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John Baez has been shared in 369 public circles

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AuthorFollowersDateUsers in CircleCommentsReshares+1Links
SJR Gamer138SHARED CIRCLE 2shared circle,  add circle , +1 circle !!compartilhando circulo 2adicionar circulo (para ter mais amigos), compartilhar circuloVamos fazer amizades, ver bons posts!!   #sharedcircle #sharedcircleoftheday #circleshared #circleshapesmonday #circle  2015-04-09 07:47:27501000
Mamy Shimomatsu2,879#CircleShare #newperson   #newpeople   #active  #UnusualShare this circle to PUBLICNew & Unusual Share Circle 5.Please, add and reshare. #topofthepeople #influencermarketing #sharedcircle #circleshare #circlethis #socialmediamarketing #socialmediastrategy #socialnetworking #circleoftheday #circleoftheweek #circleoftheday #topofthecircle #howtoenlargecircle #enlargecircle #thursdaycircle #share #weekendcircle #weekend #weekendfun #circles #circleyoushare #publiccircle #sharedcircles #addcircle #sharedpubliccircles #circleshare #publicsharedcircles #circleoftheweek #circlesharing #sharedcircle #circleshare #circ2015-03-10 07:28:11328000
Mamy Shimomatsu2,967#CircleShare #newperson   #newpeople   #active  #UnusualShare this circle to PUBLICNew & Unusual Share Circle 5.Please, add and reshare. #topofthepeople #influencermarketing #sharedcircle #circleshare #circlethis #socialmediamarketing #socialmediastrategy #socialnetworking #circleoftheday #circleoftheweek #circleoftheday #topofthecircle #howtoenlargecircle #enlargecircle #thursdaycircle #share #weekendcircle #weekend #weekendfun #circles #circleyoushare #publiccircle #sharedcircles #addcircle #sharedpubliccircles #circleshare #publicsharedcircles #circleoftheweek #circlesharing #sharedcircle #circleshare #circ2015-03-10 07:27:39328005
Mamy Shimomatsu2,879#CircleShare #newperson   #newpeople   #active  #UnusualShare this circle to PUBLICNew & Unusual Share Circle 5.Please, add and reshare. #topofthepeople #influencermarketing #sharedcircle #circleshare #circlethis #socialmediamarketing #socialmediastrategy #socialnetworking #circleoftheday #circleoftheweek #circleoftheday #topofthecircle #howtoenlargecircle #enlargecircle #thursdaycircle #share #weekendcircle #weekend #weekendfun #circles #circleyoushare #publiccircle #sharedcircles #addcircle #sharedpubliccircles #circleshare #publicsharedcircles #circleoftheweek #circlesharing #sharedcircle #circleshare #circ2015-03-10 07:27:02328000
Mamy Shimomatsu2,879#CircleShare #newperson   #newpeople   #active  #UnusualShare this circle to PUBLICNew & Unusual Share Circle 5.Please, add and reshare. #topofthepeople #influencermarketing #sharedcircle #circleshare #circlethis #socialmediamarketing #socialmediastrategy #socialnetworking #circleoftheday #circleoftheweek #circleoftheday #topofthecircle #howtoenlargecircle #enlargecircle #thursdaycircle #share #weekendcircle #weekend #weekendfun #circles #circleyoushare #publiccircle #sharedcircles #addcircle #sharedpubliccircles #circleshare #publicsharedcircles #circleoftheweek #circlesharing #sharedcircle #circleshare #circ2015-03-10 07:26:42328000
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 11:18:2598125
Robert Best2,685Google+ 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 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's please catch up!2015-03-03 01:03:39455111
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 10:33:26381172326
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 15:55:59476198138248
Crazy Cats29 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 10:25:324744411
Natasha Velicia475Get 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 13:00:35472313051
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 08:56:27466107
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 06:41:19465171632
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 Patagonia13,519┊ ☆ ┊☆ ┊ ☆ ┊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 21:11:10500454776
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 Patagonia12,536┊ ☆ ┊☆ ┊ ☆ ┊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 17:25:54499394563
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 06:30:22451423271
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 Atanasov1,746SCIENCE 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 05:47:40346607
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 Atanasov1,294SCIENCE 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   2014-11-06 08:10:3436954038
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 05:05:404783111
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 05:24:00459104
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 15:25:57463221118232
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

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Most comments: 131

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2015-03-29 16:05:41 (131 comments, 56 reshares, 269 +1s)Open 

Black holes - bigger on the inside

Guess what: black holes are bigger inside than they look - and they get bigger as they get older! 

For example, take the big black hole in the center of our galaxy, called Sagittarius A*.  It's about 2 million kilometers across.  That's pretty big - but the orbit of Mercury is 60 times bigger.  This black hole is old, roughly a billion years old.  And here's the cool part:  it's been growing on the inside  all this time!  

How is this possible?  Well, since spacetime is severely warped in a black hole, its volume can be bigger than you'd guess from outside.  And its volume can change.  Since we understand general relativity quite well, we can calculate how this works!  But nobody thought of doing it until last year, when Marios Christodoulou and my friend Carlo Rovelli did it.   

How bigis the black ho... more »

Most reshares: 99

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2015-04-03 18:31:28 (77 comments, 99 reshares, 227 +1s)Open 

Drought in California - my home

The picture shows snow in the mountains of California, 2013 and 2014.  Snow usually provides 30% of California's water, so that was bad news.  But 2015 was much worse.

"We're not only setting a new low; we're completely obliterating the previous record," said the chief of the California Department of Water Resources.  There's now only 5% as much snow as the average over the last century!

California has been hit by new weather pattern: the Ridiculously Resilient Ridge.  It's a patch of high atmospheric pressure that sits over the far northeastern Pacific Ocean and stops winter storms from reaching California.  It's been sitting there most of the time for the last 3 winters. 

We did get 2 big storms this winter.  But the water fell mainly as rain rather than snow, because ofrecord... more »

Most plusones: 302

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2015-03-16 16:05:57 (74 comments, 98 reshares, 302 +1s)Open 

Quantum superpositions

You've surely heard of Schrödinger's cat, which is both alive and dead.  It's actually in a superposition of live and dead states: in quantum mechanics you can multiply different states by numbers, and then add them up! 

On the Azimuth blog +Piotr Migdal has written an explanation of quantum superpositions:

https://johncarlosbaez.wordpress.com/2015/03/13/quantum-superposition/

He explains why in chemistry an electron likes to be in a superposition of different position states: this is a way for it to reduce its energy!

Here's what Schrödinger actually said about that cat:

One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter, there isa t... more »

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2015-04-15 13:39:58 (21 comments, 27 reshares, 91 +1s)Open 

Endless reflections

If you stood in a spherical room with mirrors for walls, what would you see?  Of course you'd need a flashlight.

This picture gives an idea of what you'd see.  It's small white marble in a mirrored spheroid, as drawn by +Refurio Anachro.  You can see almost endless reflections, forming complex patterns.  These pose many fascinating puzzles!

First, just to be clear: spheroid is a sphere that has been stretched or squashed along one axis.  This is a prolate spheroid, meaning it's been stretched: it's about 10% taller than it is wide.   The reflections in here are more complicated than in a sphere.

Refurio writes:

The pattern is made of reflections of the little white marble you can see on the right hand side. To the slightly bluish mirror it appears pure white, but I have shaded theunrefl... more »

Endless reflections

If you stood in a spherical room with mirrors for walls, what would you see?  Of course you'd need a flashlight.

This picture gives an idea of what you'd see.  It's small white marble in a mirrored spheroid, as drawn by +Refurio Anachro.  You can see almost endless reflections, forming complex patterns.  These pose many fascinating puzzles!

First, just to be clear: spheroid is a sphere that has been stretched or squashed along one axis.  This is a prolate spheroid, meaning it's been stretched: it's about 10% taller than it is wide.   The reflections in here are more complicated than in a sphere.

Refurio writes:

The pattern is made of reflections of the little white marble you can see on the right hand side. To the slightly bluish mirror it appears pure white, but I have shaded the unreflected marble afterwards to make it easier to identify.

More precisely, the marble is a sphere, with radius a tenth that of the equatorial circle of the spheroid, and touching it there from the inside. I’ve placed it 90° away from one of the two ‘straight’ positions to make the image less symmetric and more interesting.

The idea behind the marble was that we could pick a point and highlight all rays coming close to it. But the presence of the marble changes things: since it extends into the spheroid, it will catch high flying rays that might not have gotten reflected in the vicinity of our chosen point. Coloring a patch of the spheroid’s surface, or punching a hole in it, would not have produced some rather beautiful artifacts you see here.

That large, wavy, most bright reflection to the left, and all the similar ones, would resolve to a number of separate elongated images of our spot. And the smaller blots further inside, the biggest one looking like two intersecting elliptic discs, would look more like a single one. And the marble-thick, brighter appearing region all around the rim.

Aside from that, the marble works like a flashlight. Think of the pattern as a fixed, static thing, produced by all possible rays bouncing within the ellipsoid. Moving the flashlight will illuminate different parts of it. Some points will be especially hard to illuminate. Two of them are the foci of the prolate spheroid: they’re the the dark points that appear to attract reflections that can never reach them, just above and below the center.

To dig deeper into the math, visit my blog:

http://blogs.ams.org/visualinsight/2015/04/15/sphere-in-mirrored-spheroid/

#geometry  ___

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2015-04-12 05:31:26 (10 comments, 13 reshares, 115 +1s)Open 

Talks from the 8th dimension

This week I'm visiting Penn State University.  If you're nearby, you can hear 2 talks about fun geometry stuff:

Split octonions and the rolling ball, 2:30 – 3:20 p.m, Tuesday April 14th, 106 McAllister Building.

Learn what happens when you roll a ball on another ball exactly 3 times as big!   The geometry of objects rolling without slipping or twisting is always fun - but in this particular case the problem gets extra symmetries, which are best understood using an 8-dimensional number system called the split octonions.  What's so great about exactly three times as big?  I'll explain!

The exceptional Jordan algebra and the Leech lattice, 12:05 – 1:20 pm, Wednesday April 15th, 114 McAllister Building.

There's a specially beautiful way to pack balls in 24 dimensions,called ... more »

Talks from the 8th dimension

This week I'm visiting Penn State University.  If you're nearby, you can hear 2 talks about fun geometry stuff:

Split octonions and the rolling ball, 2:30 – 3:20 p.m, Tuesday April 14th, 106 McAllister Building.

