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John Baez has been shared in 364 public circles

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AuthorFollowersDateUsers in CircleCommentsReshares+1Links
Refurio Anachro4,880febulous engagers circle - Half this circle belongs to my favorite writers' circle here on g+. Thank you for welcoming me back after i've been quiet for over three months! The other half is new people. I was excited to find out that many of you do math posts, which is incredibly cool! Happy to have you here!Last month's posts:A little teaser: I'd like to write about moduli spaces sometime soonhttps://plus.google.com/+RefurioAnachro/posts/Fkwc4f1aMJuMy desire has been heard, and other fun posts about primes and field extensions came up in the following week. +wendy krieger was quick to share a nice piece on cyclotomic numbers, which are a cool and easy example for field extensions. You did it again, were this my blog i'd've invited you to do a guest post.–On the indescribable voids between humongous numbershttps://plus.google.com/+RefurioAnachro/posts/J4uze3DhTVnThe topic has been haunting me quite a while, so i was relieved to realize that disallowing to write numbers greater than 10 is key to simplify a discussion to actually expose the gaps there are between very large numbers.If you liked the style, maybe you like these 2013 posts of mine as well. I do now have some material to make one of these for Leonard Susskind or Alan Guth:"honored scientist goes crazy, video included"Roger Penrose - conformal cyclic cosmologyhttps://plus.google.com/+RefurioAnachro/posts/7U5rDDnuGrFJohn Horton Conway - the free will theoremhttps://plus.google.com/+RefurioAnachro/posts/97uRZ93wwHD–A sophisticated method to lose everythinghttps://plus.google.com/+RefurioAnachro/posts/UrRAGd2gi6UMy first serious attempt to make sense of trading tales, the maths isn't too hard, but getting a precise idea we can analyze was. Special thanks to +Mert Meral for offering to share his insider's insights, i'll come back to you soon!–Grothendieck circlehttps://plus.google.com/+RefurioAnachro/posts/2m9MWC3L7UNAs promised, i gathered people who had something to say about Alexander Grothendieck since he has passed away. To find them (find you?), i used g+'s search bar. If you're looking for more followers, or to enhance your stream: Just pick a topic and try for yourself, and don't forget to share what you've found afterwards!–reshared: A foldable papercut net for the small cubicuboctahedron (part 3)https://plus.google.com/+RefurioAnachro/posts/d3fSeSRLMYZreshared: A triangle-centered net for the small cubicuboctahedron (part 2)https://plus.google.com/+RefurioAnachro/posts/TaefxUqJiov+2015-03-04 11:18:2598011
Robert Best2,617Google+ Relevance CircleWhat people of "relevance" do we have in common? Care to share?This big shared circle contains my most "relevant" circles according to G+. It also only contains people/profiles who post publicly (Either occasionally or prolifically).Do you recognize others in the list? (I think the probability is high)As far as I know, the visual beside the "Add people" button below shows you the 8 people from this circle who are most "relevant" to you. Also, if you click that add people button, the list inside is also ordered by "relevance".Some people in this circle I know very well... others I haven't met at all (Besides interfacing with what they share publicly on G+) I'm curious, what circles do we share of high relevance? Give a few people shout outs! Let's have a random chat.You were likely notified of this post... I normally don't share to any of my specific circles, so as to avoid sending out unwanted notifications... But since I have you and your attention here... HELLO! Have we talked before? Why do you think I have you circled? What's of relevance for you? (Besides people on G+) If we do know each other... Have we been in touch lately? If not, let please catch up!2015-03-03 01:03:39455000
Paul Hutchinson2,176A G+ #FF post ... Here's my "Science" circle (at least 501 of the 1105 people in the circle)#tw #fb2015-02-27 21:06:56501000
Brian Mcquillan23,952Boost your visibility on Google+ - Share the circle!To be added to or stay in this Circle - Share the circleIf 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 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!#circles #Gif #Cute #Anime #Animals #Online #Life #CatLovers #Cat #FunnyPics #Quote #Art #CaturdayEveryday #QuoteOfTheDay #Truth #Dog #Dogs #XD #Meme #LOL #Humor #Cute #Anime #Gif #Animals #Cat #CatLovers #Art #Online #Cats #Life #FunnyPics #CaturdayEveryday #Dog #Quote #Dogs #Truth #Manga 2015-02-27 10:33:26381133
Joshua Fisher532My Math Ed circle!2015-02-19 12:29:43455203
Richard Green93,025Engagers Showcase Circle, February 1, 2015If 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.Ramanujan's nested radicalhttps://plus.google.com/101584889282878921052/posts/3WmvWEHyMNBThe exceptional symmetryhttps://plus.google.com/101584889282878921052/posts/ioQW2zGjwwMThe mystery of the missing areahttps://plus.google.com/101584889282878921052/posts/QU8aYaTCufqSunrisehttps://plus.google.com/101584889282878921052/posts/9QWbSALP2XUShakespearean Logichttps://plus.google.com/101584889282878921052/posts/MBqnMRgBiBJ“Nines” by Eric Standleyhttps://plus.google.com/101584889282878921052/posts/aR5BF9uV5n8Cherry pi (reshared from +David Richeson)https://plus.google.com/101584889282878921052/posts/bixJ7eGk3QmHappy New Year!https://plus.google.com/101584889282878921052/posts/ZJYFj1RogaSThe mathematics of card shufflinghttps://plus.google.com/101584889282878921052/posts/fAS8Y3YccfsThe sky and the fork in the path immediately preceding the arrival of the ice rinks of doomhttps://plus.google.com/101584889282878921052/posts/2MiqiTJ1FeuiPad landscape art (reshared from +Paul Haworth)https://plus.google.com/101584889282878921052/posts/27EwU49z1g8The fractional chromatic number of the planehttps://plus.google.com/101584889282878921052/posts/VbBk9JrLxqmDull (in Scotland) and Boring (in Oregon)https://plus.google.com/101584889282878921052/posts/ZuMzApfSPR4Partition and sum is fasthttps://plus.google.com/101584889282878921052/posts/Ad1ism1vJpJThe tautological clockhttps://plus.google.com/101584889282878921052/posts/diAyvxM7NuwMathematical Mr Menhttps://pl2015-02-01 15:55:59476198139246
Crazy Cats22 Public #circleshare   January 20, 2015Hope 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 #cirlce  !!!***************************************************************Boost your visibility on Google+ - Share the circle!Boost your visibility on Google+ - Share the circle!Follow me here : http://goo.gl/7rWIEVTo 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.Special Invite :+A Tech Buzz +Axel Kratel +Andrea Gervasi +Andrew Sowerby +Anette Mossbacher +Brett Szmajda +Andrew Sowerby +Irina Sadokhina +Sean Carroll +Michael Sonntag +Mighty Dragon Studios +Eric Delcour +2015-02-01 10:25:32474215
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 20:15:35473419
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 10:11:403392513
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 23:42:1339411476159
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 06:41:38487435270
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 11:32:544427557138
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 05:24:2236226837
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 06:51:44407103
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 05:34:0348613617
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

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Most comments: 244

posted image

2014-12-26 16:00:53 (244 comments, 201 reshares, 1432 +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: 240

posted image

2015-01-25 19:08:51 (117 comments, 240 reshares, 1950 +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: 1950

posted image

2015-01-25 19:08:51 (117 comments, 240 reshares, 1950 +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 »

Latest 50 posts

posted image

2015-03-04 16:36:17 (21 comments, 10 reshares, 54 +1s)Open 

Explosions make new craters in Siberia

Russian scientists have recently found more new craters in Siberia, apparently formed by explosions of methane.  3 were found last summer.  They looked for more using satellite photos... and found them.

