanged.   However, if there had been more positive charges near one pole, now there will be more positive charges near the other pole.  So the arrow d now points the other way.

T symmetry, or time reversal, is the symmetry where you reverse the direction of time: send each time t to -t.  We can't actually turn time around, but we can try to set up a neutron that's a time-reversed version of some other neutron.  It would spin the opposite way, so the arrow μ would point the other way.  But the positive charges would still be on the same side.  So d points the same way.

The picture shows that if a neutron has the μ and d arrows pointing the same way, and we apply parity or time reversal, we get another kind of neutron where the μ and d arrows point opposite  ways.  There can't be two kinds of neutrons: we'd have noticed that by now.  So, if neutrons have an electric dipole moment, they can't be symmetric under parity and time reversal. 

In fact neutrons probably aren't symmetric under parity and time reversal, because a force called the weak force doesn't have these symmetries, and it affects neutrons.  But as the name indicates, this force is very weak.  We can calculate the electric dipole moment this force creates in the neutron, and it's tiny - about 10 million times smaller than our current ability to measure it.

What's interesting is that as far as we know, the strong force could also fail to have parity and time reversal symmetry.  This is the force that holds the neutron together.  If it broke these symmetries, it could create a larger electric dipole moment than the weak force does.  

We haven't seen any sign that this happens.  People are looking because this would be one of the best ways to see if the strong force violates parity and time reversal symmetry.  If it doesn't, one of the fundamental constants of nature must be zero... and nobody knows why, though there are some fascinating theories.  This is called the strong CP problem:

http://en.wikipedia.org/wiki/Strong_CP_problem

There's a lot more to say about this, but not today!

Puzzle: I said we would have noticed by now if there are two kinds of neutrons, one where μ and d point the same way and one where they point in opposite directions.  How could we have noticed this, given that we can't yet measure the d arrow?

I'll end with some numbers, for people like me who enjoy numbers.

Right now the best upper bound on the neutron's electric dipole moment is 2.9 times 10^-26 e cm.   (Electric dipole moment is often measured in units of the electron's charge times a centimeter.)   There are at least five experiments in progress that aim at improving this limit to 10^-28 e cm.  These should be able to rule out various theories of how supersymmetry could create an electric dipole moment in the neutron. 

The weak force should create a dipole moment of about 10^-33 e cm, so detecting that is still far away.  This amount of asymmetry is so small that it's like the Earth being perfectly round except for mountains that are micron tall!___

posted image

2015-01-19 17:22:47 (13 comments, 14 reshares, 88 +1s)Open 

The arc of the moral universe is long, but it bends toward justice

On March 7, 1965, protesters seeking the right to vote tried to march from Selma to Montgomery Alabama.  State troopers and a violent posse attacked the unarmed marchers with billy clubs and tear gas.

After another try, the march finally succeeded three weeks later. After walking 54 miles, Martin Luther King gave a speech on the steps of the State Capitol of Alabama.  It began like this:

Last Sunday, more than eight thousand of us started on a mighty walk from Selma, Alabama. We have walked through desolate valleys and across the trying hills. We have walked on meandering highways and rested our bodies on rocky byways. Some of our faces are burned from the outpourings of the sweltering sun. Some have literally slept in the mud. We have been drenched by the rains. Our bodies are tired and our feetar... more »

The arc of the moral universe is long, but it bends toward justice

On March 7, 1965, protesters seeking the right to vote tried to march from Selma to Montgomery Alabama.  State troopers and a violent posse attacked the unarmed marchers with billy clubs and tear gas.

After another try, the march finally succeeded three weeks later. After walking 54 miles, Martin Luther King gave a speech on the steps of the State Capitol of Alabama.  It began like this:

Last Sunday, more than eight thousand of us started on a mighty walk from Selma, Alabama. We have walked through desolate valleys and across the trying hills. We have walked on meandering highways and rested our bodies on rocky byways. Some of our faces are burned from the outpourings of the sweltering sun. Some have literally slept in the mud. We have been drenched by the rains. Our bodies are tired and our feet are somewhat sore.

But today as I stand before you and think back over that great march, I can say, as Sister Pollard said—a seventy-year-old Negro woman who lived in this community during the bus boycott—and one day, she was asked while walking if she didn’t want to ride. And when she answered, "No," the person said, "Well, aren’t you tired?" And with her ungrammatical profundity, she said, "My feets is tired, but my soul is rested."  And in a real sense this afternoon, we can say that our feet are tired, but our souls are rested.

They told us we wouldn’t get here. And there were those who said that we would get here only over their dead bodies, but all the world today knows that we are here and we are standing before the forces of power in the state of Alabama saying, "We ain’t goin’ let nobody turn us around."

Now it is not an accident that one of the great marches of American history should terminate in Montgomery, Alabama. Just ten years ago, in this very city, a new philosophy was born of the Negro struggle. Montgomery was the first city in the South in which the entire Negro community united and squarely faced its age-old oppressors.  Out of this struggle, more than bus desegregation was won; a new idea, more powerful than guns or clubs was born. Negroes took it and carried it across the South in epic battles that electrified the nation and the world.

Yet, strangely, the climactic conflicts always were fought and won on Alabama soil. After Montgomery’s, heroic confrontations loomed up in Mississippi, Arkansas, Georgia, and elsewhere. But not until the colossus of segregation was challenged in Birmingham did the conscience of America begin to bleed. White America was profoundly aroused by Birmingham because it witnessed the whole community of Negroes facing terror and brutality with majestic scorn and heroic courage. And from the wells of this democratic spirit, the nation finally forced Congress to write legislation in the hope that it would eradicate the stain of Birmingham. The Civil Rights Act of 1964 gave Negroes some part of their rightful dignity, but without the vote it was dignity without strength.

Once more the method of nonviolent resistance was unsheathed from its scabbard, and once again an entire community was mobilized to confront the adversary.  And again the brutality of a dying order shrieks across the land. Yet, Selma, Alabama, became a shining moment in the conscience of man. If the worst in American life lurked in its dark streets, the best of American instincts arose passionately from across the nation to overcome it. There never was a moment in American history more honorable and more inspiring than the pilgrimage of clergymen and laymen of every race and faith pouring into Selma to face danger at the side of its embattled Negroes.

The confrontation of good and evil compressed in the tiny community of Selma generated the massive power to turn the whole nation to a new course. A president born in the South had the sensitivity to feel the will of the country, and in an address that will live in history as one of the most passionate pleas for human rights ever made by a president of our nation, he pledged the might of the federal government to cast off the centuries-old blight. President Johnson rightly praised the courage of the Negro for awakening the conscience of the nation.

On our part we must pay our profound respects to the white Americans who cherish their democratic traditions over the ugly customs and privileges of generations and come forth boldly to join hands with us.  From Montgomery to Birmingham, from Birmingham to Selma, from Selma back to Montgomery, a trail wound in a circle long and often bloody, yet it has become a highway up from darkness. Alabama has tried to nurture and defend evil, but evil is choking to death in the dusty roads and streets of this state. So I stand before you this afternoon with the conviction that segregation is on its deathbed in Alabama, and the only thing uncertain about it is how costly the segregationists and Wallace will make the funeral.

The whole speech is here:

http://mlk-kpp01.stanford.edu/index.php/kingpapers/article/our_god_is_marching_on/

Near the end he said:

"The arc of the moral universe is long, but it bends toward justice."

I used to wonder if this is true.  I now think it's one of those things that only becomes true if enough of us work to make it so.   A master orator, Martin Luther King was not trying to describe the world: he was trying to change it.

I saw the movie Selma, and I recommend it - a good reminder of this recent era of American history... and how powerful determination accomplished real changes.  We could use some of that spirit now.___

posted image

2015-01-19 04:26:45 (29 comments, 69 reshares, 166 +1s)Open 

How to succeed in politics, in one easy lesson

To fool adults, you need to distract them when you break the cracker.





(Thanks to +Andres M. Trianon for pointing this gif, which comes from http://imgur.com/bAGqKFS.  If you haven't circled Andres, you should!)