Learn what happens when you roll a ball on another ball exactly 3 times as big!   The geometry of objects rolling without slipping or twisting is always fun - but in this particular case the problem gets extra symmetries, which are best understood using an 8-dimensional number system called the split octonions.  What's so great about exactly three times as big?  I'll explain!

The exceptional Jordan algebra and the Leech lattice, 12:05 – 1:20 pm, Wednesday April 15th, 114 McAllister Building.

There's a specially beautiful way to pack balls in 24 dimensions, called the Leech lattice.  When physicists classified the algebras that could describe observables in quantum mechanics, they found a weird possibility: a 27-dimensional one called the exceptional Jordan algebra.   It turns out that the Leech lattice fits into the exceptional Jordan algebra in a nice way... which comes from the octonions.  So all this stuff fits together!  This talk is part of the "Geometry Luncheon Seminar", where mathematicians eat lunch and talk about mind-blowing geometry.

The first talk is about work I did with +John Huerta and James Dolan, and it will feature some fun animations made by Geoffrey Dixon.  The second is about work with Greg Egan.

The actual reason I'm at Penn State is to give a guest lecture at John Roe's undergrad course on "Mathematics for Sustainability".  I want to teach a course on math and environmental issues.  It'll be good to hear how he's been doing this.  But I thought it would be fun to talk about some other things too.

I'll also visit one of my old haunts, the Institute for Gravitation and the Cosmos, where Abhay Ashtekar, Eugenio Bianchi and others are working on loop quantum gravity.  And I'll talk to +Jason Morton about network theory.  It should be a busy, fun week.

But first I have to work on my talks...

This image here, made by Jason Hise, shows a 24-cell, a regular polytope in 4 dimensions.  There's a sculpture of this shape in the math department at Penn State!  It was designed by the mathematician Adrian Ocneanu.  I haven't been here since it was built so it will be fun to see:

https://en.wikipedia.org/wiki/Octacube_(sculpture)

https://en.wikipedia.org/wiki/24-cell#/media/File:24-cell.gif___

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2015-04-10 02:51:27 (75 comments, 63 reshares, 224 +1s)Open 

American hero

On Monday night, artists built this monument to Edward Snowden in Brooklyn.  The next day, it was taken down.   Will there be a permanent one someday?

Martin Luther King was put in jail 29 times, and now there's a monument to him in Washington DC.  But it was built only in 2011, forty-three years after King was killed.

If Snowden ever gets a monument, here are some quotes of his they can carve on it:

There can be no faith in government if our highest offices are excused from scrutiny - they should be setting the example of transparency.

I would rather be without a state than without a voice.

I don't see myself as a hero because what I'm doing is self-interested: I don't want to live in a world where there's no privacy and therefore no room for intellectual exploration andcre... more »

American hero

On Monday night, artists built this monument to Edward Snowden in Brooklyn.  The next day, it was taken down.   Will there be a permanent one someday?

Martin Luther King was put in jail 29 times, and now there's a monument to him in Washington DC.  But it was built only in 2011, forty-three years after King was killed.

If Snowden ever gets a monument, here are some quotes of his they can carve on it:

There can be no faith in government if our highest offices are excused from scrutiny - they should be setting the example of transparency.

I would rather be without a state than without a voice.

I don't see myself as a hero because what I'm doing is self-interested: I don't want to live in a world where there's no privacy and therefore no room for intellectual exploration and creativity.

After the statue was removed by park officers, a group of artists who call themselves "The Illuminator" — not related to those who built the original sculpture — used laptops and projection equipment to cast an image of Snowden in a haze of smoke at the spot where the sculpture had been.

http://mashable.com/2015/04/07/edward-snowden-hologram-statue-brooklyn/___

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2015-04-07 14:21:08 (18 comments, 10 reshares, 29 +1s)Open 

Life is a game of information and entropy.  We're having a workshop on this from Wednesday April 8th to Friday April 10th!  I hope you can  join us!  To watch live streaming videos of the workshop, go to the link here.  Go down to where it says Investigative Workshop: Information and Entropy in Biological Systems.  Then click where it says live link. There’s nothing there now - but there should be when the show starts.

You should also be able to watch videos of talks after the conference.   And you can see some talk slides now!

Here's the schedule of talks. The hours are in Eastern Daylight Time: add 4 hours to get Greenwich Mean Time. The talks start at 10 am EDT, which is 2 pm GMT. 

Wednesday April 8

• 9:45-10:00 — the usual introductory fussing around.

• 10:00-10:30 — John Baez, Information and entropyin biological syst... more »

Life is a game of information and entropy.  We're having a workshop on this from Wednesday April 8th to Friday April 10th!  I hope you can  join us!  To watch live streaming videos of the workshop, go to the link here.  Go down to where it says Investigative Workshop: Information and Entropy in Biological Systems.  Then click where it says live link. There’s nothing there now - but there should be when the show starts.

You should also be able to watch videos of talks after the conference.   And you can see some talk slides now!

Here's the schedule of talks. The hours are in Eastern Daylight Time: add 4 hours to get Greenwich Mean Time. The talks start at 10 am EDT, which is 2 pm GMT. 

Wednesday April 8

• 9:45-10:00 — the usual introductory fussing around.

• 10:00-10:30 — John Baez, Information and entropy in biological systems.  Slides here:
http://www.nimbios.org/wordpress-training/entropy/2015/03/25/introductory-talk/

• 10:30-11:00 — questions, coffee.

• 11:00-11:30 — Chris Lee, Empirical information, potential information and disinformation. Slides here:
http://www.nimbios.org/wordpress-training/entropy/2015/03/27/empirical-information-potential-information-and-disinformation/

• 11:30-11:45 — questions.

• 11:45-1:30 — lunch, conversations.

• 1:30-2:00 — John Harte, Maximum entropy as a foundation for theory building in ecology.  Slides here:
http://www.nimbios.org/wordpress-training/entropy/2015/03/25/maximum-entropy-as-a-foundation-for-theory-building-in-ecology/

• 2:00-2:15 — questions, coffee.

• 2:15-2:45 — Annette Ostling, The neutral theory of biodiversity and other competitors to the principle of maximum entropy.

• 2:45-3:00 — questions, coffee.

• 3:00-5:30 — break up into groups for discussions.

• 5:30 — reception.

Thursday April 9

• 10:00-10:30 — David Wolpert, The Landauer limit and thermodynamics of biological organisms.

• 10:30-11:00 — questions, coffee.

• 11:00-11:30 — Susanne Still, Efficient computation and data modeling.

• 11:30-11:45 — questions.

• 11:45-1:30 — lunch, conversations.

• 1:30-2:00 — Matina Donaldson-Matasci, The fitness value of information in an uncertain environment.  Paper here:
http://www.nimbios.org/wordpress-training/entropy/2015/04/02/the-fitness-value-of-information-in-an-uncertain-environment/

• 2:00-2:15 — questions, coffee.

• 2:15-2:45 — Roderick Dewar, Maximum entropy and maximum entropy production in biological systems: survival of the likeliest?

• 2:45-3:00 — questions, coffee.

• 3:00-6:00 — break up into groups for discussions.

Friday April 10

• 10:00-10:30 — Marc Harper, Information transport and evolutionary dynamics.  Slides here:
http://www.nimbios.org/wordpress-training/entropy/2015/04/02/information-transport-and-evolutionary-dynamics/

• 10:30-11:00 — questions, coffee.

• 11:00-11:30 — Tobias Fritz, Characterizations of Shannon and Rényi entropy.  Slides here:
http://www.nimbios.org/wordpress-training/entropy/2015/03/27/characterizations-of-shannon-and-renyi-entropy/

• 11:30-11:45 — questions.

• 11:45-1:30 — lunch, conversations.

• 1:30-2:00 — Christina Cobbold, Biodiversity measures and the role of species similarity.

• 2:00-2:15 — questions, coffee.

• 2:15-2:45 — Tom Leinster, Maximizing biological diversity.

• 2:45-3:00 — questions, coffee.

• 3:00-6:00 — break up into groups for discussions.___

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2015-04-06 18:01:22 (8 comments, 5 reshares, 40 +1s)Open 

I can't get this tune out of my head - and I don't want to!  Luzmila Carpio is a Bolivian singer who sings in Quechua, a native American language that was once banned in Peru.   She sings in a deeply traditional style - but here she's been remixed by El Remolón, a minimalist techno producer from Buenos Aires.   The result is striking: a sweet, delicate lark in the chilly modern world.

This tune is part of an album Luzmila Carpio Meets ZZK.  You can hear the whole thing here:

http://www.zzkrecords.com/mixtape/ZZK_Mixtape_Vol_20_-_Luzmila_Carpio_Meets_ZZK

Following the philosophy of ZZK Records, all remixes were made in collaboration with Luzmila Carpio, who had the final say over what was done.   But alas, only this one track pleases me.

As a child, Luzmila Carpio learned the songs of the Quechua and Aymara indigenouspeoples ... more »

I can't get this tune out of my head - and I don't want to!  Luzmila Carpio is a Bolivian singer who sings in Quechua, a native American language that was once banned in Peru.   She sings in a deeply traditional style - but here she's been remixed by El Remolón, a minimalist techno producer from Buenos Aires.   The result is striking: a sweet, delicate lark in the chilly modern world.

This tune is part of an album Luzmila Carpio Meets ZZK.  You can hear the whole thing here:

http://www.zzkrecords.com/mixtape/ZZK_Mixtape_Vol_20_-_Luzmila_Carpio_Meets_ZZK

Following the philosophy of ZZK Records, all remixes were made in collaboration with Luzmila Carpio, who had the final say over what was done.   But alas, only this one track pleases me.

As a child, Luzmila Carpio learned the songs of the Quechua and Aymara indigenous peoples who inhabit the mountains and valleys of Northern Potosí in Bolivia.  As a teenager, she moved to the mid-sized city of Oruro.  She soon gained fame for her voice, and her song "Siway Azucena" was the first truly indigenous tune to have widespread popular success in Bolivia.

I don't understand her career, but she later went to Paris, and in 2006 she became Bolivia's ambassador to France!  This lasted until 2010, and the next year she was made a Grand Officer of the Order of Merit of the French Republic.

El Remolón - 'the lazy one' - is really Andrés Schteingart.

There are over 10 million speakers of various related Quechan languages in Peru, Argentina, Bolivia and other countries.  The Inca were just one of the peoples who spoke these languages.  By now Quecha and Spanish have blended.  So you actually know some Quechan words: coca, condor, guano, jerky, llama, puma, quinine, quinoa, vicuña and possibly gaucho!

Apparently there are a bunch of people who speak Quechan in Queens, New York and Paterson, New Jersey.  I'm always fascinated by how people change and adapt, and this song is a metaphor for that.