“What I think is happening here is, the permafrost has been acting as a cap or seal on the ground, through which gas can’t permeate,” says Paul Overduin, a permafrost expert at the Alfred Wegener Institute in Germany. “And it reaches a particular temperature where there’s not enough ice in it to act that way anymore. And then gas can rush out.”

Since methane is a powerful greenhouse gas, some people are getting nervous.  If global warming releases the huge amounts of methane trapped under permafrost, will that create more global warming?

Yes, but Arctic methane will not be our main problem in the nextfew decades.  I... more »

Explosions make new craters in Siberia

Russian scientists have recently found more new craters in Siberia, apparently formed by explosions of methane.  3 were found last summer.  They looked for more using satellite photos... and found them.

“What I think is happening here is, the permafrost has been acting as a cap or seal on the ground, through which gas can’t permeate,” says Paul Overduin, a permafrost expert at the Alfred Wegener Institute in Germany. “And it reaches a particular temperature where there’s not enough ice in it to act that way anymore. And then gas can rush out.”

Since methane is a powerful greenhouse gas, some people are getting nervous.  If global warming releases the huge amounts of methane trapped under permafrost, will that create more global warming?

Yes, but Arctic methane will not be our main problem in the next few decades.  It's only after a century, or longer, that it could become a major cause of global warming.   Right now, by far the most important cause of global warming is human-released CO2.  If we get that under control, we can still avoid a runaway feedback loop.

David Archer of the University of Chicago, a famous expert on climate change and the carbon cycle. took a look at these craters and did some quick calculations.  He estimated that “it would take about 20,000,000 such eruptions within a few years to generate the standard Arctic Methane Apocalypse that people have been talking about.”

More importantly, people are measuring the amount of methane in the air.   We know how it's doing.

For stunning photographs of the new craters, and maps showing where they are, visit the Azimuth blog:

https://johncarlosbaez.wordpress.com/2015/03/04/melting-permafrost-part-4/

You can make graphs of methane concentration here:

http://www.esrl.noaa.gov/gmd/dv/iadv/

Click on a northern station like Svalbard.  Choose Carbon cycle gases from the menu at right and click Time series.  You'll go to another page, and choose Methane - the default choice is carbon dioxide.  Go to the bottom of the page and click Submit and you'll get a graph.

On Svalbard, an island sort of near Greenland, methane has gone up from about 1.85 to 1.9 parts per million during the years from 1994 to 2015.  That's a big increase - but not a sign of disaster anytime soon.

You can actually do lots of fun things with this data!  Play around!___

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2015-03-02 16:23:58 (29 comments, 10 reshares, 71 +1s)Open 

Climate leadership

In December, all the countries in the world will meet in Paris and try to sign a legally binding agreement to cut carbon emissions.  Right now they're on the The Road to Paris.

Last week Switzerland became the first country to officially announce what it will do.   By 2030, they will reduce their greenhouse gas emissions to half their 1990 level.

In November, the US and China jointly announced some goals.   China's greenhouse gas emissions will peak by 2030 at the latest.  The US will cut emissions by at least 25% by 2025, compared to 2005 levels.  These targets are not legally binding - but they help set the stage for Paris.

Some other news: it seems the European Union will propose to cut global greenhouse gas emissions by at least 60% by 2050, compared to their 2010 levels.

For how The Road to Paris works,see ... more »

Climate leadership

In December, all the countries in the world will meet in Paris and try to sign a legally binding agreement to cut carbon emissions.  Right now they're on the The Road to Paris.

Last week Switzerland became the first country to officially announce what it will do.   By 2030, they will reduce their greenhouse gas emissions to half their 1990 level.

In November, the US and China jointly announced some goals.   China's greenhouse gas emissions will peak by 2030 at the latest.  The US will cut emissions by at least 25% by 2025, compared to 2005 levels.  These targets are not legally binding - but they help set the stage for Paris.

Some other news: it seems the European Union will propose to cut global greenhouse gas emissions by at least 60% by 2050, compared to their 2010 levels.

For how The Road to Paris works, see this:

http://www.wclimate.com/process/

For more on The Road to Paris:

http://roadtoparis.info/___

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2015-03-01 17:01:39 (45 comments, 68 reshares, 170 +1s)Open 

The beauty of numbers

The beauty of number theory is usually not visible to the eye: we may enjoy facts about primes, but we don't often see them. 

Secretly, however, modern number theory involves a lot of geometry - and Katherine Stange has found a nice way to make some of this geometry visible!

A bigger version of this picture reveals its full beauty:

http://blogs.ams.org/visualinsight/2015/03/01/schmidt-arrangement/

and here you can read an explanation of what's going on, with a link to Katherine Stange's paper - and more of her pictures!

#spnetwork #arXiv:1410.0417 #numbertheory  

The beauty of numbers

The beauty of number theory is usually not visible to the eye: we may enjoy facts about primes, but we don't often see them. 

Secretly, however, modern number theory involves a lot of geometry - and Katherine Stange has found a nice way to make some of this geometry visible!

A bigger version of this picture reveals its full beauty:

http://blogs.ams.org/visualinsight/2015/03/01/schmidt-arrangement/

and here you can read an explanation of what's going on, with a link to Katherine Stange's paper - and more of her pictures!

#spnetwork #arXiv:1410.0417 #numbertheory  ___

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2015-02-26 16:07:21 (27 comments, 20 reshares, 98 +1s)Open 

Synthesis

A cloud of ideas coalesces to form a brilliant insight.  At first it seems big and important.  Then you start taking it for granted... and it becomes one of the ingredients of your next insight.

I got this from Maria Dubai:

https://plus.google.com/u/0/109975504645168908521/posts/RWxB9Zi4ixx

Synthesis

A cloud of ideas coalesces to form a brilliant insight.  At first it seems big and important.  Then you start taking it for granted... and it becomes one of the ingredients of your next insight.

I got this from Maria Dubai:

https://plus.google.com/u/0/109975504645168908521/posts/RWxB9Zi4ixx___

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2015-02-25 21:06:02 (44 comments, 27 reshares, 93 +1s)Open 

You can never step in the same river twice

People say this remark is from Heraclitus.  The main idea is that the river keeps changing as the water flows.  The other idea is that you keep changing, too! 

Jorge Luis Borges wrote:

… each time I recall fragment 91 of Heraclitus, "You cannot step into the same river twice," I admire his dialectical skill, for the facility with which we accept the first meaning (“The river is another”) covertly imposes upon us the second meaning (“I am another”) and gives us the illusion of having invented it…

But actually it seems Heraclitus didn't exactly say "you cannot step into the same river twice".

He lived roughly from 535 to 475 BC. Only fragments of his writings remain. Most of what we know about him comes from Diogenes Laertius, a notoriously unreliablebiographer who... more »

You can never step in the same river twice

People say this remark is from Heraclitus.  The main idea is that the river keeps changing as the water flows.  The other idea is that you keep changing, too! 

Jorge Luis Borges wrote:

… each time I recall fragment 91 of Heraclitus, "You cannot step into the same river twice," I admire his dialectical skill, for the facility with which we accept the first meaning (“The river is another”) covertly imposes upon us the second meaning (“I am another”) and gives us the illusion of having invented it…

But actually it seems Heraclitus didn't exactly say "you cannot step into the same river twice".

He lived roughly from 535 to 475 BC. Only fragments of his writings remain. Most of what we know about him comes from Diogenes Laertius, a notoriously unreliable biographer who lived 600 years later. 