How to succeed in politics, in one easy lesson

To fool adults, you need to distract them when you break the cracker.





(Thanks to +Andres M. Trianon for pointing this gif, which comes from http://imgur.com/bAGqKFS.  If you haven't circled Andres, you should!)___

posted image

2015-01-17 15:45:59 (27 comments, 53 reshares, 177 +1s)Open 

Supercell

This is not a tornado or hurricane!  It's a supercell: a thunderstorm with a deep, persistently rotating updraft. 

Supercells are one of the least common kinds of thunderstorm - but they can be the most severe!  Supercells can happen anywhere - but especially in the Great Plains of America and the Tornado Corridor of Argentina, Uruguay and southern Brazil.

They start when the wind is moving faster at one height than another: this is called wind shear, and it can create a vortex.  Thunderstorms often have a strong updraft, and this can tilt the vortex so it's vertical instead of horizontal!   This creates a mesocyclone, which you see here.  And sometimes the mesocyclone creates tornadoes.

Things always get more complicated and interesting when you study them in detail.  I find weather to be a very trickysubject... more »

Supercell

This is not a tornado or hurricane!  It's a supercell: a thunderstorm with a deep, persistently rotating updraft. 

Supercells are one of the least common kinds of thunderstorm - but they can be the most severe!  Supercells can happen anywhere - but especially in the Great Plains of America and the Tornado Corridor of Argentina, Uruguay and southern Brazil.

They start when the wind is moving faster at one height than another: this is called wind shear, and it can create a vortex.  Thunderstorms often have a strong updraft, and this can tilt the vortex so it's vertical instead of horizontal!   This creates a mesocyclone, which you see here.  And sometimes the mesocyclone creates tornadoes.

Things always get more complicated and interesting when you study them in detail.  I find weather to be a very tricky subject.  I've just skimmed the surface, and you can learn more here:

http://en.wikipedia.org/wiki/Supercell
http://en.wikipedia.org/wiki/Tornado

This animated gif seems to be created from photos taken in Nebraska by the storm chaser Mike Hollingshead in Nebraska:

http://www.corbisimages.com/stock-photo/rights-managed/42-50737080/nebraska-supercell

Google Image Search shows copies of this all over the place, with many people wrongly saying it's a hurricane.___

posted image

2015-01-16 22:08:26 (12 comments, 18 reshares, 110 +1s)Open 

A climate hero

This is Alberto Behar in Greenland with the robotic boat he designed.  How fast is Greenland melting due to global warming?  Where does the water go?  Some people sit around and argue.  Others go and find out.

It was very warm in Greenland from July 11th to 13th, 2012.  Scientists from NASA traveled by helicopter to study the melting ice.  They mapped rivers and streams over 5400 square kilometers of Greenland.   They found 523 separate drainage systems - small streams joining to form larger streams and rivers.

The water in every one of these flowed into a moulin!  A moulin is a circular, vertical shaft.  Water pours down the moulin and goes deep below the surface - sometimes forming a layer between ice and the underlying rock.  This layer can help glaciers slide down toward the ocean.  And this water reaches the ocean fast. 
In t... more »

A climate hero

This is Alberto Behar in Greenland with the robotic boat he designed.  How fast is Greenland melting due to global warming?  Where does the water go?  Some people sit around and argue.  Others go and find out.

It was very warm in Greenland from July 11th to 13th, 2012.  Scientists from NASA traveled by helicopter to study the melting ice.  They mapped rivers and streams over 5400 square kilometers of Greenland.   They found 523 separate drainage systems - small streams joining to form larger streams and rivers.

The water in every one of these flowed into a moulin!  A moulin is a circular, vertical shaft.  Water pours down the moulin and goes deep below the surface - sometimes forming a layer between ice and the underlying rock.  This layer can help glaciers slide down toward the ocean.  And this water reaches the ocean fast. 

In the area they studied, a total of between 0.13 and 0.15 cubic kilometers of water were flowing into moulins each day.  That's a lot!  That would be enough to drain 2.5 centimeters of water off the surface each day. 

To study the flow of water, Alberto Behar designed two kinds of remotely controlled boats.  One was a drone boat that measured the depth of the water and how much light it reflected, allowing the researchers to calibrate the depth of the surface water from satellite images. They used this boat on lakes and slow-flowing rivers.  But for dangerous, swift-flowing rivers, Behar developed disposable robotic drifters that measured the water's velocity, depth and temperature as they swept downstream.

Just a few days ago, Alberto Behar died in a plane crash.  The plane he was flying crashed shortly after he took off from a small airport near NASA’s Jet Propulsion Laboratory in Pasadena, California. 

So, his coauthors dedicated their paper on this research to him.  Here is is:

• Laurence C. Smith et al, Efficient meltwater drainage through supraglacial streams and rivers on the southwest Greenland ice sheet, Proc. Nat. Acad. Sci., http://www.pnas.org/content/early/2015/01/07/1413024112.full.pdf

Check out the cool images and maps.  And watch this great movie:

https://www.youtube.com/watch?v=-EMCxE1v22I  ___

2015-01-15 18:00:37 (20 comments, 19 reshares, 75 +1s)Open 

A free online course on chaos theory

Chaos theory is the study of physical systems whose motion depends very delicately on how they start out.  There's a lot of deep geometry here, and +Predrag Cvitanović has started a free online course on the subject!   

There's a lot of hype about chaos theory, but Predrag is a good physicist, and he's written a good free textbook on the subject, so this is the real deal.

To register, just go to his webpage here.  The course started a week ago but you can still join in.  It lasts 8 weeks.  It'll use his book, links to explanatory videos, and weekly homework assignments, which include some computer programming. For the assignments you can use any computational tools you want, but he'll provide you with stuff written in Python.  There are no tests.

He encourages you to register even ifyou won&#... more »

A free online course on chaos theory

Chaos theory is the study of physical systems whose motion depends very delicately on how they start out.  There's a lot of deep geometry here, and +Predrag Cvitanović has started a free online course on the subject!   

There's a lot of hype about chaos theory, but Predrag is a good physicist, and he's written a good free textbook on the subject, so this is the real deal.

To register, just go to his webpage here.  The course started a week ago but you can still join in.  It lasts 8 weeks.  It'll use his book, links to explanatory videos, and weekly homework assignments, which include some computer programming. For the assignments you can use any computational tools you want, but he'll provide you with stuff written in Python.  There are no tests.

He encourages you to register even if you won't do the homework: you can talk to other students on the course forum.  

Some administrators from his university tried to shut this course down at the last minute, probably because it's free.  I'm glad he fought them off and prevailed.  

This course is called Nonlinear dynamics 1: Geometry of chaos, and here are the topics:

Topology of flows - how to enumerate orbits, Smale horseshoes
Quantitative dynamics - periodic orbits, local stability
The role of symmetries in dynamics

There will also be a second, more advance 8-week course called Nonlinear dynamics 2: Chaos rules, with these topics:

Transfer operators - statistical distributions in dynamics
Spectroscopy of chaotic systems
Dynamical zeta functions
Dynamical theory of turbulence

The prerequisites for this first course are a basic background in linear algebra, calculus, ordinary differential equations, probability theory, classical mechanics, and statistical mechanics.  You'll need to able to work with equations involving vectors and matrices, differentiate simple functions, and understand what a probability distribution is.   You will learn to write programs in Python. ___

2015-01-14 06:15:29 (1 comments, 0 reshares, 7 +1s)Open 

Her idea of banter
Likely isn't Cantor,
Nor is she apt to murmur low
Axioms of Zermelo,
She's been kissed by geniuses,
Amateur Frobeniuses,
One by one in swank array,
Bring as any Poincaré,
And .. though she
May not care for Cauchy,
Any more than Riemann,
We'll just have to dream on ...
Let
  it occur in spots in
Whittaker and Watson --
Unforeseen converging,
Miracles emerging,
Epsilonic dances,
Small but finite chances,
For love ...