For some more traditional music by Luzmila Carpio, go here:

http://www.last.fm/music/Luzmila+Carpio

For more to read:

https://en.wikipedia.org/wiki/Quechuan_languages
https://en.wikipedia.org/wiki/Luzmila_Carpio___

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2015-04-04 16:09:15 (25 comments, 3 reshares, 90 +1s)Open 

The harvest

I live on the edge of the desert in southern California.  We tore up our lawn and planted beautiful plants that use less water.  Drip irrigation instead of sprayers! 

But we do indulge in some citrus trees.  Here's the harvest!

Satsumas in front - they're like mandarins, but different.  Meyer lemons at rear left - they're sweeter than ordinary lemons. Grapefruits at rear right - they're not very big, perhaps because our tree is still young and struggling.

What's a 'mandarin'?   It's the mandarin orange, Citrus reticulata, often marketed as a 'tangerine'.  According to DNA studies, the mandarin is one of the 4 ancestors of all other citrus species, which arose through hybridization and breeding.   The other 3 are the the citron, the pomelo, and something called apapeda. <... more »

The harvest

I live on the edge of the desert in southern California.  We tore up our lawn and planted beautiful plants that use less water.  Drip irrigation instead of sprayers! 

But we do indulge in some citrus trees.  Here's the harvest!

Satsumas in front - they're like mandarins, but different.  Meyer lemons at rear left - they're sweeter than ordinary lemons. Grapefruits at rear right - they're not very big, perhaps because our tree is still young and struggling.

What's a 'mandarin'?   It's the mandarin orange, Citrus reticulata, often marketed as a 'tangerine'.  According to DNA studies, the mandarin is one of the 4 ancestors of all other citrus species, which arose through hybridization and breeding.   The other 3 are the the citron, the pomelo, and something called a papeda. 

Among these 4 citrus ancestors, mandarins are the only really sweet ones, so they were used to create many of the fruits people like now.

For example, a Meyer lemon is probably a cross between a true lemon and a mandarin or an orange.  A grapefruit is a cross between an orange and a pomelo - a huge fruit that looks like a grapefruit on steroids.  And an orange is itself probably a cross between a pomelo and a mandarin!

It's all very complicated:

https://en.wikipedia.org/wiki/Citrus_taxonomy
https://en.wikipedia.org/wiki/Citrus_hybrids

Luckily you don't need to know this stuff to enjoy growing and eating citrus!

#citrus___

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2015-04-03 18:31:28 (77 comments, 99 reshares, 227 +1s)Open 

Drought in California - my home

The picture shows snow in the mountains of California, 2013 and 2014.  Snow usually provides 30% of California's water, so that was bad news.  But 2015 was much worse.

"We're not only setting a new low; we're completely obliterating the previous record," said the chief of the California Department of Water Resources.  There's now only 5% as much snow as the average over the last century!

California has been hit by new weather pattern: the Ridiculously Resilient Ridge.  It's a patch of high atmospheric pressure that sits over the far northeastern Pacific Ocean and stops winter storms from reaching California.  It's been sitting there most of the time for the last 3 winters. 

We did get 2 big storms this winter.  But the water fell mainly as rain rather than snow, because ofrecord... more »

Drought in California - my home

The picture shows snow in the mountains of California, 2013 and 2014.  Snow usually provides 30% of California's water, so that was bad news.  But 2015 was much worse.

"We're not only setting a new low; we're completely obliterating the previous record," said the chief of the California Department of Water Resources.  There's now only 5% as much snow as the average over the last century!

California has been hit by new weather pattern: the Ridiculously Resilient Ridge.  It's a patch of high atmospheric pressure that sits over the far northeastern Pacific Ocean and stops winter storms from reaching California.  It's been sitting there most of the time for the last 3 winters. 

We did get 2 big storms this winter.  But the water fell mainly as rain rather than snow, because of record-breaking heat.  It was enough to half fill Shasta Lake and Lake Oroville.  But it didn't help the snow pack, which holds more water.

For the first time, the governor has imposed mandatory water restrictions: a 25% cut in water use in every city and town.   This will save about 1.8 cubic kilometers of water over the next 9 months - nearly as much as Lake Oroville now holds.

He said:

People should realize we're in a new era. The idea of your nice little green grass getting lots of water every day - that's going to be a thing of the past.

But what about agriculture?   In California, about 50% of water is used by "the environment": rivers, wetlands, parks and the like.  40% is used by agriculture.  10% is left for businesses and residents. 

Brown didn't impose any cuts on agriculture!  That sounds unfair, and people are complaining.   More water is used to grow walnuts than to keep Los Angeles going!

We definitely need to improve agriculture.  But don't forget: for the second year in a row, farmers in California's big Central Valley are getting hit with big water cutbacks.  The ones who get water from the State Water Project will receive only 20% of their usual amount.  

Is all this due to climate change?  I heard a wise answer to that question:  instead of a definite yes or no, just: this is what climate change looks like.  This is the kind of thing we can expect.

And on the Road to Paris, this week the US submitted a plan to cut carbon emissions by 25% by 2030... but that's another story.  Or another part of the same big story.

What California is doing about the drought:

http://www.latimes.com/local/california/la-me-ag-water-20150403-story.html

Water used by agriculture in California:

http://www.motherjones.com/environment/2015/01/almonds-nuts-crazy-stats-charts

Make your own graphs of the California snowpack:

http://cdec.water.ca.gov/cdecapp/snowapp/swcchart.action

There's lots more water data here, too - click items on the menu above.

More on the Ridiculously Resilient Ridge or Triple R by Daniel Swain, the guy who coined the term:

http://www.weatherwest.com/archives/tag/ridiculously-resilient-ridge

In February he wrote:

In this sense, the Triple R of 2014-2015 is notably different from 2013-2014. California has certainly received more precipitation this year on a liquid equivalent basis, though we’re once again falling rapidly behind average as February turns out to be mostly dry. The extreme warmth and low snowpack, however, are very reminiscent of recent winters–as is the occurrence of infrequent but intense warm storms. It’s interesting to note that nearly the entire western United States has been exceptionally warm in recent months, while the eastern part of the country remains locked in a recurring nightmare of extreme Arctic outbreaks and almost inconceivable snow accumulations in parts of New England. This overall setup–with a big Western ridge and a deep Eastern trough–has become known as the “Warm West/Cool East” dipole pattern, and it has been a common feature of recent winters in North America. There are a number of hypotheses currently being investigated regarding the causes of an apparent recent increase in the occurrence of this pattern, though there’s not yet compelling evidence pointing to a singular cause (that’s a topic for a future blog post!).

What is more certain, at least as far as California is concerned, is that our severe long-term drought is unlikely to improve substantially until this newly-invigorated pattern of persistent West Coast high pressure is no longer dominant.___

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2015-04-02 15:48:31 (20 comments, 7 reshares, 56 +1s)Open 

+Jacob Biamonte got a grant from the Foundational Questions Institute to run a small meeting on The Categorical Foundations of Network Theory.   This is important, because we live in a world of networks.  We need to understand networks better to live wisely in this world.  

The meeting will be May 25th-28th this year in Turin, Italy - at the Institute for Scientific Interchange.   We'll make slides and/or videos available, but the main goal is to bring a few people together, exchange ideas, and push the subject forward.

There will be one talk each morning, with plenty of time for questions and interaction. We'll then break for lunch and return for an afternoon work session.  Here's the plan:

Monday May 25th  

Jacob Biamonte: opening remarks. 

For Jacob's work on quantum networks see:http:/... more »

+Jacob Biamonte got a grant from the Foundational Questions Institute to run a small meeting on The Categorical Foundations of Network Theory.   This is important, because we live in a world of networks.  We need to understand networks better to live wisely in this world.  

The meeting will be May 25th-28th this year in Turin, Italy - at the Institute for Scientific Interchange.   We'll make slides and/or videos available, but the main goal is to bring a few people together, exchange ideas, and push the subject forward.

There will be one talk each morning, with plenty of time for questions and interaction. We'll then break for lunch and return for an afternoon work session.  Here's the plan:

Monday May 25th  

Jacob Biamonte: opening remarks. 

For Jacob's work on quantum networks see:
http://www.thequantumnetwork.org/

John Baez: network theory

For my stuff see the Azimuth Project network theory page:

http://math.ucr.edu/home/baez/networks/

Tuesday May 26th 

David Spivak: operadic network design

Operads are a formalism for sticking small networks together to form bigger ones.  David wrote a 3-part series of articles sketching his ideas on networks:

https://johncarlosbaez.wordpress.com/2015/03/27/spivak-part-1/

Wednesday May 27th

Eugene Lerman: continuous time open systems and monoidal double categories

Eugene is especially interested in classical mechanics and networked dynamical systems:

https://johncarlosbaez.wordpress.com/2014/03/18/networks-of-dynamical-systems/

Thursday May 26th

Tobias Fritz: ordered commutative monoids and theories of resource convertibility

Resource theory is all about how you can or cannot turn one kind of resource or ability into another.  Here's his new paper on this subject:

http://perimeterinstitute.ca/personal/tfritz/ordered_commutative_monoids.pdf

And soon he'll be coming out with a 3-part series on the Azimuth blog!

So, there's a lot to read already, but I hope we really clarify the subject and unify different approaches.  I think I see how to do it.

For more on the workshop see:

http://math.ucr.edu/home/baez/networks_isi/___

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2015-04-01 14:39:20 (24 comments, 2 reshares, 50 +1s)Open 

April Fools!

April Fools!___

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2015-03-31 15:51:20 (81 comments, 44 reshares, 167 +1s)Open 

What it takes to understand a single proton

This computer is called JUQUEEN.    You can see 7 big boxes here.  Each box holds 4 racks.  Each rack holds 16 boards.  Each board holds 32 nodes.  Each node has 32 cores.  

A core is like the processor in your laptop.  So, this computer is roughly like half a million laptops - all connected and working together.  It can compute at a rate of 5 petaflops.  That's 5,000,000,000,000,000 floating-point operations per second!

This computer was recently used to compute the ratio of the proton and neutron masses.  A proton weighs about 1836 times as much as an electron.  A neutron is a bit heavier: about 1839 times the electron mass.

This is important, since it means a lone proton is stable, while a lone neutron is not: in about 15 minutes, it will decay into a proton and some other stuff.  Thisis why the un... more »

What it takes to understand a single proton

This computer is called JUQUEEN.    You can see 7 big boxes here.  Each box holds 4 racks.  Each rack holds 16 boards.  Each board holds 32 nodes.  Each node has 32 cores.  