For example: Diogenes said that Heraclitus became sick, tried to cure himself by smearing himself with cow manure and lying in the sun... and died, covered with poop. 

But Diogenes also said that Pythagoras died while running away from an angry mob when he refused to cross a field of beans, because beans were sacred to the Pythagoreans.  And Diogenes also said Pythagoras had a golden thigh - and was once seen in two places at the same time.

So we don't really know much about Heraclitus.  And among later Greeks he was famous for his obscurity, nicknamed “the riddler” and “the dark one”.

Nonetheless a certain remark of his has always excited people interested in the concepts of sameness and change.

In one of Plato's dialogs the Socrates character says:

Heraclitus is supposed to say that all things are in motion and nothing at rest; he compares them to the stream of a river, and says that you cannot go into the same water twice.

This is often read as saying that all is in flux; nothing stays the same. But a more reliable quote passed down through Cleanthes says:

On those stepping into rivers staying the same other and other waters flow.

That's harder to understand - read it twice!    It seems that while the river stays the same, the water does not.

No matter what the details are, to me Heraclitus was trying to pose the great mystery of time: we can only say an entity changes if it is also the same in some way — because if it were completely different, we could not speak of "an entity" that was changing.

Of course we can mentally separate the aspect that stays the same and the aspect that changes.  But we must also bind these aspects together, if we are to say that "the same thing is changing".

In category theory, we try to swim these deep waters using the concept of isomorphism.   Very roughly, two things are isomorphic if they are "the same in a way".  This lets us have our cake and eat it too: two things can be unequal yet isomorphic.

So when you step in the river the second time, it's a different but isomorphic river, and a different but isomorphic you. 

And the isomorphism itself?  That's the passage of time.

So, isomorphisms exhibit a subtle interplay between sameness and difference that may begin to do justice to Heraclitus.

None of these thoughts are new.  I'm thinking them again because I'm writing a chapter on "concepts of sameness" for Elaine Landry's book Category Theory for the Working Philosopher.  You can see a list of chapters and their authors here:

https://golem.ph.utexas.edu/category/2015/02/concepts_of_sameness_part_1.html

Here and in future articles you can watch me write my paper, and help me out.  It'll be more technical - and I hope more precise! - than my remarks here.  But it's supposed to be sort of fun, too.

In Part 2, I talk about the Chinese paradox "when is a white horse not a horse?":

https://golem.ph.utexas.edu/category/2015/02/concepts_of_sameness_part_2.html

In Part 3, I ask if you've ever used the equation x = x for anything.  And I pose a precise conjecture which claims that this equation is useless.  I would like someone to settle this conjecture!

But if x = x is a useless equation, why do mathematicians think it's fundamental to our concept of equality?

The picture here is taken from someone on G+ who is vastly more popular than me:

https://plus.google.com/104293557269756681667/posts/KGGzk5nmQ4F___

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2015-02-22 17:03:00 (64 comments, 7 reshares, 97 +1s)Open 

Crackpot email of the day

I have a webpage on the Voynich Manuscript.  It's the most mysterious manuscript in the world.  It is 7 x 10 inches in size and about 200 pages long.  It was found in a chest in a Jesuit college in Italy.  It is written in a flowing cursive script in alphabet that has never been seen elsewhere. Nobody knows what it means. During World War II some of the top military code-breakers in America tried to decipher it, but failed. A professor at the University of Pennsylvania went insane trying to figure it out.  People keep coming out with theories explaining it.

Today I got this email:

The Voynich Manuscript Book is from another World... similar to ours.  The Book was magically written by God, and the original author awakened and found the manuscript like it is. Nobody on this Earth wrote it except God.

Myperso... more »

Crackpot email of the day

I have a webpage on the Voynich Manuscript.  It's the most mysterious manuscript in the world.  It is 7 x 10 inches in size and about 200 pages long.  It was found in a chest in a Jesuit college in Italy.  It is written in a flowing cursive script in alphabet that has never been seen elsewhere. Nobody knows what it means. During World War II some of the top military code-breakers in America tried to decipher it, but failed. A professor at the University of Pennsylvania went insane trying to figure it out.  People keep coming out with theories explaining it.

Today I got this email:

The Voynich Manuscript Book is from another World... similar to ours.  The Book was magically written by God, and the original author awakened and found the manuscript like it is. Nobody on this Earth wrote it except God.

My personal opinion - I know because I have been working with God for a few years now.

Okay....

Here's my page on the Voynich, which I never finished:

http://math.ucr.edu/home/baez/voynich.html___

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2015-02-21 17:51:14 (111 comments, 39 reshares, 104 +1s)Open 

How computers can take over the world

How can computers get legal rights like people?  It sounds hard.  But in the US, it's not.  They just need to become corporations. 

You see, in the US, corporations are already persons in the legal sense, with the right to sign contracts and sue people.  In 2010, the Supreme Court said they have the right to free speech!   Since corporations are very powerful, they are likely to gain more and more rights - and not just in the US.

So, computers might take over the world by becoming corporations... or running corporations. 

Most people think computers need to be intelligent before they take over the world.  But maybe it will go like this.  First they become corporations.  Then they hire us to make them more intelligent.  

Now there's a company that's trying to speed up thisprocess!  It... more »

How computers can take over the world

How can computers get legal rights like people?  It sounds hard.  But in the US, it's not.  They just need to become corporations. 

You see, in the US, corporations are already persons in the legal sense, with the right to sign contracts and sue people.  In 2010, the Supreme Court said they have the right to free speech!   Since corporations are very powerful, they are likely to gain more and more rights - and not just in the US.

So, computers might take over the world by becoming corporations... or running corporations. 

Most people think computers need to be intelligent before they take over the world.  But maybe it will go like this.  First they become corporations.  Then they hire us to make them more intelligent.  

Now there's a company that's trying to speed up this process!  It's called Ethereum.   They want to help developers start Distributed Autonomous Corporations: corporations run by computers.

Vitalik Buterin, who runs Ethereum, explained the basic idea:

Corporations, US presidential candidate Mitt Romney reminds us, are people. Whether or not you agree with the conclusions that his partisans draw from that claim, the statement certainly carries a large amount of truth. What is a corporation, after all, but a certain group of people working together under a set of specific rules? When a corporation owns property, what that really means is that there is a legal contract stating that the property can only be used for certain purposes under the control of those people who are currently its board of directors – a designation itself modifiable by a particular set of shareholder. If a corporation does something, it’s because its board of directors has agreed that it should be done. If a corporation hires employees, it means that the employees are agreeing to provide services to the corporation’s customers under a particular set of rules, particularly involving payment. When a corporation has limited liability, it means that specific people have been granted extra privileges to act with reduced fear of legal prosecution by the government – a group of people with more rights than ordinary people acting alone, but ultimately people nonetheless. In any case, it’s nothing more than people and contracts all the way down.

However, here a very interesting question arises: do we really need the people? On the one hand, the answer is yes: although in some post-Singularity future machines will be able to survive all on their own, for the foreseeable future some kind of human action will simply be necessary to interact with the physical world. On the other hand, however, over the past two hundred years the answer has been increasingly no. The industrial revolution allowed us, for the first time, to start replacing human labor with machines on a large scale, and now we have advanced digitized factories and robotic arms that produce complex goods like automobiles all on their own. But this is only automating the bottom; removing the need for rank and file manual laborers, and replacing them with a smaller number of professionals to maintain the robots, while the management of the company remains untouched. The question is, can we approach the problem from the other direction: even if we still need human beings to perform certain specialized tasks, can we remove the management from the equation instead?