From "Against the Day" by Thomas Pynchon

I never thought I'd see "Whittaker and Watson" in a poem. It's a reference the textbook "A course in modern analysis" by E. T. Whittaker and G. N. Watson, first published in 1902. I have a copy of the 1927 edition, reprinted in 1980. Despite its age, it's still as... more »

Her idea of banter
Likely isn't Cantor,
Nor is she apt to murmur low
Axioms of Zermelo,
She's been kissed by geniuses,
Amateur Frobeniuses,
One by one in swank array,
Bring as any Poincaré,
And .. though she
May not care for Cauchy,
Any more than Riemann,
We'll just have to dream on ...
Let
  it occur in spots in
Whittaker and Watson --
Unforeseen converging,
Miracles emerging,
Epsilonic dances,
Small but finite chances,
For love ...

From "Against the Day" by Thomas Pynchon

I never thought I'd see "Whittaker and Watson" in a poem. It's a reference the textbook "A course in modern analysis" by E. T. Whittaker and G. N. Watson, first published in 1902. I have a copy of the 1927 edition, reprinted in 1980. Despite its age, it's still a standard reference for special functions.___

posted image

2015-01-13 16:38:31 (33 comments, 7 reshares, 60 +1s)Open 

Ditrigonal dodecadodecahedron

That's the crazy name of this crazy shape!  It's called a 'dodecadodecahedron' because supposedly it has 12 pentagons and 12 pentagrams as faces. 

It's easy to see the pentagrams - they're the red stars.   But what about the 12 pentagons?  Those must be the yellow stuff.  But do you see how to get this yellow stuff from 12 pentagons? 

At first I didn't see how.  Now maybe I do.  But maybe you can help.

This shape is an example of a uniform star polyhedron.  A uniform polyhedron has regular polygons as faces, with enough symmetries that every vertex looks like every other.  In a uniform star polyhedron we also allow regular stars as faces.

I like how these shapes look, but I've never been sure the math of them is deep enough to be worth studying.  That may soundsnobbish.  ... more »

Ditrigonal dodecadodecahedron

That's the crazy name of this crazy shape!  It's called a 'dodecadodecahedron' because supposedly it has 12 pentagons and 12 pentagrams as faces. 

It's easy to see the pentagrams - they're the red stars.   But what about the 12 pentagons?  Those must be the yellow stuff.  But do you see how to get this yellow stuff from 12 pentagons? 

At first I didn't see how.  Now maybe I do.  But maybe you can help.

This shape is an example of a uniform star polyhedron.  A uniform polyhedron has regular polygons as faces, with enough symmetries that every vertex looks like every other.  In a uniform star polyhedron we also allow regular stars as faces.

I like how these shapes look, but I've never been sure the math of them is deep enough to be worth studying.  That may sound snobbish.  But you see, a lot of uniform polyhedra come from Coxeter groups.   These are discrete symmetry groups that are closely connected to lots of other great math - so these are very interesting.  The uniform star polyhedra, on the other hand, don't seem connected to other math in such a strong way.  Or maybe I just haven't learned how.

Still, they're pretty.  There are 57 of them - not counting an infinite number of prisms and antiprisms, star prisms and star antiprisms.  You can see them all here:

http://en.wikipedia.org/wiki/Uniform_star_polyhedron

Puzzle 1: Why does it say "57 varieties" on a bottle of Heinz ketchup?  Is it really because there are 57 uniform star polyhedra?

Puzzle 2: What's the most important appearance of the number 57 in group theory?

Puzzle  3: Why is this shape called "ditrigonal"?  I don't know.

This picture was made by +Tom Ruen using Robert Webb's Stella software and put on Wikimedia Commons.  Webb demands a link to his website:

 http://www.software3d.com/Stella.php

#geometry   ___

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2015-01-12 20:17:58 (61 comments, 54 reshares, 153 +1s)Open 

Doggerland

As recently as 6500 BC, Great Britain was connected to Europe!  And if you go back further in your time machine, you'll see a huge plain called Doggerland between Britain and Denmark.

Why?  Because the sea level is lower during ice ages.  More water is locked up in ice!

The last ice age, the Wisconsin glaciation, reached its peak a bit before 18000 BC.  Back then, there were huge ice sheets going down to the Great Lakes and the mouth of the Rhine.  The north of Britain was covered with ice, and the south was a polar desert!

The light green stuff in this map shows the land a bit later, in 16000 BC.  Back then Doggerland was a wide undulating plain full of complicated meandering river systems.

As the ice age ended, the sea level rose rather quickly.  Doggerland shrank to the medium green stuff in 8000 BC and thedark g... more »

Doggerland

As recently as 6500 BC, Great Britain was connected to Europe!  And if you go back further in your time machine, you'll see a huge plain called Doggerland between Britain and Denmark.

Why?  Because the sea level is lower during ice ages.  More water is locked up in ice!

The last ice age, the Wisconsin glaciation, reached its peak a bit before 18000 BC.  Back then, there were huge ice sheets going down to the Great Lakes and the mouth of the Rhine.  The north of Britain was covered with ice, and the south was a polar desert!

The light green stuff in this map shows the land a bit later, in 16000 BC.  Back then Doggerland was a wide undulating plain full of complicated meandering river systems.

As the ice age ended, the sea level rose rather quickly.  Doggerland shrank to the medium green stuff in 8000 BC and the dark green stuff in 7000 BC.  One of the last parts to survive was the Dogger Bank.  You can see it on the map if you look close.  It was an island until 5000 BC.

A new theory says that Doggerland was flooded by a huge tsunami around 6200 BC, thanks to a submarine landslide off the coast of Norway!  It's called the Storegga Slide.  There's geological evidence of sediments washed up onto land then.  Maybe an earthquake triggered a catastrophic expansion of methane hydrates underwater.

This tsunami would have devastated a rich hunting and fishing ground populated by Mesolithic humans.   People of some sort have lived on the British Isles, on and off, for much longer!  There are flint tools dating back to 815,000 BC.  These would not be made by Homo sapiens, since our species only came into existence around 250,000 BC. 

But there were Homo sapiens in Britain by 40,000 BC, in the middle of the last ice age.  And when that ice age ended and treeless tundra slowly turned into forests of birch trees, more of us moved in.  Instead of eating reindeer and wild horses, the ancient Britons started eating pigs, elk, deer, wild boar and wild cattle - hunting them with ever more sophisticated stone tools.  So by 6200 BC, when the tsunami crashed over Doggerland, there would have been lots of people living quite well.

Puzzle 1: Why is it called "Doggerland"? 

Puzzle 2: What's a "dogger"?

Puzzle 3: When did people start building Stonehenge - how does that fit into the chronology here?

http://en.wikipedia.org/wiki/Doggerland
http://en.wikipedia.org/wiki/Storegga_Slide
http://en.wikipedia.org/wiki/Prehistoric_Britain

I got this map from Jamie Woodward:

https://twitter.com/Jamie_Woodward_/status/554662957339402240/photo/1

thanks to +Susan Stone.  I don't know who made it.___

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2015-01-11 17:51:42 (38 comments, 25 reshares, 136 +1s)Open 

Blue Hole

We've all heard about black holes, but here's a blue hole. 

It's the Great Blue Hole, off the coast of Belize in Central America. It's over 300 meters across and 120 meters deep.  It's in the Belize Barrier Reef Reserve System, which is considered a World Heritage Site by UNESCO.

What made this hole?

Sinkholes form when underground limestone slowly gets dissolved and washed away by slightly acidic rainwater, forming caves, and then the surface of the ground collapses.  The Great Blue Hole formed during two separate ice ages.  During an ice age, the sea level is much lower!  So, this sinkhole didn't form under the ocean.  It formed on land.

The Great Blue Hole began to form 150,000 years ago, during the second to last ice age: the Wolstonian glaciation.   That ice age ended around130,000 ... more »

Blue Hole

We've all heard about black holes, but here's a blue hole. 

It's the Great Blue Hole, off the coast of Belize in Central America. It's over 300 meters across and 120 meters deep.  It's in the Belize Barrier Reef Reserve System, which is considered a World Heritage Site by UNESCO.

What made this hole?