A core is like the processor in your laptop.  So, this computer is roughly like half a million laptops - all connected and working together.  It can compute at a rate of 5 petaflops.  That's 5,000,000,000,000,000 floating-point operations per second!

This computer was recently used to compute the ratio of the proton and neutron masses.  A proton weighs about 1836 times as much as an electron.  A neutron is a bit heavier: about 1839 times the electron mass.

This is important, since it means a lone proton is stable, while a lone neutron is not: in about 15 minutes, it will decay into a proton and some other stuff.  This is why the universe is mainly made of hydrogen, not neutrons!

Why is the neutron a bit heavier?  People have been wondering for a long time.

The answer lies in the Standard Model, our best theory of particles and all the forces except gravity.  Protons and neutrons are made of quarks, and the Standard Model says exactly how this works.  So, we can use the Standard Model to compute the ratio of proton and neutron masses.

But it's not easy!  As anyone who has studied quantum field theory will tell you, this problem is a nightmare.  For the course I took, in the final exam we had to compute how two electrons scatter off each other. I probably screwed up, because I only got a B+.  But that problem is really, really easy compared to computing the mass of a proton or neutron.

The problem is that the strong force, which holds the quarks together, interacts with itself in a complicated way.  The strong force is carried by particles called gluons.   Quarks emit and absorb gluons.  But gluons also emit and absorb gluons!   So, a proton or neutron is like a blob containing 3 quarks - but a blob made of gluons, virtual quark-antiquark pairs, and some other virtual particles, all held together by their interactions. 

To accurately compute the total energy of this blob, and thus its mass, you basically need to simulate it.  And even though we know the basic rules, that takes a lot of computing.

But now it's been done! 

• Sz. Borsanyi, S. Durr, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, L. Lellouch, T. Lippert, A. Portelli, K. K. Szabo, and B. C. Toth, Ab initio calculation of the neutron-proton mass difference, Science  347 (27 March 2015), 1452-1455.  Free version available at http://arxiv.org/abs/1406.4088.

It makes me simultaneously relieved that I didn't go deeper into this subject, and jealous - because it's so beautiful, intricate and demanding.

You can see the world in a grain of sand... or even a single proton.

The abstract gives a tiny taste:

The existence and stability of atoms rely on the fact that neutrons are more massive than protons. The measured mass difference is only 0.14% of the average of the two masses. A slightly smaller or larger value would have led to a dramatically different universe. Here, we show that this difference results from the competition between electromagnetic and mass isospin breaking effects. We performed lattice quantum-chromodynamics and quantum-electrodynamics computations with four nondegenerate Wilson fermion flavors and computed the neutron-proton mass-splitting with an accuracy of 300 kilo–electron volts, which is greater than 0 by 5 standard deviations. We also determine the splittings in the Σ, Ξ, D, and Ξcc isospin multiplets, exceeding in some cases the precision of experimental measurements.

If you're a quantum field theory geek, you'll want to read the 'supplementary material', because that's where all the details are: 

http://www.sciencemag.org/content/suppl/2015/03/25/347.6229.1452.DC1/Borsanyi-SM.pdf

Here you'll see particle physics jargon blending with computing jargon in a marvelous symphony:

Starting with Sec. 6 we present the details of the many simulations that are performed and summarized here. The use of Rational Hybrid Monte-Carlo method is discussed with a special emphasis on the lowest eigenvalues of the Dirac operator. Autocorrelations are under control for our choice of parameters in the QCD part of our work. However, due to the zero mass of the photon and the correspondingly large correlation lengths, a standard Hybrid Monte-Carlo integration of the photon fields results in large autocorrelation times. We show how we solved this problem by developing a Fourier accelerated algorithm. For the propagator calculations we used a 2-level multi-grid approach to have several hundred source positions and significantly improve our statistics.

And here's a vastly harder challenge: do these calculations in a way where you can prove they are accurate up to some tolerance.  We can't do this yet because we haven't even proved the Standard Model is mathematically consistent.  Until we do, and until we develop a rigorous approach to computing things like the proton-neutron mass difference, there's always the danger that researchers are subconsciously choosing certain approximations because they seem to make the answer come out closer to what we observe.

Puzzle: why are there just 7 big boxes here, not 8?  Everything else comes in powers of 2.   If it had 8 boxes, JUQUEEN would have

2^19 = 524288

cores.  But it has just 458752.  Budget cuts?

#spnetwork arXiv:1406.4088 #particlePhysics   #qcd  
#bigness  ___

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2015-03-29 16:05:41 (131 comments, 56 reshares, 269 +1s)Open 

Black holes - bigger on the inside

Guess what: black holes are bigger inside than they look - and they get bigger as they get older! 

For example, take the big black hole in the center of our galaxy, called Sagittarius A*.  It's about 2 million kilometers across.  That's pretty big - but the orbit of Mercury is 60 times bigger.  This black hole is old, roughly a billion years old.  And here's the cool part:  it's been growing on the inside  all this time!  

How is this possible?  Well, since spacetime is severely warped in a black hole, its volume can be bigger than you'd guess from outside.  And its volume can change.  Since we understand general relativity quite well, we can calculate how this works!  But nobody thought of doing it until last year, when Marios Christodoulou and my friend Carlo Rovelli did it.   

How bigis the black ho... more »

Black holes - bigger on the inside

Guess what: black holes are bigger inside than they look - and they get bigger as they get older! 

For example, take the big black hole in the center of our galaxy, called Sagittarius A*.  It's about 2 million kilometers across.  That's pretty big - but the orbit of Mercury is 60 times bigger.  This black hole is old, roughly a billion years old.  And here's the cool part:  it's been growing on the inside  all this time!  

How is this possible?  Well, since spacetime is severely warped in a black hole, its volume can be bigger than you'd guess from outside.  And its volume can change.  Since we understand general relativity quite well, we can calculate how this works!  But nobody thought of doing it until last year, when Marios Christodoulou and my friend Carlo Rovelli did it.   

How big is the black hole at the center of our galaxy?  On the inside, it can hold a million solar systems!  Its volume is about 10^34 cubic kilometers!   And it's growing at a rate of about 10^25 cubic kilometers per year!

Or suppose you have an ordinary star that turns into a black hole.  This black hole will last a long time before it evaporates due to Hawking radiation.  Christodolou and Rovelli estimate how big its volume will get before this happens.  And it gets really big - bigger than the current-day observable universe!

Before you get too excited, remember: people falling into the black hole will not have time to do anything fun inside.  They will hit the singularity in a short time.  Very very roughly speaking, the problem is not the shortage of space inside the black hole, it's the shortage of time.  

If you fall into the black hole at the center of our galaxy, it will be about 1 minute, at most, before you hit the singularity.   You will not get to see most of the space inside the black hole!   The singularity is not in the 'middle' of the black hole - it's in your future.  You will hit it before you can reach the 'middle'.  So, you will only get to see part of the 'edge regions' inside the black hole.

The 'middle regions' can only be seen by people who fell in much earlier.  And they can't see the 'edge', where you are!

And now for the serious part. 

The hard part of this problem is defining the volume inside a black hole. 

If you choose a moment in time, the black hole's event horizon at that moment is a sphere.  There are infinitely many ways to extend this sphere to a solid ball.  In other words: there are many ways to choose a slice of space inside the black hole whose boundary is your chosen sphere. 

The slice can bend forwards in time, or backwards in time.  We can choose a wiggly slice or a smooth one.  Each slice has its own volume.  

How do you choose one, so you can calculate its volume? Christodoulou and Rovelli choose the one with the largest volume. This may sound like it's cheating.  But it's not.

Think of a simpler problem one dimension down.  You have a loop of wire.  You ask me: "What's the area of the surface whose boundary is this loop?" 

I say: "That's a meaningless question!  Which surface?  There are lots!"  

You say: "Pick the best one!"

So, it's up to me.   I take some soapy water and make a soap film whose boundary is that loop.  That's the surface I use.   If the loop of wire is not too crazy in its shape, this surface is uniquely defined.   In some sense it's the "least wiggly" surface I could choose.

This surface minimizes the area.  A more wiggly surface would have more area.

Christodoulou and Rovelli are doing the same thing.  But spacetime is different than space!   If you choose a wiggly 3-dimensional spatial surface in spacetime, it will have less volume than a flatter surface with the same boundary!  

So, the way to pick the flattest, nicest spatial surface inside our black hole is to pick the one that maximizes the volume. 

If you tried to minimize the volume, you could get it as close to zero as you wanted.  And this would have nothing to do with black holes!   This would be true even in your living room.

Puzzle: why?

Here's the paper:

• Marios Christodoulou and Carlo Rovelli, How big is a black hole?, http://arxiv.org/abs/1411.2854.

#spnetwork arXiv:1441.2854 #generalRelativity  ___

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2015-03-28 22:29:31 (54 comments, 45 reshares, 181 +1s)Open 

News in the USA

News in the USA___

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2015-03-28 00:29:19 (96 comments, 10 reshares, 64 +1s)Open 

Puzzle: if the Earth became a black hole, how big would this black hole be?

Please tell me your first guess, without calculating or looking it up.  And then,  after a few hours have gone by, some of you can calculate the answer. 

Warning: the film here is NOT AT ALL REALISTIC!!!  It's just funny.

Puzzle: if the Earth became a black hole, how big would this black hole be?

Please tell me your first guess, without calculating or looking it up.  And then,  after a few hours have gone by, some of you can calculate the answer. 

Warning: the film here is NOT AT ALL REALISTIC!!!  It's just funny.___

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2015-03-27 14:39:57 (24 comments, 1 reshares, 40 +1s)Open 

Where in the world?

Where in the world?___

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2015-03-26 15:30:39 (127 comments, 73 reshares, 153 +1s)Open 

Big data

A petabyte is a lot of information.  But how many petabytes does it take to completely describe one gram of water?  

Let's see:

A bit is the information in one binary decision — a no or yes, a 0 or 1.

• 5 bits: approximate information in one letter of the Roman alphabet.

A byte is 8 bits.

A kilobyte is about a thousand bytes (actually 1024 of them).

• 2 kilobytes: a typewritten page.
• 100 kilobytes: a low-resolution photograph.

A megabyte is about a million bytes.

• 1 megabyte: a small novel or a 3.5 inch floppy disk.
• 2 megabytes: a high-resolution photograph.
• 5 megabytes: the complete works of Shakespeare.
• 500 megabytes: a CD-ROM.

A gigabyte is about a billion bytes.