Most companies have some kind of mission statement; often it's about making money for shareholders; at other times, it includes some moral imperative to do with the particular product that they are creating, and other goals like helping communities sometimes enter the mix, at least in theory. Right now, that mission statement exists only insofar as the board of directors, and ultimately the shareholders, interpret it. But what if, with the power of modern information technology, we can encode the mission statement into code; that is, create an inviolable contract that generates revenue, pays people to perform some function, and finds hardware for itself to run on, all without any need for top-down human direction?

And then he went on to explain a plan to do this:

https://bitcoinmagazine.com/7050/bootstrapping-a-decentralized-autonomous-corporation-part-i/

https://bitcoinmagazine.com/7119/bootstrapping-an-autonomous-decentralized-corporation-part-2-interacting-with-the-world/

https://bitcoinmagazine.com/7235/bootstrapping-a-decentralized-autonomous-corporation-part-3-identity-corp/

The fascinating technical details of Ethereum are here:

https://github.com/ethereum/wiki/wiki/White-Paper

For more on decentralized autonomous corporations, or DACs, see:

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

For the American legal doctrine of corporate personhood, see:

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

Does this make you want to rebel?  It may be too late.   I, for one, welcome our new robot overlords. 

I thank +Daniel Estrada for pointing out this article on DACs:

http://io9.com/how-much-longer-before-companies-start-to-run-themselve-1687015200

The picture here was made by TheMarex:

http://themarex.deviantart.com/art/All-hail-our-new-robot-overlords-292510016___

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2015-02-19 16:33:58 (13 comments, 7 reshares, 71 +1s)Open 

Binary star zips through solar system 70,000 years ago

100,000 years ago, some of my ancestors came out of Africa and arrived in the Middle East.  50,000 years ago, some of them reached Asia.  But between those dates, two stars passed through the outer reaches of the Solar System, where icy comets float in dark space! 

One is a tiny red dwarf called Scholz's star.  It's only 90 times as heavy as Jupiter.  Right now it's 20 light years from us, so faint that it was only discovered in 2013, by Ralf-Dieter Scholz - an expert on nearby stars, high-velocity stars, and dwarf stars. 

The other star is a brown dwarf: a star so small that it doesn't produce energy by fusion, at least not after some easily fused isotopes get used up.  This guy is only 65 times the mass of Jupiter, and it orbits its companion at a distance of 80 AU. 
more »

Binary star zips through solar system 70,000 years ago

100,000 years ago, some of my ancestors came out of Africa and arrived in the Middle East.  50,000 years ago, some of them reached Asia.  But between those dates, two stars passed through the outer reaches of the Solar System, where icy comets float in dark space! 

One is a tiny red dwarf called Scholz's star.  It's only 90 times as heavy as Jupiter.  Right now it's 20 light years from us, so faint that it was only discovered in 2013, by Ralf-Dieter Scholz - an expert on nearby stars, high-velocity stars, and dwarf stars. 

The other star is a brown dwarf: a star so small that it doesn't produce energy by fusion, at least not after some easily fused isotopes get used up.  This guy is only 65 times the mass of Jupiter, and it orbits its companion at a distance of 80 AU. 

(An AU, or astronomical unit, is the distance between the Earth and the Sun.)

A team of scientists has just computed that while some of my ancestors were making their way to Asia, these stars passed about 0.8 light years from our Sun.  That's not very close.  But it's close enough to penetrate the large cloud of comets surrounding the Sun: the Oort cloud.

They say this event didn't affect the comets very much.  But if it shook some comets loose from the Oort cloud, they would take about 2 million years to get here!  So, they won't arrive for a long time.

At its closest approach, Scholz's star would have had an apparent magnitude of about 11.4.  This is a bit too faint to see, even with binoculars.  So, don't look for it myths and legends!

As usual, the paper that made this discovery is expensive in journals but free on the arXiv:

• Eric E. Mamajek, Scott A. Barenfeld, Valentin D. Ivanov, Alexei Y. Kniazev, Petri Vaisanen, Yuri Beletsky, Henri M. J. Boffin, The closest known flyby of a star to the Solar System, http://arxiv.org/abs/1502.04655.

It must be tough being a scientist named 'Boffin', especially in England.

#spnetwork arXiv:1502.04655 #astronomy #brownDwarf___

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2015-02-17 17:31:06 (99 comments, 18 reshares, 96 +1s)Open 

Understanding ISIS

You need to understand your enemies.  You may discover you can coexist with them.  You may discover you cannot: you may need to defeat them.   Either way, it's crucial to understand their motivations and their goals.

By "understanding" I don't mean "sympathy".   Even if you want to destroy something, you need to understand how it works - or you may fail, unless you're vastly more powerful. 

We've been complaining a lot about ISIS, but not working hard enough to understand them.   Perhaps this is because we believe we're so powerful that understanding is not necessary?  Or that "evil" is a sufficient explanation? 

Major General Michael K. Nagata, the Special Operations commander for the United States in the Middle East, is wiser:

We do not understand the movement,and until ... more »

Understanding ISIS

You need to understand your enemies.  You may discover you can coexist with them.  You may discover you cannot: you may need to defeat them.   Either way, it's crucial to understand their motivations and their goals.

By "understanding" I don't mean "sympathy".   Even if you want to destroy something, you need to understand how it works - or you may fail, unless you're vastly more powerful. 

We've been complaining a lot about ISIS, but not working hard enough to understand them.   Perhaps this is because we believe we're so powerful that understanding is not necessary?  Or that "evil" is a sufficient explanation? 

Major General Michael K. Nagata, the Special Operations commander for the United States in the Middle East, is wiser:

We do not understand the movement, and until we do, we are not going to defeat it.  We have not defeated the idea. We do not even understand the idea.

The article here is an attempt to understand ISIS, so it's worth reading even if it's not completely right.

Here's something Sheikh Abu Muhammad al-Adnani, the chief spokesman for ISIS, recently said:

We will conquer your Rome, break your crosses, and enslave your women.  If we do not reach that time, then our children and grandchildren will reach it, and they will sell your sons as slaves at the slave market.

The article argues that he means this quite literally, and tries to explain where this goal comes from.___

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2015-02-15 17:14:27 (41 comments, 13 reshares, 62 +1s)Open 

Pentagons and decagons

The science fiction writer Greg Egan and I have been exploring the forbidden tilings.  These are ways to stick together shapes that look like they should tile the plane, but don't.

Two regular pentagons and a regular decagon fit snugly at a point: their interior angles sum to 360°. Despite this, you cannot tile the plane with regular pentagons and decagons! 

But you can do some other things.  This picture by Egan shows one of them.

The idea here is to start drawing regular pentagons and decagons on the plane, and make sure that:

• each decagon touches 10 pentagons along its edges

• each pentagon touches 2 decagons and 3 pentagons along its edges

• there are two kinds of vertices: at some, 2 pentagons and a decagon meet, while at others, 10 pentagons meet.

The pentagonsand decag... more »

Pentagons and decagons

The science fiction writer Greg Egan and I have been exploring the forbidden tilings.  These are ways to stick together shapes that look like they should tile the plane, but don't.

Two regular pentagons and a regular decagon fit snugly at a point: their interior angles sum to 360°. Despite this, you cannot tile the plane with regular pentagons and decagons! 

But you can do some other things.  This picture by Egan shows one of them.

The idea here is to start drawing regular pentagons and decagons on the plane, and make sure that:

• each decagon touches 10 pentagons along its edges

• each pentagon touches 2 decagons and 3 pentagons along its edges

• there are two kinds of vertices: at some, 2 pentagons and a decagon meet, while at others, 10 pentagons meet.