Sinkholes form when underground limestone slowly gets dissolved and washed away by slightly acidic rainwater, forming caves, and then the surface of the ground collapses.  The Great Blue Hole formed during two separate ice ages.  During an ice age, the sea level is much lower!  So, this sinkhole didn't form under the ocean.  It formed on land.

The Great Blue Hole began to form 150,000 years ago, during the second to last ice age: the Wolstonian glaciation.   That ice age ended around 130,000 years ago at the start of the Eemian interglacial.  The sea rose, and the Great Blue Hole was flooded.

The most recent ice age, the Wisconsin glaciation, started about 110,000 years ago.  The sea level dropped.  The Great Blue Hole continued to get bigger, in several separate stages.

And then the most recent ice age ended.  Sea levels rose by about 120 meters!  This was mostly finished about 10,000 years ago. 

There are also many sinkholes on land in Belize and the Yucatán Peninsula, where they are known as cenotes.  Often they're connected to underwater cave systems... but apparently not to the Great Blue Hole.

http://en.wikipedia.org/wiki/Great_Blue_Hole___

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2015-01-10 17:24:16 (79 comments, 61 reshares, 190 +1s)Open 

The Pythagorean theorem says the sum of the squares of the sides of a right triangle is the square of the hippopotamus.  For example, there's a right triangle with sides of length 3, 4, and 5, since

9 + 16 = 25

so

3² + 4² = 5²

We call three integers with these properties a Pythagorean triple.  There are infinitely many!  For example, the next ones are

5² + 12² = 13²              (25 + 144 = 169)

8² + 15² = 17²              (64 + 225 = 289)

There's a nice recipe to get all the Pythagorean triples!  Just take integers n < m and let

a = m² - n²
b = 2mn
c = m² + n²

Then you get

a² + b² = c²

This doesn't give all the Pythagorean triples yet - but you can get the rest by taking a, b, and c and multiplying them all bythe same number.

All this has been... more »

The Pythagorean theorem says the sum of the squares of the sides of a right triangle is the square of the hippopotamus.  For example, there's a right triangle with sides of length 3, 4, and 5, since

9 + 16 = 25

so

3² + 4² = 5²

We call three integers with these properties a Pythagorean triple.  There are infinitely many!  For example, the next ones are

5² + 12² = 13²              (25 + 144 = 169)

8² + 15² = 17²              (64 + 225 = 289)

There's a nice recipe to get all the Pythagorean triples!  Just take integers n < m and let

a = m² - n²
b = 2mn
c = m² + n²

Then you get

a² + b² = c²

This doesn't give all the Pythagorean triples yet - but you can get the rest by taking a, b, and c and multiplying them all by the same number.

All this has been known for a long time - Euclid wrote about it around 300 BC.  There's a lot more to say, but not now!

Yesterday the guy who fixes my computers, David Scharffenberg, told me that

3³ + 4³ + 5³ = 6³        (27 + 64 + 125 = 216)

That's great!  It looks like a generalization of

3² + 4² = 5²

But it's not really a generalization in any way that I know.  As far as I know, it's just a wonderfully cute, meaningless coincidence.  I could be wrong.

When is the sum of 3 cubes a cube?  I don't know, but there's a conjecture saying that any number except for those of the form 9k+4 and 9k-4 is the sum of 3 cubes. 

Puzzle 1: why can't numbers of the form 9k+4 or 9k-4 for some integer k be written as the sum of 3 cubes of integers?

For example, 29 is the sum of 3 cubes:

3³ + 1³ + 1³ = 29

But cubes can be negative!   This makes it harder to find all the solutions.  For example, we also have

4³ + (-2)³ + (-3)³ = 29

So, was only rather recently that someone showed that 30 is the sum of 3 cubes!

Puzzle 2: write 30 as the sum of 3 cubes of integers.___

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2015-01-07 21:36:29 (19 comments, 21 reshares, 75 +1s)Open 

My friend +Simon Willerton writes:

A mathematician hands out a pack of cards to a group of five people. They repeatedly cut the deck and then take a card each.

The mathematician tries to use telepathy to divine the cards that the people are holding - but unfortunately due to solar disturbances, the mind waves are a bit scrambled and the mathematician has to ask a few questions to help unscramble the images being received:

“Who had porridge for breakfast?”

“Who is holding a red card?”

“Is anyone a Pisces?”

“Who has a dog called Stanley?”

The answers to these questions are sufficient to allow the mathematician to name the card that each person is holding. The audience applauds wildly.

How does this trick work?  It uses a lot of math - and it's explainedin a book by the... more »

My friend +Simon Willerton writes:

A mathematician hands out a pack of cards to a group of five people. They repeatedly cut the deck and then take a card each.

The mathematician tries to use telepathy to divine the cards that the people are holding - but unfortunately due to solar disturbances, the mind waves are a bit scrambled and the mathematician has to ask a few questions to help unscramble the images being received:

“Who had porridge for breakfast?”

“Who is holding a red card?”

“Is anyone a Pisces?”

“Who has a dog called Stanley?”

The answers to these questions are sufficient to allow the mathematician to name the card that each person is holding. The audience applauds wildly.

How does this trick work?  It uses a lot of math - and it's explained in a book by the famous mathemagicians +Persi Diaconis and Ron Graham.  Diaconis is the one who ran away from home to join the circus and wound up as a math grad student at Harvard and then a professor at Stanford.

It sounds like a fun book!  But you can learn how this trick works by reading Simon's blog post:

https://golem.ph.utexas.edu/category/2015/01/mathematics_and_magic_the_de_b.html

Beware: it uses some serious mathematics!

#cardtricks #mathematics___

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2015-01-05 16:31:59 (209 comments, 34 reshares, 98 +1s)Open 

Why Google Gave Up On Global Warming

I was disappointed when Google gave up. In 2007, they announced a bold initiative to fight global warming.  They wanted to replace a gigawatt of coal power by renewable energy, in less than a decade.  In 2011, they gave up.

Now two engineers in the project have said why.  It's important to read this! 

http://johncarlosbaez.wordpress.com/2015/01/05/why-google-gave-up/

Why Google Gave Up On Global Warming

I was disappointed when Google gave up. In 2007, they announced a bold initiative to fight global warming.  They wanted to replace a gigawatt of coal power by renewable energy, in less than a decade.  In 2011, they gave up.

Now two engineers in the project have said why.  It's important to read this! 

http://johncarlosbaez.wordpress.com/2015/01/05/why-google-gave-up/___

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2015-01-04 17:20:34 (109 comments, 50 reshares, 267 +1s)Open 

The most dangerous animal in the world

An adult male grizzly bear can stand 3 meters tall (almost 10 feet) on its hind legs.  A big one can weigh 360 kilograms (almost 800 pounds).

But that's not the really dangerous animal in this picture.  A human being won this contest — with a gun. 

Luckily it was a dart gun.  This bear, near Vancouver, is sedated, about to be tagged by scientists.  It will be fine, losing only a bit of its dignity.

Derrick Jensen wrote a book Thought to Exist in the Wild: Awakening from the Nightmare of Zoos.   Here are some quotes:

-------

The bear takes seven steps, her claws clicking on concrete. She dips her head, turns, and walks toward the front of the cage. Another dip, another turn, another three steps. When she gets back to where she started, she begins all over. This is what’s left of herlife. 
more »

The most dangerous animal in the world

An adult male grizzly bear can stand 3 meters tall (almost 10 feet) on its hind legs.  A big one can weigh 360 kilograms (almost 800 pounds).

But that's not the really dangerous animal in this picture.  A human being won this contest — with a gun. 

Luckily it was a dart gun.  This bear, near Vancouver, is sedated, about to be tagged by scientists.  It will be fine, losing only a bit of its dignity.

Derrick Jensen wrote a book Thought to Exist in the Wild: Awakening from the Nightmare of Zoos.   Here are some quotes:

-------

The bear takes seven steps, her claws clicking on concrete. She dips her head, turns, and walks toward the front of the cage. Another dip, another turn, another three steps. When she gets back to where she started, she begins all over. This is what’s left of her life. 