• 1.25 gigabytes:the human genome, or... more »

Big data

A petabyte is a lot of information.  But how many petabytes does it take to completely describe one gram of water?  

Let's see:

A bit is the information in one binary decision — a no or yes, a 0 or 1.

• 5 bits: approximate information in one letter of the Roman alphabet.

A byte is 8 bits.

A kilobyte is about a thousand bytes (actually 1024 of them).

• 2 kilobytes: a typewritten page.
• 100 kilobytes: a low-resolution photograph.

A megabyte is about a million bytes.

• 1 megabyte: a small novel or a 3.5 inch floppy disk.
• 2 megabytes: a high-resolution photograph.
• 5 megabytes: the complete works of Shakespeare.
• 500 megabytes: a CD-ROM.

A gigabyte is about a billion bytes.

• 1.25 gigabytes: the human genome, or a pickup truck full of books.
• 20 gigabytes: a good collection of the works of Beethoven.
• 100 gigabytes: a library floor of academic journals.

A terabyte is about a trillion bytes.

• 2 terabytes: an academic research library.
• 6 terabytes: all academic journals printed in 2002.
• 10 terabytes: the print collections of the U.S. Library of Congress.
• 40 terabytes: all books printed in 2002.
• 60 terabytes: all audio CDs released in 2002.
• 80 terabytes: capacity of all floppy discs produced in 2002.
• 140 terabytes: all newspapers printed in 2002.
• 170 terabytes: the searchable part of the World-Wide Web in 2002.
• 250 terabytes: capacity of all zip drives produced in 2002.

A petabyte is about 10^15 bytes.

• 1.5 petabytes: all office documents generated in 2002.
• 2 petabytes: all U.S academic research libraries.
• 6 petabytes: all cinema release films in 2002.
• 90 petabytes: the "Deep Web" in 2002
• 130 petabytes: capacity of all audio tapes produced in 2002.
• 400 petabytes: all photographs taken in 2002.
• 440 petabytes: all emails sent in 2002.

An exabyte is about 10^18 bytes.

• 1.3 exabytes: capacity of all videotapes produced in 2002.
• 2 exabytes: capacity of all hard disks produced in 2002.
• 5 exabytes: all the words ever spoken by human beings.
• 9 exabytes: all the genomes of every living person in 2015.

A zettabyte is about 10^21 bytes.

• 500 zettabytes: the information needed to completely describe the state of a gram of water at room temperature.

So, the answer is:

It takes 500,000,000 petabytes to completely describe one gram of water, down to the positions and velocities of the individual subatomic particles...

... limited, of course, by the Heisenberg uncertainty principle!   That's what makes the amount of information finite.

How can we calculate this?  It sounds hard, but it's not if you look up a few numbers.

First of all, the entropy of water! At room temperature (25 degrees Celsius) and normal pressure (1 atmosphere), the entropy of a mole of water is 69.91 joules per kelvin.

To understand this, first you need to know that chemists like moles — and by a 'mole', I don't mean that fuzzy creature that ruins your lawn: I mean a certain ridiculously large number of molecules or atoms, invented to deal with the fact that even a tiny little thing is made of lots of atoms.  By definition, a mole is about the number of atoms in one gram of hydrogen.

A guy named Avogadro figured out that this number is about 6.023 × 10^23. People now call this Avogadro's number. So, a mole of water means 6.023 × 10^23 molecules of water. And since a water molecule is 18 times heavier than a hydrogen atom, this is 18 grams of water.

So, if we prefer grams to moles, the entropy of a gram of water is is 69.91/18 = 3.88 joules per kelvin.  By the way, I don't want to explain why entropy is measured in joules per kelvin — that's another fun story.

But what does all this have to do with information? Well, Boltzmann, Shannon and others figured out how entropy and information are related, and the formula is pretty simple: one nat of information equals 1.3808 × 10^(-23) joules per kelvin of entropy. This number is called Boltzmann's constant.

What's a 'nat' of information?  Well, bits of information are a good unit when you're using binary notation — 0's and 1's — but trits would be a good unit if you were using base 3, and so on.  For physics  the most natural unit is a nat, where we use base e.  So, 'nat' stands for 'natural'.

Don't get in a snit over the fact that we can't actually write numbers using base e — if you do, I'll just say you're nitpicking, or natpicking! The point is, information in the physical world is not binary — so base e turns out to be the best.

Okay: so, by taking the reciprocal of Boltzmann's constant we see that one joule per kelvin of entropy equals 7.24 × 10^22 nats of information. 

That's all we need to look up.  We can now just multiply and see that a gram of water (at room temperature and pressure) holds

3.88 × 7.24 × 10^23 = 2.81 × 10^24 nats

of information. In other words, this is how much information it takes to completely specify the state of one gram of water.

Or if you prefer bits, use the fact that a bit equals ln(2) or .693 nats. Dividing by this, we see a gram of water holds

4.05 × 10^24 bits

of information.  And amazingly, this is something we know quite precisely!  I've rounded off the numbers, but we could actually work it out to more decimal places if we wanted. 

If you want to learn more about this, study statistical mechanics - that's where physics meets information theory. 

A bunch of my figures came from here:

• Peter Lyman, Hal R. Varian, Kirsten Swearingen, et al, How much information? 2003, http://www.sims.berkeley.edu/research/projects/how-much-info-2003/

and the chart originally came from here:

http://mozy.com/blog/misc/how-much-is-a-petabyte/

though it was edited by folks at Gizmodo.  All this stuff and more is on this page of mine:

http://math.ucr.edu/home/baez/information.html

#informationtheory    #bigness  ___

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2015-03-24 19:51:26 (81 comments, 27 reshares, 130 +1s)Open 

Integrals from hell

When my uncle first tried to teach me calculus I thought it was confusing.  I already wanted to be a mathematician.  So I decided to be a mathematician who wouldn't use calculus.

A few weeks later I wasn't scared of calculus anymore, thanks to the wonderful book Calculus Made Easy by Silvanus Thompson, which he gave me.

Later, when I took calculus in high school, it became fun to tackle tough integrals.  The reason it's fun is that a limited set of rules lets you do a lot of integrals, yet there's still some art involved in doing them well.  It's like a game.

(This was before computers were programmed to do integrals much better than people can.)

Later, I learned that only a small fraction of the integrals you can write down can be done using the rules you learn in school.  For most, the answer is somefunc... more »

Integrals from hell

When my uncle first tried to teach me calculus I thought it was confusing.  I already wanted to be a mathematician.  So I decided to be a mathematician who wouldn't use calculus.

A few weeks later I wasn't scared of calculus anymore, thanks to the wonderful book Calculus Made Easy by Silvanus Thompson, which he gave me.

Later, when I took calculus in high school, it became fun to tackle tough integrals.  The reason it's fun is that a limited set of rules lets you do a lot of integrals, yet there's still some art involved in doing them well.  It's like a game.

(This was before computers were programmed to do integrals much better than people can.)

Later, I learned that only a small fraction of the integrals you can write down can be done using the rules you learn in school.  For most, the answer is some function you can't even write down using the usual kit of high-school functions: addition, subtraction, multiplication, division, exponentials, logarithms, trig and inverse trig functions. 

I also learned that most professional mathematicians consider it uncool to get really good at doing integrals: it's just one step up from memorizing digits of pi.  Professional mathematicians want to learn about stuff like Shimura varieties, or motivic cohomology - stuff you can't even begin to explain to ordinary folks.  This is what it takes to impress other mathematicians.

Still, integrals can be fun.

Sometimes I teach calculus at U.C. Riverside - though now it's mainly poorly-paid "lecturers" who have to do this job.  When I teach calculus, I usually focus on the students who are having trouble.  I want everyone to learn the stuff!   Unfortunately this means I never spend time showing the good students fun tricks.

I realize now that I should spend a little time doing "integrals from hell" like this one here.  First of all, it would be fun for the better students.  It shows there's a kind of athletic element to math, where you don't just learn to walk: you learn to run insanely fast!  Second of all, it makes the easy integrals seem easier.

This particular integral is fun because at first glance it looks horrible, yet it falls quickly to high-school tricks.  It's fun to see how these tricks make it simpler!  Then you get something that's a bit grungy and boring.  It's just the first steps that are fun.

Puzzle: what's a nice way to start doing this integral?

I got this integral from here:

https://plus.google.com/u/0/+Icalculusblog/posts

They post lots of fun integrals - good puzzles if you still remember your high school calculus and you're not too much of a professional mathematician to enjoy this sort of thing.  You can see solutions here:

http://www.i-calculus.com/___

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2015-03-21 15:31:37 (24 comments, 26 reshares, 179 +1s)Open 

Seven caves on a Martian volcano

In 2007, NASA discovered seven very dark circles on a mountain in Mars.  Using infrared cameras, they checked the temperature of these circles - and discovered that they didn't change much from day to night.

"They are cooler than the surrounding surface in the day and warmer at night," said Glen Cushing of the U.S. Geological Survey's Astrogeology Team. "Their thermal behavior is not as steady as large caves on Earth that often maintain a fairly constant temperature, but it is consistent with these being deep holes in the ground."

In short, they could be windows into caves!

They're called the Seven Sisters: Dena, Chloe, Wendy, Annie, Abby, Nikki and Jeanne.   They range in diameter from about 100 to 250 meters.

They're on one of the highest places on Mars: a volcanona... more »

Seven caves on a Martian volcano

In 2007, NASA discovered seven very dark circles on a mountain in Mars.  Using infrared cameras, they checked the temperature of these circles - and discovered that they didn't change much from day to night.

"They are cooler than the surrounding surface in the day and warmer at night," said Glen Cushing of the U.S. Geological Survey's Astrogeology Team. "Their thermal behavior is not as steady as large caves on Earth that often maintain a fairly constant temperature, but it is consistent with these being deep holes in the ground."

In short, they could be windows into caves!

They're called the Seven Sisters: Dena, Chloe, Wendy, Annie, Abby, Nikki and Jeanne.   They range in diameter from about 100 to 250 meters.

They're on one of the highest places on Mars: a volcano named Arsia Mons near Mars' tallest mountain, Olympus Mons.

But the picture here is part of another story.   A few years ago, an artist named Ron Guyatt started making "solar system travel posters."  This was one of the first.  I want some!

You can see them all here:

http://ron-guyatt.deviantart.com/gallery/36677786/Space-Travel-Posters

There are 3 pages of them.  Click to make them bigger.

He says:

Space tourism is still a long ways off, but it's not hard to imagine that someday, tourists will visit the natural geological landmarks of other worlds much like they tour the Grand Canyon, Mount Everest or Ayers Rock. Each of these great tourist destinations needs a classic retro travel poster to entice visitors. Until the day people settle off-world and make their own destinations many of these may be the places that people will want to travel to. I hope that these posters can inspire people to think beyond our world to the limitless possibilities of the Universe.