The pentagons and decagons will overlap, and the picture will get very confusing.  So here Egan shows just one stage of drawing the picture.  To see more stages, go here:

http://blogs.ams.org/visualinsight/2015/02/15/pentagon-decagon-branched-covering/

What's really going on here?  It's secretly all about non-Euclidean geometry!   There's a way to tile the hyperbolic plane by somewhat warped pentagons and decagons.  It's very symmetrical, and the shapes don't overlap... but their edges are not straight lines.

Then, there's a way to map the hyperbolic plane down to the ordinary Euclidean plane.  This map has infinitely many branch points.   If you walk around a branch point in the hyperbolic plane, your shadow down in the Euclidean plane will walk three times around a point down there.

This map sends the warped pentagons and decagons in the hyperbolic plane to perfectly regular pentagons and decagons in the Euclidean plane.

This is how we get 10 pentagons to meet at a point.  Up in the hyperbolic plane, they don't overlap.  Down in the Euclidean plane, they do: they wrap three times around a point.  That's what you see in the very middle of this picture!

It's a bit mind-blowing, and that's what I like.  You can learn more of the math at the link above.

For a cool-looking failed attempt to tile the Euclidean plane with regular pentagons and decagons, try this:

http://blogs.ams.org/visualinsight/2015/02/01/pentagon-decagon-packing/

This also has a list of some other forbidden tilings.

#geometry___

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2015-02-14 17:23:46 (34 comments, 20 reshares, 148 +1s)Open 

Our nearest neighbor

Proxima Centauri, the nearest star to the Sun, is a red dwarf.  If we ever explore interstellar space, we may stop by this star.  So, it's worth knowing a bit about it.

The planet in this picture is just a guess - we don't know if it has planets.   But it could be part of a triple star system!  The closest neighboring stars, Alpha Centauri A and B, orbit each other every 80 years.  One is a bit bigger than the Sun, the other a bit smaller.  They orbit in a fairly eccentric ellipse.  At their closest, their distance is like the distance from Saturn to the Sun.  At their farthest, it's more like the distance from Pluto to the Sun.  

Proxima Centauri is fairly far from both: a quarter of a light year away.  That's about 350 times the distance from Pluto to the Sun!  We're not even sure Proxima Centauri isgravitatio... more »

Our nearest neighbor

Proxima Centauri, the nearest star to the Sun, is a red dwarf.  If we ever explore interstellar space, we may stop by this star.  So, it's worth knowing a bit about it.

The planet in this picture is just a guess - we don't know if it has planets.   But it could be part of a triple star system!  The closest neighboring stars, Alpha Centauri A and B, orbit each other every 80 years.  One is a bit bigger than the Sun, the other a bit smaller.  They orbit in a fairly eccentric ellipse.  At their closest, their distance is like the distance from Saturn to the Sun.  At their farthest, it's more like the distance from Pluto to the Sun.  

Proxima Centauri is fairly far from both: a quarter of a light year away.  That's about 350 times the distance from Pluto to the Sun!  We're not even sure Proxima Centauri is gravitationally bound to the other stars.  If it is, its orbital period could easily exceed 500,000 years. 

If Proxima Centauri had an Earth-like planet,  could we set up a base there?  Well, there's a bit of a problem: it's a flare star.

You see, convection stirs up this star's whole interior, unlike the Sun.  Convection of charged plasma makes strong magnetic fields.  Magnetic fields get tied in tight knots... and the energy gets released through enormous flares!  They can become as large as the star itself, and get so hot that they radiate lots of X-rays.

This could be bad for life on nearby planets... especially since an Earth-like planet would have to be very close.  You see, Proxima Centauri is very faint: just 0.17% the brightness of our Sun!

In fact many red dwarfs are flare stars, for the same reasons.  Proxima Centauri is actually fairly tame as red dwarfs go, because it's 4.9 billion years old.  Younger ones are more lively, with bigger flares.

Proxima Centauri is just 4.24 light-years away.  It's actually getting closer: it'll come within about 3 light-years of us in roughly 27,000 years, and then drift past us.  We should take advantage of this and go visit it soon... like in a few centuries!  Once we get our own house in order, it'll be time to look around.

#astronomy  ___

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2015-02-13 08:00:08 (46 comments, 40 reshares, 125 +1s)Open 

Can red dwarf stars have Earth-like planets with life?

This is an important question because 80% of stars are red dwarfs, even though none are visible to the naked eye.  Proxima Centauri, the nearest star to the Sun, is a red dwarf!  We don't know if it has planets!  20 of the 30 nearest stars are red dwarfs.  It would be nice to know if they can have planets with life.

Also, red dwarf stars live a long time!   They're small - and the smaller a star is, the longer it lives.  Calculations show that a red dwarf one tenth the mass of our Sun should last for 10 trillion years! 

So if life is possible on planets orbiting red dwarf stars - or if life could get there - we could someday have very, very old civilizations.  That idea excites me.   Imagine what 10 trillion years of thought could discover!  

(No: you can't really imagine it.)
Si... more »

Can red dwarf stars have Earth-like planets with life?

This is an important question because 80% of stars are red dwarfs, even though none are visible to the naked eye.  Proxima Centauri, the nearest star to the Sun, is a red dwarf!  We don't know if it has planets!  20 of the 30 nearest stars are red dwarfs.  It would be nice to know if they can have planets with life.

Also, red dwarf stars live a long time!   They're small - and the smaller a star is, the longer it lives.  Calculations show that a red dwarf one tenth the mass of our Sun should last for 10 trillion years! 

So if life is possible on planets orbiting red dwarf stars - or if life could get there - we could someday have very, very old civilizations.  That idea excites me.   Imagine what 10 trillion years of thought could discover!  

(No: you can't really imagine it.)

Since a planet needs to be close to a red dwarf to be warm enough for liquid water, such planets will be tidally locked, with one side facing their sun all the time.  This made scientists believe the day side of the planet would be hot and dry, with all the water locked in the form of ice on the night side.  That's the water-trapped world shown at right.  Not so good for life!

But a new paper argues that other worlds are likely too!  Those are the thin ice waterworld and the ice sheet-ocean world.

What's fun about this new paper is that it uses detailed climate models outrageously tweaked to deal with a red dwarf star.

We perform climate simulations with the Community Climate System Model version 3.0 (CCSM3) which was originally developed by the National Center for Atmospheric Research to study the climate of Earth. The model contains four coupled components: atmosphere, ocean, sea ice, and land. The atmosphere component calculates atmospheric circulation and parameterizes sub-grid processes such as convection, precipitation, clouds, and boundary- layer mixing. The ocean component computes ocean circulation using the hydrostatic and Boussinesq approximations. The sea-ice component predicts ice fraction, ice thickness, ice velocity, and energy exchanges between the ice and the atmosphere/ocean. The land component calculates surface temperature, soil water content, and evaporation.

We modify CCSM3 to simulate the climate of habitable planets around M-stars following Rosenbloom et al., Liu et al., and Hu & Yang. The stellar spectrum we use is a blackbody with an effective temperature of 3400 K. We employ planetary parameters typical of a super-Earth: a radius of 1.5 R⊕, gravity of 1.38 g⊕, and an orbital period of 37 Earth-days. The orbital period of habitable zone planets around M stars is roughly 10–100 days. We set the insolation to 866 watts per square meter and both the obliquity and eccentricity to zero. The atmospheric surface pressure is 1.0 bar, including N2, H2O, and 355 parts per million CO2.

And so on.   Way cool!  They consider a variety of different kinds of continents and oceans... including one where they're just like those here on Earth - just because the data for that is easy to get.