Outside the cage, people pass by on a sidewalk. Parents stop strollers until they realize there’s nothing here to see. A pair of teenagers approach, wearing Walkmans and holding hands; one glance inside is enough, and they’re off to the next cage. Still the bear paces; three steps, head dip, turn.

My fingers are wrapped tightly around the metal railing outside the enclosure. I notice they’re sore. I look at the silver on the bear’s back, the concave bridge of her nose. I wonder how long she’s been here. I release the rail, and as I walk away, the rhythmic clicking of claws on concrete slowly fades.

Unfortunately most of us by now have been to enough zoos to be familiar with the archetype of the creature who has been driven insane by confinement: the bear pacing a precise rectangle; the ostrich incessantly clapping his bill; the elephants rhythmically swaying. But the bear I describe is no archetype. She is a bear. She is a bear who, like all other bears, at one time had desires and preferences all her own, and who may still, beneath the madness.

Or at this point she may not.

[...]

If you see an animal in a zoo, you are in control. You can come, and you can go. The animal cannot. She is at your mercy; the animal is on display for you.

In the wild, the creature is there for her own purposes. She can come, and she can go. So can you. Both of you can display as much of yourselves to the other as you wish. It is a meeting of equals. And that makes all the difference in the world.

One of the great delights of living far from the city is getting to know my nonhuman neighbors — the plants, animals, and others who live here. Although we’ve occasionally met by chance, I’ve found that it is usually the animals who determine how and when they reveal themselves to me. The bears, for example, weren’t shy, showing me their scat immediately and their bodies soon after, standing on hind legs to put muddy paws on windows and look inside; or offering glimpses of furry rumps that disappeared quickly whenever I approached on a path through the forest; or walking slowly like black ghosts in the deep gray of predawn. Though I am used to their being so forward, it is always a gift when they reveal themselves, as one did recently when he took a swim in the pond in front of me.

Robins, flickers, hummingbirds, and phoebes all present themselves, too. Or rather, like the bear, they present the parts of themselves they want seen. I see robins often, and a couple of times I’ve seen fragments of blue eggshells long after the babies have left, but I’ve never seen their nests.

These encounters — these introductions — are on terms chosen by those who were on this land long before I was: they choose the time, place, and duration of our meetings. Like my human neighbors and friends, they show me what they want of themselves, when they want to show it, how they want to show it, and for that I am glad. To demand they show me more — and this is as true for nonhumans as it is for humans — would be unconscionably rude. It would destroy any potential our relationship may once have had. It would be unneighborly.

I am fully aware that even a young bear can kill me. I am also fully aware that humans have coexisted with bears and other wild animals for tens of thousands of years. Nature is not scary. It is not a den of fright and horrors. For almost all of human existence, it has been home, and the wild animals have been our neighbors.

Right now, worldwide, more than 1 million people die each year in road accidents. In the United States alone, there are about forty-two thousand traffic fatalities a year. Yet I am not afraid of cars — though perhaps I should be. Around the world, nearly 2 million people per year are killed through direct violence by other people. Almost 5 million people die each year from smoking. And how many people do bears kill? About one every other year in all of North America.

We are afraid of the wrong things.

[...]

I’m at a zoo. Everywhere I see consoles atop small stands. Each console has a cartoonish design aimed at children, and each has a speaker with a button. When I push the button, I hear a voice begin the singsong: “All the animals in the zoo are eagerly awaiting you.” The song ends by reminding the children to be sure to “get in on the fun.”

I look at the concrete walls, the glassed-in spaces, the moats, the electrified fences. I see the expressions on the animals’ faces, so different from the expressions of the wild animals I’ve seen. The central conceit of the zoo, and in fact the central conceit of this whole culture, is that all of these “others” have been placed here for us, that they do not have any existence independent of us, that the fish in the oceans are waiting there for us to catch them, that the trees in the forests stand ready for us to cut them down, that the animals in the zoo are there for us to be entertained by them.

It may be flattering to believe that everything is here to serve you, but in the real world, where real creatures exist and real creatures suffer, it’s narcissistic and dangerous to pretend nobody matters but you.

-------

For more of Derrick Jensen's book, see:

http://thesunmagazine.org/issues/383/thought_to_exist_in_the_wild

I got the photo from Sean Sparling's Twitter feed:

https://twitter.com/sasagronomy

I do not know who took it.___

posted image

2015-01-03 17:57:01 (11 comments, 0 reshares, 52 +1s)Open 

Rock Point

After visiting Canyon de Chelly, my wife and I drove up north to Mesa Verde, another famous cliff dwelling up in Colorado.   We never quite made it to Mesa Verde - it became too snowy, and they wouldn't let us in the park without snow tires or chains.  But the drive was still worthwhile.

We went back out through Chinle, the main town near Canyon de Chelly. Then we drove north through Arizona along Route 191, through the lonely lands of the Navajo Nation.   Small isolated hogans, groups of two or three houses, and occasional Navajo chapter houses for meetings. 

As we approached Colorado the landscape became dramatic, with weird red sandstone formations poking up through the flat plains.  We passed the town of Rock Point, shown here.  Well, okay, we'd already passed it by the time I took this shot - it went by pretty fast.
Then ... more »

Rock Point

After visiting Canyon de Chelly, my wife and I drove up north to Mesa Verde, another famous cliff dwelling up in Colorado.   We never quite made it to Mesa Verde - it became too snowy, and they wouldn't let us in the park without snow tires or chains.  But the drive was still worthwhile.

We went back out through Chinle, the main town near Canyon de Chelly. Then we drove north through Arizona along Route 191, through the lonely lands of the Navajo Nation.   Small isolated hogans, groups of two or three houses, and occasional Navajo chapter houses for meetings. 

As we approached Colorado the landscape became dramatic, with weird red sandstone formations poking up through the flat plains.  We passed the town of Rock Point, shown here.  Well, okay, we'd already passed it by the time I took this shot - it went by pretty fast.

Then Route 191 hit Route 160, and we headed east through crazy landscapes until the land flattened out and we reached Teec Nos Pos.   This is a big city for these parts, with about 800 inhabitants.   The name comes from the Navajo T’iis Názbąs, meaning 'cottonwoods in a circle'.

Teec Nos Pos appears in the wonderful mystery novels by Tony Hillerman, so we had to stop and check out the trading post!  Trading posts serve as grocery stores, drug stores, pawn shops, places to buy and sell jewelry, blankets and saddles, and more.   Lisa got into a long conversation with John McCullough, owner of the trading post, who showed us the turquoise and silver necklaces made by the Navajo, called squash blossom necklaces for their design.

The picture here is mine; you can see some better pictures of Rock Point here:

https://www.google.com/maps/place/Rock+Point,+AZ+86545/@36.714115,-109.6286595,17719m/data=!3m2!1e3!4b1!4m2!3m1!1s0x8730a122fbed1265:0xe52b0ff39f31a2dd___

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2015-01-02 18:05:23 (47 comments, 54 reshares, 231 +1s)Open 

Shadows of higher dimensions

The icosidodecahedron can be built by truncating either a regular icosahedron or a regular dodecahedron.  It's a beautiful, highly symmetrical shape with 30 vertices.

But it's just a shadow of an even more symmetrical shape with twice as many vertices, living in a space with twice as many dimensions!  The D6 root polytope is a 6-dimensional shape with 60 vertices.  When you project it down to 3 dimensions in the right way, you get the picture here, made by Greg Egan. 

In this 3d picture, the 60 vertices of the D6 root polytope lie on two icosidodecahedra, one larger than the other.  How much larger?  The golden ratio!

For the details of how this works, visit my blog Visual Insight:

http://blogs.ams.org/visualinsight/2015/01/01/icosidodecahedron-from-projected-d6-root-polytope/
The... more »

Shadows of higher dimensions

The icosidodecahedron can be built by truncating either a regular icosahedron or a regular dodecahedron.  It's a beautiful, highly symmetrical shape with 30 vertices.

But it's just a shadow of an even more symmetrical shape with twice as many vertices, living in a space with twice as many dimensions!  The D6 root polytope is a 6-dimensional shape with 60 vertices.  When you project it down to 3 dimensions in the right way, you get the picture here, made by Greg Egan. 