I like this idea! 

The posters on the website are more abstract than the one here.  Compare it to this:

http://ron-guyatt.deviantart.com/art/Mars-Seven-Sisters-491470969

Which do you like better?  I like the less  abstract picture here.

Puzzle: There are lots of things called "The Seven Sisters".  Which ones do you know?

Here's some more about Arsia Mons, including photos of the Seven Sisters:

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

#astronomy  ___

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2015-03-20 16:19:07 (21 comments, 12 reshares, 84 +1s)Open 

Zooming in

If you only see a dot, click on the picture!

As you zoom in, this will stretch out to become a line segment.  As you zoom in further, you see its thickness.  It's really a long thin rectangle.

But wait longer and you see it's really a field of dots.  And zooming into any one of these dots, this process repeats... forever!

Each long thin rectangle is 10,000 times longer than the next smaller one.

So, you're looking at a very complicated set of points in the plane, whose dimension seems to depend on how closely you zoom in. 

In this example, created by +Simon Willerton, the dimension keeps cycling: 0, 1, 2, 0, 1, 2, ...  But you can make examples that do other things.

The moral?  Mathematicians have various ways of defining the dimension of a set of points in the plane, or even more general sets. A poin... more »

Zooming in

If you only see a dot, click on the picture!

As you zoom in, this will stretch out to become a line segment.  As you zoom in further, you see its thickness.  It's really a long thin rectangle.

But wait longer and you see it's really a field of dots.  And zooming into any one of these dots, this process repeats... forever!

Each long thin rectangle is 10,000 times longer than the next smaller one.

So, you're looking at a very complicated set of points in the plane, whose dimension seems to depend on how closely you zoom in. 

In this example, created by +Simon Willerton, the dimension keeps cycling: 0, 1, 2, 0, 1, 2, ...  But you can make examples that do other things.

The moral?  Mathematicians have various ways of defining the dimension of a set of points in the plane, or even more general sets.  A point, or a finite set of points, is 0-dimensional.  A line, or a smooth curve, is 1-dimensional.  A solid rectangle, or a disk, is 2-dimensional. 

But sometimes it's more complicated!  There are fractals whose dimension is not an integer... at least if we use the right definition of 'dimension'.  The old Lebesgue dimension is always an integer, but the Hausdorff-Besicovich dimension or Minkowski dimension can be fractional, or even irrational. 

And there are also sets whose dimension seems to depend on how closely you look at them!  That's what we have here.

Simon is working on a theory of scale-dependent dimension, to make this precise.  He's writing a series of blog articles on it - and the first is here:

https://golem.ph.utexas.edu/category/2015/03/a_scaledependent_notion_of_dim.html

There's a lot of nice math here, but a lot of open questions... which is good if you're a mathematician!  More puzzles to work on!

For the hard-core details, go here:

•+Simon Willerton, Spread: a measure of the size of metric spaces, http://arxiv.org/abs/1209.2300.

#spnetwork arXiv:1209.2300 #fractal #dimension  ___

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2015-03-18 15:28:24 (65 comments, 93 reshares, 252 +1s)Open 

A planet in the fourth dimension

I bet you know that planets go around the sun in elliptical orbits. But do you know why?

In fact, they’re moving in circles in 4 dimensions. But when these circles are projected down to 3-dimensional space, they become ellipses!  This animation by Greg Egan shows the idea.

The plane here represents 2 of the 3 space dimensions we live in. The vertical direction is the mysterious fourth dimension. The planet goes around in a circle in 4-dimensional space. But down here in 3 dimensions, its ‘shadow’ moves in an ellipse!

What’s this fourth dimension I’m talking about here? It’s a lot like time. But it’s not exactly time. It’s the difference between ordinary time and another sort of time, which flows at a rate inversely proportional to the distance between the planet and the sun.

Egan's animationuses this other ... more »

A planet in the fourth dimension

I bet you know that planets go around the sun in elliptical orbits. But do you know why?

In fact, they’re moving in circles in 4 dimensions. But when these circles are projected down to 3-dimensional space, they become ellipses!  This animation by Greg Egan shows the idea.

The plane here represents 2 of the 3 space dimensions we live in. The vertical direction is the mysterious fourth dimension. The planet goes around in a circle in 4-dimensional space. But down here in 3 dimensions, its ‘shadow’ moves in an ellipse!

What’s this fourth dimension I’m talking about here? It’s a lot like time. But it’s not exactly time. It’s the difference between ordinary time and another sort of time, which flows at a rate inversely proportional to the distance between the planet and the sun.

Egan's animation uses this other sort of time. Relative to this other time, the planet is moving at constant speed around a circle in 4 dimensions. But in ordinary time, its shadow in 3 dimensions moves faster when it’s closer to the sun.

All this sounds crazy, but it’s not some new physics theory. It’s just a different way of thinking about Newtonian physics!  Of course you can see that planets move in elliptical orbits without resorting to the 4th dimension.  But it becomes a lot more obvious if you do!

Physicists have known about this viewpoint at least since 1980, thanks to a paper by the mathematical physicist Jürgen Moser. Some parts of the story are much older. A lot of papers have been written about it.

But I only realized how simple it is when I got a paper in my email from someone I didn't know: an amateur mathematician  named Jesper Göransson.  I get a lot of papers by crackpots, but the occasional gem like this makes up for all those.

The best thing about Göransson’s 4-dimensional description of planetary motion is that it gives a clean explanation of an amazing fact. You can take any elliptical orbit, apply a rotation of 4-dimensional space, and get another valid orbit!

Of course we can rotate an elliptical orbit about the sun in the usual 3-dimensional way and get another elliptical orbit. The interesting part is that we can also do 4-dimensional rotations. This can make a round ellipse look skinny: when we tilt a circle into the fourth dimension, its ‘shadow’ in 3-dimensional space becomes thinner!

In fact, you can turn any elliptical orbit into any other elliptical orbit with the same energy by a 4-dimensional rotation of this sort. All elliptical orbits with the same energy are really just circular orbits on the same sphere in 4 dimensions!

For the details, see the Azimuth blog:

https://johncarlosbaez.wordpress.com/2015/03/17/planets_in_the_4th_dimension/

I had to go through Göransson’s calculations to convince myself that they were right. 

And here is his paper:

• Jesper Göransson, Symmetries of the Kepler problem, 8 March 2015, http://math.ucr.edu/home/baez/mathematical/Goransson_Kepler.pdf.

#physics  ___

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2015-03-16 16:05:57 (74 comments, 98 reshares, 302 +1s)Open 

Quantum superpositions

You've surely heard of Schrödinger's cat, which is both alive and dead.  It's actually in a superposition of live and dead states: in quantum mechanics you can multiply different states by numbers, and then add them up! 

On the Azimuth blog +Piotr Migdal has written an explanation of quantum superpositions:

https://johncarlosbaez.wordpress.com/2015/03/13/quantum-superposition/

He explains why in chemistry an electron likes to be in a superposition of different position states: this is a way for it to reduce its energy!

Here's what Schrödinger actually said about that cat:

One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter, there isa t... more »

Quantum superpositions

You've surely heard of Schrödinger's cat, which is both alive and dead.  It's actually in a superposition of live and dead states: in quantum mechanics you can multiply different states by numbers, and then add them up! 

On the Azimuth blog +Piotr Migdal has written an explanation of quantum superpositions:

https://johncarlosbaez.wordpress.com/2015/03/13/quantum-superposition/

He explains why in chemistry an electron likes to be in a superposition of different position states: this is a way for it to reduce its energy!

Here's what Schrödinger actually said about that cat:

One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter, there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer that shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The psi-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts.

It is typical of these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into macroscopic indeterminacy, which can then be resolved by direct observation.

He wrote this in 1935, in a paper "The present situation in quantum mechanics".  Actually it was in German: "Die gegenwärtige Situation in der Quantenmechanik". 

He wrote this in response to a famous 1935 paper by Einstein, Podolsky and Rosen.  Their paper raised the "paradox" of how quantum mechanics lets the spins of distant electrons be correlated in ways that are impossible in classical mechanics.  We now say their spins are entangled

In fact, this article by Schrödinger was the first to use the term "entanglement" - or in German, "Verschränkung".  Einstein thought Schrödinger was one of the few people who understood what he was trying to get at:

You are the only contemporary physicist, besides Laue, who sees that one cannot get around the assumption of reality, if only one is honest. Most of them simply do not see what sort of risky game they are playing with reality—reality as something independent of what is experimentally established. Their interpretation is, however, refuted most elegantly by your system of radioactive atom + amplifier + charge of gunpowder + cat in a box, in which the psi-function of the system contains both the cat alive and blown to bits. Nobody really doubts that the presence or absence of the cat is something independent of the act of observation.

"Nobody really doubts it."   The "appeal to common sense" is often a bad move when dealing with fundamental physics.  It would have stopped Einstein from realizing that the rate at which time passes for you depends on your motion... or that gravity is the curvature of space and time.  It may have actually stopped Einstein from realizing that matter can form black holes, and that the universe is expanding. 

Einstein realized that quantum mechanics would change our basic ideas about reality, but he seems to have concluded this meant it couldn't be right.  Too bad!

When I was a student I spent a lot of time thinking about the interpretation of quantum mechanics... and when the internet was invented and people around the world started chatting on the newsgroup sci.physics, I spent a lot of time discussing this. 

I eventually worked through this phase.  I got a deep intuitive understanding of how quantum mechanics changes our picture of reality.  Unfortunately, this understanding doesn't fit into ordinary English very well, since ordinary English evolved from dealing with the macroscopic world, where quantum effects aren't apparent.  Anything you say in ordinary English about the meaning of quantum mechanics is going to be quite misleading.

Much later, when I got a job at the +Centre for Quantum Technologies, I was surprised and delighted to discover that everyone there was just like me.  We'd all worked through this stuff.  People there were not busy arguing: they were busy with getting quantum mechanics to do amazing things.  I think this will eventually happen to more and more people.  We'll just get used to quantum mechanics.

The cat here seems to be half awake, half asleep.  I got this picture from:

http://heinakroon.com/2011/12/28/save-schrodingers-cat-dead-or-alive/

#quantumphysics  
 ___

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2015-03-14 17:38:54 (36 comments, 30 reshares, 161 +1s)Open 

So you want to know what pi equals???