Read it here:

• Jun Yang, Yonggang Liu, Yongyun Hu and Dorian S. Abbot, Water trapping on tidally locked terrestrial planets requires special conditions, http://arxiv.org/abs/1411.0540.

Abstract: Surface liquid water is essential for standard planetary habitability. Calculations of atmospheric circulation on tidally locked planets around M stars suggest that this peculiar orbital configuration lends itself to the trapping of large amounts of water in kilometers-thick ice on the night side, potentially removing all liquid water from the day side where photosynthesis is possible. We study this problem using a global climate model including coupled atmosphere, ocean, land, and sea-ice components as well as a continental ice sheet model driven by the climate model output. For a waterworld we find that surface winds transport sea ice toward the day side and the ocean carries heat toward the night side. As a result, night-side sea ice remains about 10 meters thick and night-side water trapping is insignificant. If a planet has large continents on its night side, they can grow ice sheets about a kilometer thick if the geothermal heat flux is similar to Earth's or smaller. Planets with a water complement similar to Earth's would therefore experience a large decrease in sea level when plate tectonics drives their continents onto the night side, but would not experience complete day-side dessication. Only planets with a geothermal heat flux lower than Earth's, much of their surface covered by continents, and a surface water reservoir about 10% of Earth's would be susceptible to complete water trapping.

#spnetwork arxiv:1411.0540 #mustread #xenobiology #exoplanet   #astronomy  ___

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2015-02-11 15:59:02 (56 comments, 17 reshares, 66 +1s)Open 

The dilemma

The National Academy of Sciences has a new study on ways to fight global warming by deliberately intervening in the Earth's climate. 

They find that putting stuff in the upper atmosphere to make it reflect more sunlight could rapidly and cheaply change the Earth's climate.   It could be done unilaterally by a single country!  But we don't understand its effects very well, so it poses new risks. 

On the other hand, removing carbon dioxide from the atmosphere is currently expensive and slow.  It would be hard for a single country to do enough to matter.  But it poses few new risks.

The chart here, taken from their report, shows the dilemma in detail.  Of course the third, more conventional approach is reduce carbon emissions!  And the fourth, even more popular, is wait and let others figure out how to adapt!  

Inpractice w... more »

The dilemma

The National Academy of Sciences has a new study on ways to fight global warming by deliberately intervening in the Earth's climate. 

They find that putting stuff in the upper atmosphere to make it reflect more sunlight could rapidly and cheaply change the Earth's climate.   It could be done unilaterally by a single country!  But we don't understand its effects very well, so it poses new risks. 

On the other hand, removing carbon dioxide from the atmosphere is currently expensive and slow.  It would be hard for a single country to do enough to matter.  But it poses few new risks.

The chart here, taken from their report, shows the dilemma in detail.  Of course the third, more conventional approach is reduce carbon emissions!  And the fourth, even more popular, is wait and let others figure out how to adapt!  

In practice we will do a mixture of these things.  We already know a lot about how to reduce carbon emissions, and how to adapt.  But we need more research on climate interventions.

The report says:

CLIMATE INTERVENTION IS NO SUBSTITUTE for reductions in carbon dioxide emissions and adaptation efforts aimed at reducing the negative consequences of climate change. However, as our planet enters a period of changing climate never before experienced in recorded human history, interest is growing in the potential for deliberate intervention in the climate system to counter climate change. This study assesses the potential impacts, benefits, and costs of two different proposed classes of climate intervention:

(1) carbon dioxide removal

(2) albedo modification (reflecting sunlight).

Carbon dioxide removal strategies address a key driver of climate change, but research is needed to fully assess if any of these technologies could be appropriate for large-scale deployment. Albedo modification strategies could rapidly cool the planet’s surface but pose environmental and other risks that are not well understood and therefore should not be deployed at climate-altering scales; more research is needed to determine if albedo modification approaches could be viable in the future.

Here are their main recommendations:

Recommendation 1: Efforts to address climate change should continue to focus most heavily on mitigating greenhouse gas emissions in combination with adapting to the impacts of climate change because these approaches do not present poorly defined and poorly quantified risks and are at a greater state of technological readiness.

Recommendation 2: The Committee recommends research and development investment to improve methods of carbon dioxide removal and disposal at scales that would have a global impact on reducing greenhouse warming, in particular to minimize energy and materials consumption, identify and quantify risks, lower costs, and develop reliable sequestration and monitoring.

Recommendation 3: Albedo modification at scales sufficient to alter climate should not be deployed at this time.

Recommendation 4: The Committee recommends an albedo modification research program be developed and implemented that emphasizes multiple benefit research that also furthers both basic understanding of the climate system and its human dimensions.

Recommendation 5: The Committee recommends that the United States improve its capacity to detect and measure changes in radiative forcing and associated changes in climate.

Recommendation 6: The Committee recommends the initiation of a serious deliberative process to examine:

(a) What types of research governance, beyond those that already exist, may be needed for albedo modification research;

(b) The types of research that would require such governance, potentially based on the magnitude of their expected impact on radiative forcing, their potential for detrimental direct and indirect effects, and other considerations.

You can get the whole 2-part report and short summary here:

http://nas-sites.org/americasclimatechoices/public-release-event-climate-intervention-reports/___

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2015-02-08 03:03:39 (48 comments, 30 reshares, 182 +1s)Open 

Guesstimates

Part of being a scientist is not just making up numbers.  Sometimes the best you can do is guess a rough estimate.  But doing this well is a real art!   It takes practice and work.

Here are some puzzles from the book Guesstimation 2.0 by Lawrence Weinstein.  If you answer one, please show your work!  That's the most interesting part.  Wild guesses aren't worth much.

The first one is about fighting climate change by throwing money at the problem:

Puzzle 1: Suppose we could launch a trillion one-dollar bills into the air and keep them there.  What fraction of the sunlight hitting the Earth could we block with all that paper? 

The second is about recycling:

Puzzle 2: How much energy does it take to melt a glass beer bottle?  How much would this energy cost, in the form of electricity?
And th... more »

Guesstimates

Part of being a scientist is not just making up numbers.  Sometimes the best you can do is guess a rough estimate.  But doing this well is a real art!   It takes practice and work.

Here are some puzzles from the book Guesstimation 2.0 by Lawrence Weinstein.  If you answer one, please show your work!  That's the most interesting part.  Wild guesses aren't worth much.

The first one is about fighting climate change by throwing money at the problem:

Puzzle 1: Suppose we could launch a trillion one-dollar bills into the air and keep them there.  What fraction of the sunlight hitting the Earth could we block with all that paper? 

The second is about recycling:

Puzzle 2: How much energy does it take to melt a glass beer bottle?  How much would this energy cost, in the form of electricity?

And the third is about walking versus driving:

Puzzle 3: Suppose you could digest gasoline and use its energy as efficiently as you do with food.  Suppose you used this energy to walk.  How many kilometers per liter would you get?  Or, for Americans: how many miles per gallon?

(To help people compare results: a kilometer per liter equals 2.35 miles per gallon.)___

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2015-02-06 05:37:52 (15 comments, 10 reshares, 42 +1s)Open 

Check out this Royal Society study on "Geoengineering the climate: science, governance and uncertainty".

The relative affordability/effectiveness/risk assessment for some of the leading geoengineering proposals.  This doesn't tell the whole story, as there are other considerations that matter when evaluating the alternatives (most prominently the difficulty of implementing each procedure from both a technical and a political perspective).  It's an interesting plot nonetheless.  

From https://royalsociety.org/policy/publications/2009/geoengineering-climate/

#climatechange   #geoengineering   #climateengineering  ___Check out this Royal Society study on "Geoengineering the climate: science, governance and uncertainty".