In this 3d picture, the 60 vertices of the D6 root polytope lie on two icosidodecahedra, one larger than the other.  How much larger?  The golden ratio!

For the details of how this works, visit my blog Visual Insight:

http://blogs.ams.org/visualinsight/2015/01/01/icosidodecahedron-from-projected-d6-root-polytope/

The vertices of the D6 root polytope are

(±1, ±1, 0, 0, 0, 0)

and all vectors you can get by permuting the coordinates.  You can map 6-dimensional space down to 3-dimensional space in a way that sends these to the vertices of two icosidodecahedra, one larger than the other by a factor of the golden ratio. 

But there's a deep underlying reason why you can do it, which my blog article explains!    And the same idea lets you map a beautiful polytope in 8 dimensions down to one in 4 dimensions.   The 8-dimensional one is called the E8 root polytope, and it has 240 vertices.  The 4-dimensional one is called the 600-cell, because it has 600 tetrahedral faces - and it has 120 vertices.  Unfortunately all this stuff is harder to draw.

Puzzle: why is it called an 'icosidodecahedron' instead of an 'icosadodecahedron'?   I've never understood that.

#geometry  ___

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2015-01-01 18:56:17 (91 comments, 28 reshares, 161 +1s)Open 

Happy New Year!

The Earth has successfully completed another revolution about the Sun!   Thanks to all of you for making it an interesting orbit. 

I really like talking to smart people, so please keep posting articles where you actually say something - not just links.  And please keep commenting on my posts, offering your thoughts - not just +1s.

New Year's Day is a traditional excuse for a moment of reflection, so let's try that.  What have you been doing this year?  Do you feel you're making progress, doing good stuff?   Struggling?

It's been an interesting and stressful year for me.  I'm struggling to do some more practical things for the health of our planet.  I believe global warming is a serious problem and we're facing a mass extinction event.  I can't just sit around.   But my love for the beauty of pure mathand theoreti... more »

Happy New Year!

The Earth has successfully completed another revolution about the Sun!   Thanks to all of you for making it an interesting orbit. 

I really like talking to smart people, so please keep posting articles where you actually say something - not just links.  And please keep commenting on my posts, offering your thoughts - not just +1s.

New Year's Day is a traditional excuse for a moment of reflection, so let's try that.  What have you been doing this year?  Do you feel you're making progress, doing good stuff?   Struggling?

It's been an interesting and stressful year for me.  I'm struggling to do some more practical things for the health of our planet.  I believe global warming is a serious problem and we're facing a mass extinction event.  I can't just sit around.   But my love for the beauty of pure math and theoretical physics keeps pulling me back to the things I used to think about.  I feel torn and frustrated.

With my pals at the Azimuth Project, we've reached the point of understanding a bit about El Niño prediction - I gave a talk about this to about 1000 people at the Neural Information Processing Seminar, a big annual conference on machine learning.  We made some good progress.  But we've only just dipped our toes into a very deep subject.  To go further I'd need to learn a lot more, get serious about programming, and start attending the annual conference on Climate Informatics.  I'd need to get better at working with folks in the Azimuth Project, and pull more experts into it.   And most of all: I'd need to think harder about climate science and the art of prediction, and come up with some new ideas. 

By comparison, it seems easy to come up with new ideas in pure math and theoretical physics - because I spent decades doing it.  Unfortunately, it feels a bit pointless.  I don't think the world urgently needs to understand more about the fundamental laws of physics, not right now.  Someday it will be important.  But fundamental physics doesn't hold the 'magic bullet' for the problems we face today.  And anyway, we've already got a lot of very smart people banging their heads against that wall.  We need something a bit different.  I'm in a lucky position where I can afford to thrash around trying to figure out what that is.  If 1000 of us try, some will succeed, and we may do a bit better finding our way through the ecological bottleneck. 

That's what I tell myself, anyway.  But I also just love pure math regardless of whether it's good for anything.  So right now I'm pursuing it as a kind of 'hobby'.  It helps me relax.  I've stepped aside from the great mathematical challenge of our time - developing the theory of infinity-categories and the new world of math this opens up.  Instead, I'm thinking about 'exceptional structures' in algebra, and their role in physics: things like the octonions, the group called E8, and the Leech lattice.  I've put enough time into these over the years that I can come up with cute ideas without a massive investment of effort.... thanks to help from Greg Egan, who is great at proving or disproving my conjectures.

As a kind of middle road, I'm also working with my grad students on 'network theory' - basically, applying category theory to comlex systems made of interacting parts, as we see in biology, chemistry, electrical engineering, and the like.  This is not instantly useful; it will take years to develop.  But I have a good feeling about it!  This might be a place where fancy abstract math can do some good.

So I guess it's a 3-pronged approach to life.   It gets to be a bit much at times!  And then there's the job I actually get paid for: teaching.  I may be doing too many things to do any of them well.  

But I'm rarely bored.  When I was a kid, I was often bored.  I didn't know how to do the cool things I dreamt of doing.  I hated it.  Those days are gone.  I'm very happy about that.

In case you're wondering, this is the Higman-Sims graph, an exceptional structure lurking in the Leech lattice, animated by David Madore here:

https://www.youtube.com/watch?v=neUd794Gbg0

He writes:

The Higman-Sims graph is the unique graph with 100 vertices such that each is adjacent to 22 others and no two adjacent vertices have a common neighbor (i.e., the graph has no triangle) and any two non-adjacent vertices have exactly six common neighbors. It has 88704000 automorphism, forming an extension of 2 by the unique simple group of order 44352000 (the Higman-Sims group, a sporadic group).

The Higman-Sims graph occurs inside the 24-dimensional Leech lattice (if X,Y,Z are Leech lattice points at distances 3,3,2 from each other, then there are 100 Leech lattice points at distance 2,2,2 from X,Y,Z, and if we connect those at distance 3 from another, we obtain the Higman-Sims graph).

This animation displays various orthogonal projections of the Higman-Sims graph inside the Leech lattice, chosen so as to reveal an 11-fold symmetry (there is only one conjugacy class of order 11 in the Conway group ·0, which is in the Higman-Sims group).

For more, try:

https://en.wikipedia.org/wiki/Higman-Sims_graph
https://en.wikipedia.org/wiki/Higman-Sims_group___

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2014-12-30 16:56:52 (14 comments, 24 reshares, 172 +1s)Open 

White House Ruin

I took this photo from the bottom of Canyon de Chelly, in Arizona. These buildings, called the White House Ruin, were built around 1200 AD.  A bit later, the civilization that built them disappeared!

They're called the Anasazi, or Ancient Pueblo People.  Starting around 800 AD, they started building great houses: multi-storied buildings with high ceilings, rooms much larger than you'd see in houses, and elaborate subterranean rooms called kivas. And around 900 AD, they started building houses with stone roofs. We call this the start of the Pueblo II Era.

For a long time their civilization was centered in Chaco Canyon, in New Mexico, 125 kilometers east of Canyon de Chelly.  Their biggest great house was founded in the 800s.  Starting in 1020 it grew immensely, and it kept growing until 1120. By this timeit ... more »

White House Ruin

I took this photo from the bottom of Canyon de Chelly, in Arizona. These buildings, called the White House Ruin, were built around 1200 AD.  A bit later, the civilization that built them disappeared!

They're called the Anasazi, or Ancient Pueblo People.  Starting around 800 AD, they started building great houses: multi-storied buildings with high ceilings, rooms much larger than you'd see in houses, and elaborate subterranean rooms called kivas. And around 900 AD, they started building houses with stone roofs. We call this the start of the Pueblo II Era.

For a long time their civilization was centered in Chaco Canyon, in New Mexico, 125 kilometers east of Canyon de Chelly.  Their biggest great house was founded in the 800s.  Starting in 1020 it grew immensely, and it kept growing until 1120. By this time it had 700 rooms, nearly half devoted to grain storage. It also had 33 kivas.