Sometimes I think mathematics has a built-in sense of humor.  This is a good approximation to pi:

22/7 = 3.142857142857142857....

but the hilarious part is that the difference

22/7 - π = 0.00126448926...

is given by the elegant integral shown here!

Why is this true?  I don't know any good way to answer that.  I'm sure with work I could do the integral and see that it is true, and that would be one answer.  But the question "why should there be such a cute formula for the difference between pi and everybody's favorite approximation to pi?" would remain.

Who first discovered this formula?  I don't know that either. Wolfram MathWorld says:

This integral was known by Kurt Mahler in the mid-1960s and appears in an exam at the University of Sydneyin No... more »

So you want to know what pi equals???

Sometimes I think mathematics has a built-in sense of humor.  This is a good approximation to pi:

22/7 = 3.142857142857142857....

but the hilarious part is that the difference

22/7 - π = 0.00126448926...

is given by the elegant integral shown here!

Why is this true?  I don't know any good way to answer that.  I'm sure with work I could do the integral and see that it is true, and that would be one answer.  But the question "why should there be such a cute formula for the difference between pi and everybody's favorite approximation to pi?" would remain.

Who first discovered this formula?  I don't know that either. Wolfram MathWorld says:

This integral was known by Kurt Mahler in the mid-1960s and appears in an exam at the University of Sydney in November 1960.

So, maybe Mahler discovered it, or maybe not.

Kurt Mahler did other cool things.  One of the cute things he proved was that like pi, the Champernowne constant

0.1234567891011121314151617181920...

is a transcendental number.  In other words: it's not the root of any polynomial with integer coefficients!

But he also did more important things.

For example, he proved Mahler's inequality: the geometric mean of the sum of two lists of n positive numbers is greater than or equal to the sum of their geometric means!

That's pretty easy.  Mahler's theorem is harder, and I'll throw it in here just for people who need something stronger for their daily dose of math.  Mahler's theorem says that any continuous function from the p-adic integers to the p-adic numbers can be expressed in terms of difference operators using the same formula that works for polynomial functions from the integers to the real numbers.

I won't write down the formula, but Newton probably knew it, and you should too - you can see it here:

https://en.wikipedia.org/wiki/Mahler's_theorem

So, p-adic integers are in some ways better than ordinary integers!

You can see more shocking formulas for pi here:

http://mathworld.wolfram.com/PiFormulas.html

In the comments, it turned out that this formula goes back at least to this paper:

• D. P. Dalzell, On 22/7, J. London Math. Soc. 19 (1944), 133–134,

It gives a nice proof that pi is between 22/7 and 22/7 - 1/630.  For more on this formula, including a proof, see:

http://mathoverflow.net/questions/67384/source-and-context-of-frac227-pi-int-01-x-x24-dx-1x2

and

https://en.wikipedia.org/wiki/Proof_that_22/7_exceeds_pi

#pi  ___

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2015-03-12 16:13:52 (45 comments, 17 reshares, 117 +1s)Open 

Ion drive

This is the kind of thruster that powered the spacecraft Dawn to the asteroids Vesta and Ceres.  It's beautiful!  It creates a beam of xenon ions.  These ions blast into space at 40 kilometers per second - 90,000 miles per hour! - pushing the ship forward. 

Dawn is solar powered.  It sucks up 10 kilowatts of solar power and uses this to run the ion thruster.  It started out with 275 kilograms of the noble gas xenon.  It takes atoms of this gas and strips off some the electrons, leaving the atoms positively charged.  These are called ions.  

It accelerates these ions with an electric field, and they shoot out of thousands of tiny holes - which I think you can see here.  Each hole acts as a lens that electrically focuses the ions.

Because the ion thruster puts out positive ions, an equal amount of negative charge must beexpelled t... more »

Ion drive

This is the kind of thruster that powered the spacecraft Dawn to the asteroids Vesta and Ceres.  It's beautiful!  It creates a beam of xenon ions.  These ions blast into space at 40 kilometers per second - 90,000 miles per hour! - pushing the ship forward. 

Dawn is solar powered.  It sucks up 10 kilowatts of solar power and uses this to run the ion thruster.  It started out with 275 kilograms of the noble gas xenon.  It takes atoms of this gas and strips off some the electrons, leaving the atoms positively charged.  These are called ions.  

It accelerates these ions with an electric field, and they shoot out of thousands of tiny holes - which I think you can see here.  Each hole acts as a lens that electrically focuses the ions.

Because the ion thruster puts out positive ions, an equal amount of negative charge must be expelled to keep the spacecraft from getting a huge electric charge.  So, a small gadget called the neutralizer shoots out electrons.

The force produced by Dawn's thrusters is tiny: just 40 millinewtons.  A newton is the force it takes to accelerate one kilogram one meter per second each second.  Dawn's thrusters push as hard as a sheet of paper pushes down on your hand!  

So, this spacecraft takes four days to accelerate from 0 to 100 kilometers per hour, while a good car can do it in 3-6 seconds.  The advantage of Dawn is that it can keep up this acceleration for years without running out of propellant.  This is what made it the first spacecraft able to slow down and orbit one body in our Solar System, then take off and go to another, then slow down and orbit that!

Puzzle 1:  Why does Dawn use xenon?  Is it just because this gas has a really cool-sounding name?

Puzzle 2: What bad things might happen if Dawn built up a big electric charge?

Puzzle 3: Why does the ion beam glow?  Why is it blue?
 
Puzzle 4: Another elegant form of propulsion is a solar sail, where sunlight pushes a spacecraft directly.  Why isn't this better than converting sunlight to energy and using that to run an ion drive?  After all, converting energy from one form to another tends to create waste heat.

#dawn  ___

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2015-03-11 19:43:52 (101 comments, 33 reshares, 159 +1s)Open 

Let's go!

The ancient game of go still holds many challenges.   Compared to go, chess is like tac-tac-toe.   There are 255168 possible games of tic-tac-toe.  There are about 10^120 possible chess games.  But there are about 10^761 possible go games!

The main challenge with go is playing it well.  But there's also counting the number of legal positions.

A full-fledged go board has 19 × 19 squares, and approximately this many legal positions:

208168199381982000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

That's about 2 × 10^170.

John Tromp wants to know the exact number.  Why?  Because it's a fun challenge.   He knows an algorithm to calculate this number.  It's very clever.  But it still takes alot of computi... more »

Let's go!

The ancient game of go still holds many challenges.   Compared to go, chess is like tac-tac-toe.   There are 255168 possible games of tic-tac-toe.  There are about 10^120 possible chess games.  But there are about 10^761 possible go games!

The main challenge with go is playing it well.  But there's also counting the number of legal positions.

A full-fledged go board has 19 × 19 squares, and approximately this many legal positions:

208168199381982000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

That's about 2 × 10^170.

John Tromp wants to know the exact number.  Why?  Because it's a fun challenge.   He knows an algorithm to calculate this number.  It's very clever.  But it still takes a lot of computing power.

He recently calculated the number of legal moves on an 18 × 18 board.   It took 9 months, and 4 petabytes of disk IO on a Dell PowerEdge R820 server.  He did it at the Institute for Advanced Studies in Princeton... that place where Einstein used to hang out.  The answer is:

669723114288829212892740188841706543509937780640178732810318337696945624428547218105214326012774371397184848890970111836283470468812827907149926502347633

To get the answer for a 19 × 19 board will take more work.  Using some very clever math, the task can be split up into 9 jobs that each compute 64 bits of the 566-bit result.   To do this and be sure the answer is right, he needs about 10 to 13 servers, each with at least 8 cores, 512 gigabytes of RAM, and 10-15 terabytes of disk space.  The job will  take about 5-9 months.

If you want to help out, email him at john.tromp@gmail.com.

You can read more about this here:

http://tromp.github.io/go/legal.html

including the answers for all square boards of size up to 18 × 18.   The most interesting part is the algorithm to compute these numbers:

http://tromp.github.io/go/gostate.ps

Puzzle 1: Can you turn this animated gif into a looped animated gif?   It's from Wikicommons.  It's sad that it plays just once and then stops:

http://commons.wikimedia.org/wiki/File:Go-board-animated.gif

Puzzle 2: This animated gif is just 11 kilobytes.  We could easily afford a looped gif that plays a whole game - say, a famous game.  Wikicommons deserves it!   Can you make one?

I thank +David Tweed for pointing out Tromp's work.  A calculation of the number of tic-tac-toe games is here:

http://www.se16.info/hgb/tictactoe.htm

There seems to be some controversy here: another person computed the number of games counting to as the same if they're related by symmetries, and got an answer of 26830.  Because this is less than 1/8 times the answer above, they can't both be right.

Ever since I got a smart phone in September, I've been playing go with a program called "Go Free".  Someday soon I'll fork over the $1.99 to get the version that lets you play on a 19 × 19 board.  So far I'm just trying to learn to crush the computer on a 13 × 13 board.  I like playing a computer because I get annoyed at human opponents when they beat me, and I don't like that.  At some point I should be able to beat the computer consistently on a 19 × 19 board and then I'll have to either quit or... something else.

Go is full of general lessons about life, which become more profound as you get better... at least for a while.  Maybe at some point it devolves into being "merely technical" - I don't know. 

Unfortunately these general lessons mostly apply to situations where cooperation is not an option.  So it's possible my personality is being warped by playing go.  Sometimes I can sort of feel that happening.

#bigness  ___

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2015-03-10 23:13:11 (31 comments, 51 reshares, 187 +1s)Open 

What is the big bad publisher Elsevier doing now?

1) Selling blank pages. 

That's right: some of their articles, costing $30 to access, consist solely of one blank page!  You can see one here:

https://twitter.com/fxcoudert/status/521675319322112000

It's a paper called 'Verified synthesis of zeolitic materials'.  Just a blank page.

When these pages were discovered, Elsevier deleted them.  That's sort of weird in itself: deleting nothingness so people can't see it.

2) Exempting themselves from their own copyright agreements. 

You can pay them to make your papers open-access.  In return you get a Creative Commons copyright which says nobody can charge money for them. 

But if you read the fine print, which is hidden somewhere else, you'll see that Elsevier excludesitself... more »

What is the big bad publisher Elsevier doing now?

1) Selling blank pages. 

That's right: some of their articles, costing $30 to access, consist solely of one blank page!  You can see one here:

https://twitter.com/fxcoudert/status/521675319322112000

It's a paper called 'Verified synthesis of zeolitic materials'.  Just a blank page.

When these pages were discovered, Elsevier deleted them.  That's sort of weird in itself: deleting nothingness so people can't see it.

2) Exempting themselves from their own copyright agreements. 