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2015-02-04 19:47:24 (121 comments, 26 reshares, 71 +1s)Open 

The utility monster

How to be good?  Some people think we should maximize the total happiness of all people, or perhaps all conscious beings. 

They usually call it utility instead of happiness, and of course we can argue all day about how it's defined.  But a guy named Robert Nozick posed a problem for this approach to ethics.

Suppose each time I eat a cookie it makes me 100 times as happy as you or anyone else would be if you ate it.   If our aim is to maximize total happiness, everyone should give their cookies to me.  Lots of people would lose a bit of happiness, but I would gain a lot!

Now suppose we're trying to maximize the total happiness of all conscious beings.  What if there were a being - a utility monster - that feels happiness much, much more strongly than us?  Then we should all do whatever it wants... even ifit'... more »

The utility monster

How to be good?  Some people think we should maximize the total happiness of all people, or perhaps all conscious beings. 

They usually call it utility instead of happiness, and of course we can argue all day about how it's defined.  But a guy named Robert Nozick posed a problem for this approach to ethics.

Suppose each time I eat a cookie it makes me 100 times as happy as you or anyone else would be if you ate it.   If our aim is to maximize total happiness, everyone should give their cookies to me.  Lots of people would lose a bit of happiness, but I would gain a lot!

Now suppose we're trying to maximize the total happiness of all conscious beings.  What if there were a being - a utility monster - that feels happiness much, much more strongly than us?  Then we should all do whatever it wants... even if it's a sadist and it gets happiness from destroying us!

Maybe such a being doesn't exist yet.  But maybe we could create it!

For more, try +Eric Schwitzgebel's blog:

http://schwitzsplinters.blogspot.com/2014/03/our-moral-duties-to-monsters.html

He asks:

-------
(1) Should we work to create artificially conscious beings who are capable of superhuman heights of pleasure? On the face of it, it seems like a good thing to do, to bring beings capable of great pleasure into the world! On the other hand, maybe we have no general obligation to bring happy beings into the world. (Compare: many people think we have no obligation to increase the number of human children even if we think they would be happy.)

(2) If we do create such beings, ought we immiserate ourselves for their happiness? It seems unintuitive to say that we should, but I can also imagine a perspective on which it makes sense to sacrifice ourselves for superhumanly great descendants.
-------

Of course, one can take these puzzles as arguments that maximizing total happiness is the wrong goal.   But personally I'd go further: I think that "total happiness" - or total utility - doesn't even make sense. 

I can imagine an experiment to see if chocolate ice cream makes me twice as happy as vanilla: if they cost the same amount, would I prefer to buy one small scoop of chocolate or two of vanilla?  ("Small" because none of this stuff is really linear.) 

But what experiment can we do to see if chocolate ice cream makes me twice as happy as you?  I can't think of one. 

Until we can decide how to estimate "total happiness" - or "total utility" -  it doesn't make much sense to base our moral decisions on it.

And of course, even if we can estimate it, it might not be the best basis for moral decisions. 

Indeed, we can only decide on "the best basis for moral decisions" if we have already chosen what counts as "best".  Maximizing total happiness would certainly be the best if what counts as "best" is... maximizing total happiness.  This limits my enthusiasm for discussing the foundations of ethics. 

However, I like the idea of the "utility monster", even though I don't think it makes sense.

The cartoon here is part of a bigger one, made by Ryan Lake, who "has a PhD in philosophy, and all the wealth, power, and glamor that goes with it."

http://chaospet.com/230-utility-monster/

#ethics  ___

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2015-02-03 17:01:13 (20 comments, 34 reshares, 126 +1s)Open 

Ancient alien mathematics

“So you traveled the whole twenty light years?”

“More than that,” Joan said truthfully, “from my original home. I've spent half my life traveling.”

“Faster than light?” Pirit suggested hopefully.

“No. That's impossible.”

They circled around the question a dozen more times, before Pirit finally changed her tune from how to why?

“I'm a xenomathematician,” Joan said. “I've come here in the hope of collaborating with your archaeologists in their study of Niah artifacts.”

Pirit was stunned. “What do you know about the Niah?”

“Not as much as I'd like to.” Joan gestured at her Noudah body. “As I'm sure you've already surmised, we've listened to your broadcasts for some time, so we know pretty much what an ordinary Noudahknows. That includes the basic facts ... more »

Ancient alien mathematics

“So you traveled the whole twenty light years?”

“More than that,” Joan said truthfully, “from my original home. I've spent half my life traveling.”

“Faster than light?” Pirit suggested hopefully.

“No. That's impossible.”

They circled around the question a dozen more times, before Pirit finally changed her tune from how to why?

“I'm a xenomathematician,” Joan said. “I've come here in the hope of collaborating with your archaeologists in their study of Niah artifacts.”

Pirit was stunned. “What do you know about the Niah?”

“Not as much as I'd like to.” Joan gestured at her Noudah body. “As I'm sure you've already surmised, we've listened to your broadcasts for some time, so we know pretty much what an ordinary Noudah knows. That includes the basic facts about the Niah. Historically they've been referred to as your ancestors, though the latest studies suggest that you and they really just have an earlier common ancestor. They died out about a million years ago, but there's evidence that they might have had a sophisticated culture for as long as three million years. There's no indication that they ever developed space flight. Basically, once they achieved material comfort, they seem to have devoted themselves to various artforms, including mathematics.”

“So you've traveled twenty light years just to look at Niah tablets?” Pirit was incredulous.

“Any culture that spent three million years doing mathematics must have something to teach us.”

“Really?” Pirit's face became blue with disgust. “In the ten thousand years since we discovered the wheel, we've already reached halfway to the Cataract. They wasted their time on useless abstractions.”

Joan said, “I come from a culture of spacefarers myself, so I respect your achievements. But I don't think anyone really knows what the Niah achieved. I'd like to find out, with the help of your people.”

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

“Jown! Jown! Come and look at this!” Surat called to her. Joan switched off the tomography unit and jogged toward the archaeologists, suddenly conscious of her body's strangeness. Her legs were stumpy but strong, and her balance as she ran came not from arms and shoulders but from the swish of her muscular tail.

“It's a significant mathematical result,” Rali informed her proudly when she reached them. He'd pressure-washed the sandstone away from the near-indestructible ceramic of the tablet, and it was only a matter of holding the surface at the right angle to the light to see the etched writing stand out as crisply and starkly as it would have a million years before.

Rali was not a mathematician, and he was not offering his own opinion on the theorem the tablet stated; the Niah themselves had a clear set of typographical conventions which they used to distinguish between everything from minor lemmas to the most celebrated theorems. The size and decorations of the symbols labelling the theorem attested to its value in the Niah's eyes.

Joan read the theorem carefully. The proof was not included on the same tablet, but the Niah had a way of expressing their results that made you believe them as soon as you read them; in this case the definitions of the terms needed to state the theorem were so beautifully chosen that the result seemed almost inevitable.

The theorem itself was expressed as a commuting hypercube, one of the Niah's favorite forms. You could think of a square with four different sets of mathematical objects associated with each of its corners, and a way of mapping one set into another associated with each edge of the square. If the maps commuted, then going across the top of the square, then down, had exactly the same effect as going down the left edge of the square, then across: either way, you mapped each element from the top-left set into the same element of the bottom-right set. A similar kind of result might hold for sets and maps that could naturally be placed at the corners and edges of a cube, or a hypercube of any dimension. It was also possible for the square faces in these structures to stand for relationships that held between the maps between sets, and for cubes to describe relationships between those relationships, and so on.