But this was just one of a dozen great houses built in Chaco Canyon by 1120. About 215 thousand ponderosa pine trees were cut down in this building spree!  Building these houses probably took over 2 million man-hours of work. They also built about 650 kilometers of roads!  Most of these connect one great house to another… but some mysteriously seem to go to ‘nowhere’.

By 1080, however, the summer rainfall had started to decline. And by 1090 there were serious summer drought lasting for five years. We know this sort of thing from tree rings: there are enough ponderosa logs and the like that archaeologists have built up a detailed year-by-year record.

Starting around 1100 AD, many of the ancient Pueblo people left the Chaco Canyon area. Many moved upland, to places with more rain and snow. Instead of great houses, many returned to building the simpler pit houses of old.

By 1150 AD, some of the ancient Pueblo people began building cliff dwellings at higher elevations—like Mesa Verde in Colorado  This marks the start of the Pueblo III Era.  The settlements in Canyon de Chelly, shown here, date to 1200.   But this era lasted a short time. By 1350, all these cliff dwellings were abandoned!

The people didn't leave... they're still around.  But they stopped building large settlements.   Why?  For some answers, read my article:

http://johncarlosbaez.wordpress.com/2013/01/24/anasazi-america-part-2/

#archaeology #history  ___

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2014-12-29 18:16:34 (20 comments, 0 reshares, 51 +1s)Open 

Canyon de Chelly

Over Christmas, Lisa and I drove to Naabeehó Bináhásdzo - a semi-autonomous territory sprawling for 71 thousand square kilometers inside northeastern Arizona, southeastern Utah, and northwestern New Mexico.  English speakers call it the Navajo Nation

At its heart is Canyon de Chelly, home to a wonderful ancient cliff dwelling called the White House Ruin.  We'd been there before, but we couldn't resist another look.  It was a great day, so we hiked down in.  This is the view from on top.

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

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

Canyon de Chelly

Over Christmas, Lisa and I drove to Naabeehó Bináhásdzo - a semi-autonomous territory sprawling for 71 thousand square kilometers inside northeastern Arizona, southeastern Utah, and northwestern New Mexico.  English speakers call it the Navajo Nation

At its heart is Canyon de Chelly, home to a wonderful ancient cliff dwelling called the White House Ruin.  We'd been there before, but we couldn't resist another look.  It was a great day, so we hiked down in.  This is the view from on top.

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

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

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2014-12-26 16:00:53 (244 comments, 203 reshares, 1443 +1s)Open 

Pals

This photo is almost unbearably cute!

It was taken by Barry Bland at TIGERS - The Institute for Greatly Endangered and Rare Species, in Myrtle Beach, Florida.

It's interesting to think about why this photo is so cute.

First of all, obviously, the young wolf and tiger seem like pals, walking in step - and the wolf is even smiling!  But more deeply, I think we like the idea that animals of different species, even fierce ones, could be friends.  The lamb may not lie down with the lion, but at least the tiger can play with the wolf!  It gives us hope.

Finally, these are young animals, and thus more friendly, playful and inquisitive than their adult versions... and more cute.  We seem to be innately fond of baby animals, perhaps thanks to our instinct to care for human babies. 

Dogs are neotenized wolves - adult dogs,espe... more »

Pals

This photo is almost unbearably cute!

It was taken by Barry Bland at TIGERS - The Institute for Greatly Endangered and Rare Species, in Myrtle Beach, Florida.

It's interesting to think about why this photo is so cute.

First of all, obviously, the young wolf and tiger seem like pals, walking in step - and the wolf is even smiling!  But more deeply, I think we like the idea that animals of different species, even fierce ones, could be friends.  The lamb may not lie down with the lion, but at least the tiger can play with the wolf!  It gives us hope.

Finally, these are young animals, and thus more friendly, playful and inquisitive than their adult versions... and more cute.  We seem to be innately fond of baby animals, perhaps thanks to our instinct to care for human babies. 

Dogs are neotenized wolves - adult dogs, especially of certain breeds, resemble young wolves, not only in looks (a more round head, etcetera) but in behavior.  Dogs are now considered to be the same species as wolves, just a different subspecies.  We clearly got along best with wolf puppies that stayed friendly and submissive.

We may ourselves be neotenized apes.  I'm not sure what the current thinking on this.  But it could be that intelligence, playfulness and curiosity are traits of youth that proved, in certain social contexts, to be adaptive even for adults.   If so, there could be something profound about 'cuteness'.  Perhaps our attraction to youthful, friendly, playful things helped spawn art, music, science, more merciful codes of morality, and more.___

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2014-12-23 16:23:45 (20 comments, 28 reshares, 141 +1s)Open 

Cages made of ice

Water can freeze to form 'cages' that trap other molecules.  They're called clathrates.  There are several kinds, all beautiful.  Nature is a great geometer!

This one, animated by Isaac Calder here on G+, shows a type I clathrate.  The oxygen atoms in the water are at the corners of 12-sided and 14-sided shapes. 

The 12-sided shapes have pentagons as sides, but they are not regular dodecahedra - if you look carefully, you'll see the pentagons are a bit off.  The 14-sided shapes also have two hexagonal sides! 

All this is very similar to the Weaire-Phelan structure, the best known solution to an old puzzle raised by Kelvin.  He asked how space could be partitioned into cells of equal volume with the least area of surface between them.  He proposed a solution, and for a long time people thought it wasthe best... more »

Cages made of ice

Water can freeze to form 'cages' that trap other molecules.  They're called clathrates.  There are several kinds, all beautiful.  Nature is a great geometer!

This one, animated by Isaac Calder here on G+, shows a type I clathrate.  The oxygen atoms in the water are at the corners of 12-sided and 14-sided shapes. 

The 12-sided shapes have pentagons as sides, but they are not regular dodecahedra - if you look carefully, you'll see the pentagons are a bit off.  The 14-sided shapes also have two hexagonal sides! 

All this is very similar to the Weaire-Phelan structure, the best known solution to an old puzzle raised by Kelvin.  He asked how space could be partitioned into cells of equal volume with the least area of surface between them.  He proposed a solution, and for a long time people thought it was the best possible, but in 1993 Weaire and Phelan found one where the area is 0.3% less.  It looks a lot like this, but the surfaces are curved:

https://en.wikipedia.org/wiki/Weaire-Phelan_structure

For a great view of different clathrate structures, go here:

http://www1.lsbu.ac.uk/water/clathrate_hydrates.html

It's worth learning how to enable Java applets just to see these in motion!   Nowadays Windows makes it really hard to use Java applets that aren't registered in a certain way. 

For Isaac's original post, go here:

https://plus.google.com/u/0/+IsaacCalder/posts/exsVgRTbKT8

#geometry  ___

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2014-12-21 16:43:01 (39 comments, 52 reshares, 178 +1s)Open 

Quantum ants

They're not really quantum, and they're not really ants - but they're cute, and Alexander Vlasov calls them qu-ants.   Here's his explanation:

There are four states: 0 (empty, white), 1 (red), 2 (green), 3 (blue). A step may be divided into two stages:

First stage. Mark all cells satisfying two conditions:

1) the total number of red and blue cells in four closest positions is one or two

2) the cells in the four diagonal positions are either white (empty) or green.

Second stage. Change unmarked red cells to green, unmarked green cells to red, marked empty cells to red, marked red cells to blue, marked green cells to empty, and marked blue cells to green.

----

This is a cellular automaton.  In other words, we've got a regular grid of cells, each colored from somefi... more »

Quantum ants

They're not really quantum, and they're not really ants - but they're cute, and Alexander Vlasov calls them qu-ants.   Here's his explanation:

There are four states: 0 (empty, white), 1 (red), 2 (green), 3 (blue). A step may be divided into two stages:

First stage. Mark all cells satisfying two conditions:

1) the total number of red and blue cells in four closest positions is one or two

2) the cells in the four diagonal positions are either white (empty) or green.

Second stage. Change unmarked red cells to green, unmarked green cells to red, marked empty cells to red, marked red cells to blue, marked green cells to empty, and marked blue cells to green.