You can pay them to make your papers open-access.  In return you get a Creative Commons copyright which says nobody can charge money for them. 

But if you read the fine print, which is hidden somewhere else, you'll see that Elsevier excludes itself from this restriction! 

This lets them charge money for open-access papers.  Read the story here:

http://www.laurenbcollister.com/well-well-look-whos-at-it-again/

3) Letting editors publish hundreds of their own papers in journals they edit. 

You may remember the case of a physicist who did this in the Elsevier journal Chaos, Solitons and Fractals.  

But now a medical researcher named Johnny Matson has been caught publishing hundreds of his own papers in two Elsevier journals he edits: Research in Developmental Disabilities and Research in Autism Spectrum Disorders. 

That means nobody can really trust these papers to have been properly refereed!  And this matters more for autism than for theoretical physics.  Nobody will get sick from a bad theory about fractal spacetime.

Check out the story here:

http://deevybee.blogspot.com/2015/02/journals-without-editors-what-is-going.html

and join the Elsevier boycott if you haven't yet:

http://thecostofknowledge.com/___

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2015-03-07 22:25:30 (37 comments, 23 reshares, 110 +1s)Open 

14-year-old guitarist shreds Paganini

Tina S is is blowing everyone away on YouTube.  She started playing classical guitar at the age of 6 - but now she enjoys intense electric guitar solos!

I really love to see a young girl playing a traditionally macho instrument better than most men will ever do - and looking calm, almost bored, while doing it. 

Some rock guitarists twist about and contort their faces while playing a simple melody really loud.  Others, like Robert Fripp, channel all their emotion into the music and look bland and businesslike while ripping your brain to pieces with complex patterns.  Tina S is in the latter school.

I'm amused at how this upsets some people.  Some comments on her videos tell her to play with more 'feeling'.   You even see this on her latest treat, a great version of the solo from Pink Floyd'ssong &... more »

14-year-old guitarist shreds Paganini

Tina S is is blowing everyone away on YouTube.  She started playing classical guitar at the age of 6 - but now she enjoys intense electric guitar solos!

I really love to see a young girl playing a traditionally macho instrument better than most men will ever do - and looking calm, almost bored, while doing it. 

Some rock guitarists twist about and contort their faces while playing a simple melody really loud.  Others, like Robert Fripp, channel all their emotion into the music and look bland and businesslike while ripping your brain to pieces with complex patterns.  Tina S is in the latter school.

I'm amused at how this upsets some people.  Some comments on her videos tell her to play with more 'feeling'.   You even see this on her latest treat, a great version of the solo from Pink Floyd's song 'Comfortably Numb':

https://www.youtube.com/watch?v=2_P-t0idgqI

They're probably watching her face instead of listening to the music.

I find that solo more musically satisfying than this one here - but I posted this one because she did it when she was only 14, and it demonstrates her speed better.

She has a nice matter-of-fact attitude.  From an interview:

Q : What has so far been the most memorable of your early career stages?

Tina: It is a mistake to talk about career for me.  [....]   Of course, the buzz on the Internet with the Van Halen solo and the follow-up of the buzz with my second video is very impressive. But for me, I have not yet started my career as a guitarist.

In case you don't know what shred means, try guitarist Steve Vai's definition:

The terminology used for someone who can play an instrument, and has such a tremendous amount of technique that what they do just seems completely effortless and absurd.

I guess this version of Paganini's 5th Caprice was written by Steve Vai.  I paraphrased part of her interview.  For the whole thing go here:

http://theguitarchannel.biz/2013/08/tina-s-tina_guitare-interview-14-year-old-girl-7-millions-views-youtube-video-eruption/

For more on 'shred guitar' see:

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

I'll quote some for you:

In 1974, the German band Scorpions used their new guitarist Ulrich Roth for their album Fly to the Rainbow, for which the title track features Roth performing "... one of the most menacing and powerful whammy-bar dive bombs ever recorded".  A year later, Roth's solo guitar playing for the album In Trance "... would become the prototype for shred guitar. Everything associated with the genre can be found on this brilliant collection of songs — sweep-picked arpeggios, harmonic minor scales, finger-tapping and ... jaw-dropping whammy-bar abuse".

In 1979, Roth left Scorpions to begin his own power trio, named Electric Sun. His debut album Earthquake contained "... heaps of spellbinding fret gymnastics ... and nimble-fingered classical workouts."  In 1978, a "heretofore unknown guitarist named Eddie Van Halen" from Los Angeles released 'Eruption', a blistering aural assault of solo electric guitar" which featured rapid "tapping", which "had rarely been heard in a rock context before". Chris Yancik argues that it is this "record, above any other, that spawned the genre of Shred."

10 million people have watched Tina S play "Eruption" here:

https://www.youtube.com/watch?v=rV6SmY04WdE

Check it out!

Guitar Player's article "Blast Into Hyperspace With The Otherworldly Power Of Shred" reviews the book Shred! and states that the pioneers were "Eddie Van Halen, Al Di Meola and Ritchie Blackmore". This fast playing style combined with melody and technique and a heavily distorted tone of heavy metal music resulted in a new nickname "shred".  Randy Rhoads and Yngwie Malmsteen advanced this style further with the infusion of Neo-classical elements. Progressive rock, heavy metal, hard rock, and jazz fusion have all made use of and adapted the style successfully over the years. In general, the phrase "shred guitar" has been traditionally associated with instrumental rock and heavy metal guitarists. This association has become less common now that modern forms of metal have adopted shredding as well. In the 1990s, its mainstream appeal diminished with the rise of grunge and nu metal, both of which eschewed flashy lead guitar solos. Underground acts like Shawn Lane and Buckethead continued to develop the genre further.

Shred guitar has advanced rapidly. What was once 'fast' playing during the 1990s has been rapidly replaced by a new breed of shred guitar style twice or up to four times faster. The most recent ascent in speed seems to be centered around speed competitions using the classical music piece 'Flight of the Bumblebee.' Many shred guitarists demonstrate their mastery of the piece and others on sites like YouTube. Tom Hard, Philip Taylor and a few others demonstrate similar speed.___

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2015-03-06 16:22:11 (43 comments, 46 reshares, 168 +1s)Open 

Confirmed: I am in orbit around #Ceres

That's what the spacecraft Dawn said on Twitter today.  After more than 7 years and a visit to the asteroid Vesta, Dawn has reached its goal!

I hope the white spots on Ceres are a solar power plant for an alien base, or at least a cryovolcano - a volcano that shoots up liquid water instead of lava.  But probably they're just ice or snow. 

Still, this is pretty cool!  Ceres is probably a protoplanet whose growth got stopped by the huge gravitational pull of Jupiter.  We know it has water vapor in its tiny atmosphere.  It may have a lot of ice inside, unlike most of the rocky asteroids.  So we can learn a bit about how planets first formed by visiting this world.

It's also cool how we got there.  NASA couldn't explore Ceres without ion propulsion.   Dawn started outits jour... more »

Confirmed: I am in orbit around #Ceres

That's what the spacecraft Dawn said on Twitter today.  After more than 7 years and a visit to the asteroid Vesta, Dawn has reached its goal!

I hope the white spots on Ceres are a solar power plant for an alien base, or at least a cryovolcano - a volcano that shoots up liquid water instead of lava.  But probably they're just ice or snow. 

Still, this is pretty cool!  Ceres is probably a protoplanet whose growth got stopped by the huge gravitational pull of Jupiter.  We know it has water vapor in its tiny atmosphere.  It may have a lot of ice inside, unlike most of the rocky asteroids.  So we can learn a bit about how planets first formed by visiting this world.

It's also cool how we got there.  NASA couldn't explore Ceres without ion propulsion.   Dawn started out its journey with 425 kilograms of the noble gas xenon.   It turns this gas into ions and shoots them out the back to accelerate through space.  So, it rides through space on a blue-green beam of ions!

The force of this beam is tiny: 90 millinewtons, like a piece of paper pushing your hand down.  But it's a way to use very little mass to push the spacecraft for a long time.  Dawn has accelerated to a speed of 38,300 kilometers per hour this way! 

This is the kind of thing we can do if we stop fussing and fighting for a while.  Are we really ready to explore the universe?  Maybe not yet.   But someday....

#ceres #astronomy  ___

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2015-03-05 17:10:01 (17 comments, 37 reshares, 138 +1s)Open 

Good news: we're beating Ebola!

Today Liberia released its last Ebola patient after a week without any new cases of the virus.

Bad news is loud.  Good news comes quietly.  If you forget this, you'll think the world is always getting worse.  It's not.  When it gets better, people don't yell about it.

I got this graph from The Economist, and you can see more charts there:

http://www.economist.com/blogs/graphicdetail/2015/03/ebola-graphics

The news about Liberia is here:

http://www.bbc.com/news/world-africa-31744616

There were 132 new cases in Guinea and Sierra Leone in the week before March 1st.   A World Health Organization spokesman said "We have to get down to zero in all three countries before we can consider this thing beaten."

My wife and I gave $500 to Doctors WithoutBorde... more »

Good news: we're beating Ebola!

Today Liberia released its last Ebola patient after a week without any new cases of the virus.

Bad news is loud.  Good news comes quietly.  If you forget this, you'll think the world is always getting worse.  It's not.  When it gets better, people don't yell about it.

I got this graph from The Economist, and you can see more charts there:

http://www.economist.com/blogs/graphicdetail/2015/03/ebola-graphics

The news about Liberia is here:

http://www.bbc.com/news/world-africa-31744616

There were 132 new cases in Guinea and Sierra Leone in the week before March 1st.   A World Health Organization spokesman said "We have to get down to zero in all three countries before we can consider this thing beaten."

My wife and I gave $500 to Doctors Without Borders.  Money well spent!

#ebola  ___

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2015-03-04 16:36:17 (21 comments, 12 reshares, 78 +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 (39 comments, 10 reshares, 74 +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 (51 comments, 71 reshares, 182 +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, 23 reshares, 108 +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 (43 comments, 26 reshares, 95 +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 (72 comments, 7 reshares, 100 +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 (125 comments, 39 reshares, 108 +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 (97 comments, 18 reshares, 97 +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, 16 reshares, 65 +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, 19 reshares, 149 +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 (53 comments, 39 reshares, 123 +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, 18 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, 181 +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, 33 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/.___

posted image

2015-02-02 16:28:05 (8 comments, 19 reshares, 50 +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, 24 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?___

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2015-01-27 17:53:00 (50 comments, 35 reshares, 133 +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  ___

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2015-01-26 21:45:51 (26 comments, 18 reshares, 67 +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  ___

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