That a theorem took this form didn't guarantee its importance; it was easy to cook up trivial examples of sets and maps that commuted. The Niah didn't carve trivia into their timeless ceramic, though, and this theorem was no exception. The seven dimensional commuting hypercube established a dazzlingly elegant correspondence between seven distinct, major branches of Niah mathematics, intertwining their most important concepts into a unified whole. It was a result Joan had never seen before: no mathematician anywhere in the Amalgam, or in any ancestral culture she had studied, had reached the same insight.

She explained as much of this as she could to the three archaeologists; they couldn't take in all the details, but their faces became orange with fascination when she sketched what she thought the result would have meant to the Niah themselves.

“This isn't quite the Big Crunch,” she joked, “but it must have made them think they were getting closer.” The Big Crunch was her nickname for the mythical result that the Niah had aspired to reach: a unification of every field of mathematics that they considered significant. To find such a thing would not have meant the end of mathematics — it would not have subsumed every last conceivable, interesting mathematical truth — but it would certainly have marked a point of closure for the Niah's own style of investigation.

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

These are two quotes from Greg Egan's story "Glory".  Read the whole story here:

http://outofthiseos.typepad.com/blog/files/GregEganGlory.pdf

The start is quite dramatic!  

The image here is not alien mathematics; it's from an article about a codes for communicating with extraterrestrial civilizations:

• Brandon Keim,  Building a better alien-calling code, Wired, 23 November 2009, http://www.wired.com/2009/11/better-seti-code/.___

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2015-02-02 16:28:05 (8 comments, 18 reshares, 48 +1s)Open 

Geometry mashup

This is Greg Egan's clue to a puzzle. 

The red, yellow and blue shape is called the small cubicuboctahedron.   It has 8 blue triangles, 6 red squares and 6 yellow octagons that go through the shape and are mostly hidden from view.  The octagons cross each other in some 'false edges'.

The green and purple shape is made of 48 isosceles triangles.  They wrap around a sphere twice, and there are 8 'branch points' where the two sheets meet, as well as some 'false edges' where they cross.

Amazingly, both these shapes have the topology of a 3-holed torus!  (You have to ignore the false edges for this to be true.)

The puzzle is: how, exactly, can you turn one of these shapes into the other?

If you missed the story of these two shapes, try this:
htt... more »

Geometry mashup

This is Greg Egan's clue to a puzzle. 

The red, yellow and blue shape is called the small cubicuboctahedron.   It has 8 blue triangles, 6 red squares and 6 yellow octagons that go through the shape and are mostly hidden from view.  The octagons cross each other in some 'false edges'.

The green and purple shape is made of 48 isosceles triangles.  They wrap around a sphere twice, and there are 8 'branch points' where the two sheets meet, as well as some 'false edges' where they cross.

Amazingly, both these shapes have the topology of a 3-holed torus!  (You have to ignore the false edges for this to be true.)

The puzzle is: how, exactly, can you turn one of these shapes into the other?

If you missed the story of these two shapes, try this:

https://plus.google.com/117663015413546257905/posts/CgSR1NW9sDY

and this:

https://plus.google.com/117663015413546257905/posts/TcyGYfAvcY7

#geometry___

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2015-01-31 17:37:38 (26 comments, 22 reshares, 77 +1s)Open 

This is a 3-holed pretzel

This looks like a jewel, not a pretzel.  But Greg Egan made this beautiful shape by warping the surface of a 3-holed pretzel so that it crosses over itself in 8 places and drawing 48 triangles on it - half of them green, and half of them purple.

Mathematicians would call this a branched cover of the sphere.  The surface of the Earth is roughly a sphere.  But suppose you lived on an artificial planet like this.  In most places there would be 2 levels: an upper level where you could see outside, and an inner level underground.  But at 8 points, these levels cross over each other!

As you walk around one of these points, the upper level switches to being the lower level - and the lower level becomes the upper one!  You have to walk around these points twice to get back where you started.  Your path will form the edge of a Möbiusstrip!<... more »

This is a 3-holed pretzel

This looks like a jewel, not a pretzel.  But Greg Egan made this beautiful shape by warping the surface of a 3-holed pretzel so that it crosses over itself in 8 places and drawing 48 triangles on it - half of them green, and half of them purple.

Mathematicians would call this a branched cover of the sphere.  The surface of the Earth is roughly a sphere.  But suppose you lived on an artificial planet like this.  In most places there would be 2 levels: an upper level where you could see outside, and an inner level underground.  But at 8 points, these levels cross over each other!

As you walk around one of these points, the upper level switches to being the lower level - and the lower level becomes the upper one!  You have to walk around these points twice to get back where you started.  Your path will form the edge of a Möbius strip!

Mathematicians call the two levels sheets.  They call the special points where levels trade places branch points.   So, if you want to show off, tell your friends you spent time on G+ looking at a branched cover of the sphere with 2 sheets and 8 branch points... while they were looking at cat pictures on Facebook.

And if you really want to show off, tell them this:

A Schwarz triangle is a triangle you can draw on the sphere, whose three angles are all rational multiples of pi.  If you take such a triangle and keep flipping it over - reflecting it across any edge - you'll get a lot of triangles that cover the sphere.  Typically they will cover the sphere many times.  You get a branched cover of the sphere!

You could get infinitely many triangles if you start with one and keep flipping it over - but it's especially fun when you only get finitely many.  In this picture there are 48. 

Egan started with a triangle drawn on the sphere.  Its interior angles are pi/4, pi/4, and 2pi/3.  He repeatedly flipped it over and got 48 copies of this triangle, covering the sphere twice.  Then he made these triangles flat, so they don't exactly lie on the surface of the sphere.  They poke up, so it's easier to see you've got a branched cover of the sphere.

If you massage this shape a bit further, you get the small cubicuboctahedron I showed you before:

https://plus.google.com/u/0/117663015413546257905/posts/CgSR1NW9sDY

Puzzle 1: what can you do to this shape, to get the small cubicuboctahedron?

Puzzle 2: how can you get this shape starting from an octahedron?

Hint: if you take each triangle in the octahedron, put a dot in the middle, and use this to chop that triangle into 8 right triangles, you get a total of 48 right triangles.   How are those related to the 48 triangles here?

Puzzle 3: count the vertices, edges and faces of this shape and use Euler's formula

2 - 2g = V - E + F

to find g, the number of holes in this shape - which is secretly a g-holed torus!

Taken together these form a good introduction to Schwarz triangles, with lots of pictures:

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

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

#geometry  ___

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2015-01-29 19:09:16 (29 comments, 12 reshares, 68 +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!  So, we call it a uniform star 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 V - E + F.    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  ___

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2015-01-28 16:30:00 (23 comments, 62 reshares, 210 +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, 36 reshares, 136 +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 (26 comments, 18 reshares, 65 +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 (117 comments, 240 reshares, 1950 +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, 63 +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, 51 reshares, 204 +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, 30 reshares, 83 +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 (29 comments, 20 reshares, 74 +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 (15 comments, 14 reshares, 89 +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, 70 reshares, 168 +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, 181 +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, 77 +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, 8 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 (62 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 (41 comments, 26 reshares, 137 +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 (77 comments, 61 reshares, 191 +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 (208 comments, 34 reshares, 99 +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/___

posted image

2015-01-04 17:20:34 (110 comments, 48 reshares, 269 +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, 56 reshares, 228 +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, 31 reshares, 165 +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, 171 +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, 201 reshares, 1432 +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, 140 +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, 53 reshares, 177 +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 (67 comments, 27 reshares, 226 +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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