----

This is a cellular automaton.  In other words, we've got a regular grid of cells, each colored from some finite set of colors, with a rule for updating all cells simultaneously based on the colors of their neighbors.   But it's also reversible: the previous color of any cell before an update can be determined uniquely from the updated colors of all the cells.  If you've got a reversible cellular automaton, you can run it backwards in time using another cellular automaton rule.

Vlasov actually constructed his qu-ants as a second-order cellular automaton.  This is a different kind of thing, where the color of each square depends on what's going on in its neighborhood in the previous two time steps.  It's easy to make reversible second-order cellular automata... just like how Newton's laws of physics are reversible and given by second-order differential equations.  But the description above conceals this fact, and describes the qu-ants as an ordinary cellular automaton.

For more information and more examples, see:

• Alexander Vlasov, Qu-ants, https://ayvlasov.wordpress.com/2012/07/23/qu-ants/

• B. Schumacher and R.F. Werner, Reversible quantum cellular automata, http://arxiv.org/abs/quant-ph/0405174.

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

#spnetwork #cellularAutomata arXiv:quant-ph/0405174___

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2014-12-20 18:02:12 (68 comments, 27 reshares, 227 +1s)Open 

Alien structure on Mars

Astronomers recently photographed this hole on Mars!  There's no way to explain it by natural processes, and it's very regular in shape, so they believe it was produced by intelligent life.  Since there's no life on Mars now, it must have been made by visitors from some other planet!

This hole is 1.6 centimeters across and 6.6 centimeters deep.  It's in a rock in Gale Crater.  It was drilled by the NASA rover Curiosity on May 19, 2013.

The rock, which NASA dubbed 'Cumberland', is interesting because it's made of ancient mud.  NASA found that the ratio of deuterium to ordinary hydrogen in this rock is half the ratio seen in the water vapor in the Martian atmosphere. This suggests that Mars has lost a lot of water since the formation of Cumberland, probably about 3.6 billion years ago during the HesperianPeri... more »

Alien structure on Mars

Astronomers recently photographed this hole on Mars!  There's no way to explain it by natural processes, and it's very regular in shape, so they believe it was produced by intelligent life.  Since there's no life on Mars now, it must have been made by visitors from some other planet!

This hole is 1.6 centimeters across and 6.6 centimeters deep.  It's in a rock in Gale Crater.  It was drilled by the NASA rover Curiosity on May 19, 2013.

The rock, which NASA dubbed 'Cumberland', is interesting because it's made of ancient mud.  NASA found that the ratio of deuterium to ordinary hydrogen in this rock is half the ratio seen in the water vapor in the Martian atmosphere. This suggests that Mars has lost a lot of water since the formation of Cumberland, probably about 3.6 billion years ago during the Hesperian Period - the period when Mars dried out and its atmosphere thinned to its current density.

Puzzle: Why would water on Mars have more deuterium now? 

A bunch of the clay in Cumberland is smectite.  I had to look that up.  Clay turns out to be quite interesting - like most other things, if you dig deep enough.   Clay minerals are made of tetrahedral sheets of silica and octahedral sheets of hydroxide.  There are two kinds: 1:1 clays and 2:1 clays.   A 1:1 clay consists of alternating layers with one tetrahedral sheet followed by one octahedral sheet: examples are kaolinite and serpentine. A 2:1 clay consists of an octahedral sheet sandwiched between two tetrahedral sheets, and examples are talc, vermiculite and those in the smectite groups.  I should include some pictures of these clay structures... maybe another day.

For more on what they discovered by drilling this hole, read:

• P. R. Mahaffy et al., The imprint of atmospheric evolution in the D/H or Hesperian clay minerals on Mars, Science, 16 December 2014, http://www.sciencemag.org/content/early/2014/12/15/science.1260291.full.pdf?ijkey=rJnJhjOGsS5S.&keytype=ref&siteid=sci

The photo is from this NASA webpage:

http://photojournal.jpl.nasa.gov/catalog/PIA16936

#mars #astronomy #spnetwork doi:10.1126/science.1260291___

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2014-12-19 16:45:53 (15 comments, 27 reshares, 114 +1s)Open 

Knotted vortex tubes

Dolphins blow rings of bubbles and play with them.   Like smoke rings, these are examples of vortex tubes.   You can also make vortex tubes that are knotted! 

Long ago, the physicist Kelvin conjectured that for any kind of knot, you can create a vortex tube shaped like that knot.  He even guessed that atoms were 'knotted vortex tubes in the aether'.   That wasn't true, but his conjecture is still interesting - at least if we state it a bit more precisely.

Say you had a incompressible fluid with no viscosity.   Then its velocity vector field would obey the Euler equations.    The Euler equations have lots of steady solutions where the velocity of the fluid doesn't change with time.  The fluid is still moving, but the same way all the time.

In these steady solutions, the fluid flows alongcurves.   T... more »

Knotted vortex tubes

Dolphins blow rings of bubbles and play with them.   Like smoke rings, these are examples of vortex tubes.   You can also make vortex tubes that are knotted! 

Long ago, the physicist Kelvin conjectured that for any kind of knot, you can create a vortex tube shaped like that knot.  He even guessed that atoms were 'knotted vortex tubes in the aether'.   That wasn't true, but his conjecture is still interesting - at least if we state it a bit more precisely.

Say you had a incompressible fluid with no viscosity.   Then its velocity vector field would obey the Euler equations.    The Euler equations have lots of steady solutions where the velocity of the fluid doesn't change with time.  The fluid is still moving, but the same way all the time.

In these steady solutions, the fluid flows along curves.   These curves can be very complicated!  

Kelvin conjectured that for any knot, there's a stationary solution of the Euler equations where the fluid flows along a curve shaped like that knot.  

This was recently proved to be true!

• Alberto Enciso, Daniel Peralta-Salas, Existence of knotted vortex tubes in steady Euler flows, http://arxiv.org/abs/1210.6271.

This paper will appear in the prestigious math journal Acta Mathematica.   The paper is deep: in addition to a lot of work on topology and differential equations, it even uses some number theory!  To get 'invariant tori' - that is, surfaces of vortex tubes - they use the Kolmogorov-Arnold-Moser theorem.  This requires checking that some flow lines spiral around a torus with a slope that's an irrational number that's hard to approximate by rational numbers!  This condition makes the torus robust against small perturbations.

Here's the abstract:

We prove the existence of knotted and linked thin vortex tubes for steady solutions to the incompressible Euler equation in R^3. More precisely, given a finite collection of (possibly linked and knotted) disjoint thin tubes in R^3, we show that they can be transformed with a C^m-small diffeomorphism into a set of vortex tubes of a Beltrami field that tends to zero at infinity. The structure of the vortex lines in the tubes is extremely rich, presenting a positive-measure set of invariant tori and infinitely many periodic vortex lines. The problem of the existence of steady knotted vortex tubes can be traced back to Lord Kelvin.

A Beltrami field is a vector field with no divergence:

∇ · v = 0

whose curl is proportional to itself:

∇ × v = c v

Any Beltami field gives a steady solution of Euler's equation!

William Irvine at the University of Chicago makes knotted vortex tubes in his lab, and this picture is from there:

http://irvinelab.uchicago.edu/

Real-world fluids have viscosity, and then things get even more complicated and interesting.  Vortex tubes can crash into each other and reconnect, as shown here! 

#spnetwork arXiv:1210.6271 #hydrodynamics #fluiddynamics #knots  ___

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2014-12-18 16:38:45 (36 comments, 18 reshares, 147 +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, 16 reshares, 67 +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  ___

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2014-12-16 18:37:18 (26 comments, 24 reshares, 117 +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 (40 comments, 16 reshares, 92 +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, 2 reshares, 34 +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, 34 reshares, 127 +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 (57 comments, 47 reshares, 147 +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, 31 reshares, 112 +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 (19 comments, 3 reshares, 81 +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, 43 reshares, 149 +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___

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2014-12-05 22:27:34 (39 comments, 19 reshares, 121 +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, 49 +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, 61 reshares, 239 +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, 76 +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, 26 reshares, 188 +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, 167 +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, 91 +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, 136 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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