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John Baez has been shared in 372 public circles

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AuthorFollowersDateUsers in CircleCommentsReshares+1Links
Refurio Anachro5,295April engagers' circle - This month has been all about #billiards in #spheroids. It all started ten months ago, when +John Baez first asked me about spheroids in the comments of his post:https://plus.google.com/117663015413546257905/posts/RGasKB4eYe1Since he is the "ubiquitous John Baez", I greatly appreciated the chance to have a nice party with fans of his work, that's you, and I'd like to tell that all of it has been great fun for me. I'm glad to see so many enjoyed the piece, thank you!I've been giving quite some thought wether I should batten onto the reactions John's announcement here on g+ has attracted. The rules I normally follow (see below) would have gotten about 85 people into this collection, but I figured it would be shame to lock you other people out from the benefits, just because I'm smug 'n square.So this circle is full of engaging lovers of mathematics, you won't regret adding us!Last month's posts:Appendix 1https://plus.google.com/+RefurioAnachro/posts/KbcDG6zbSEwAppendix 2 - Sphere vs spheroid - Some high res views from a pole.https://plus.google.com/b/115434895453136495635/+RefurioAnachro/posts/Y6fLoy9HQhDAppendix 3 - A prolate spheroid's domains - A ball position exposing a mysterious "shadow" of a hyperboloid.https://plus.google.com/+RefurioAnachro/posts/jUQWPxq9jUMAppendix 4 - A negative perspective on spheroids - +John Valentine came up with his own renderings. There are more, look at his profile!https://plus.google.com/+RefurioAnachro/posts/ZgMjn4miDE6Appendix 5 - Computing the black curves - Here's an approach to get at he black curves in the picture, but...https://plus.google.com/+RefurioAnachro/posts/YxSsvRV7fAkCorrection to Appendix 5 - Something went wrong with puzzle 2 - ...it didn't quite work out...https://plus.google.com/+RefurioAnachro/posts/9t1SeKzjH5f...and I have some ideas to fix it, and I'm still confident that I'll be able to describe the black lines better. Also, +Michael Nelson has come up with an unexpected idea (see appendix 3): Some of the lines appear to follow the contours of a polylogarithm! So the story is not at its end!Previous month's circlehttps://plus.google.com/+RefurioAnachro/posts/Y7amEot5kAxYou're in this circle because you reshared, plussed or commented on one of my posts (possibly via +Sp2015-05-01 17:58:22130001
Refurio Anachro5,301March engagers' circle - I'm always a tad sad when i don't manage to post more. Talking to you helps me get my thoughts organized, and, more importantly, fix silly misconceptions of which i harbour many. Not being able to do that, on the other hand, raises the pressure in my head.Knowing these cool people improved the quality of my life, have a look, tune in, add us all, and join in chatting maths now!The posts in march were:" " (This study is intentionally left blank)https://plus.google.com/+RefurioAnachro/posts/ifDjvqDmuQMRegrettably, there's been only few posts last month. The first gap cleared the carpet for this hilarious paper, clearly making a point about publishers!–She is a mathematicianhttps://plus.google.com/+RefurioAnachro/posts/Roa69xjWk7uI don't know what drove me to write the above, but i still do think it could be lovely if we all started spicing our words with other aspects. And when it comes to gender: Why restrict to have it only one way per word? Let's have variants for all of them! Or am i just reinventing hungarian...–Moduli spaces teaser part 2https://plus.google.com/+RefurioAnachro/posts/gAhomesjycg+wendy krieger wrote a nice playful about cyclotomic numbers. It's actually so nice that, even though it isn't directly about field theory, it's just the cup of tea that'll make one feel at home in the weird jungle of algebraic geometry i still owe you a tour of.–Diving for Calabi-Yau manifoldshttps://plus.google.com/+RefurioAnachro/posts/WqSgU8cDayaDespite the title, the above post isn't hard to understand. Maybe that's no surprise, because it did not cover Calabi-Yau manifolds. The journey ended one step short in the thin and clear air you breathe when you've climbed K3 surfaces. I wonder, were some of you able to use the swing of the post to appreciate the links about Calabi-Yau manifolds with new understanding?If you found that unsatisfying, i stand by my promises, and will write more on it in the future.–My activites, the "+1" tab on my profilehttps://plus.google.com/+RefurioAnachro/plusonesMoreover, i've seen many cool other posts by cool people, and I'm publishing the links to these here, disappointingly they appear without the titles. There's a solution to that: More reshares! What do you think?–Previous month's circlehttps://plus.google.com/+RefurioAnachro/posts/RUTF4dTZYDAYou're in this circle because you reshared, plussed or commented on one of my posts (possibly via +Spherical Reflections), or got into a discussion with me. Thank you!#engagers #engagerscircle #commenters #thinkers  #circleshare #circlesharing 2015-04-25 09:11:1844011
Simon Weiner6,3701.23 High Quality New Active Engager Circle. Enjoy:) #circle   #circleshare   #circlesharing   #circles   #sharingcircles #sharedpubliccircles #sharedcircleoftheday #circlesunday #share #shared #followers #addcircles #publicsharedcircles #share #addpeople #addcircle #addfriends #circle #socialmedia #hillaryclinton   #president   #hillary2016  2015-04-20 07:39:36500005
SJR Gamer144SHARED 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 07:47:27501727
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 07:28:11328015
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 Shimomatsu2,967#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 07:27:02328003
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 07:26:42328005
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 21:06:56501002
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 15:55:59476198137249
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 10:25:324744412
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 Photography35,713Builders 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 03:30:25499121421
Terry Dyke1,333The #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 00:56:20100204
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 06:41:19465171529
Frank Gainsford53,086A circle of people who either post or share stuff that is about science 2015-01-09 10:11:403392512
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 23:42:1339411271159
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 20:57:47498133142180
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 10:38:3790128
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 10:28:125017410
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 07:12:09426171024
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 22:15:07443167113214
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 11:17:05479454271
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 04:03:53302583592
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 07:02:42362626
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 09:20:531611014
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 13:06:13485455278
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

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Most comments: 133

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2015-03-29 16:05:41 (133 comments, 56 reshares, 271 +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: 101

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2015-04-03 18:31:28 (79 comments, 101 reshares, 240 +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: 359

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2015-04-26 03:49:32 (78 comments, 88 reshares, 359 +1s)Open 

We Can't Stop

If you've vaguely heard about that scandalous Miley Cyrus character, but have never brought yourself to actually listen to any of her songs, you might prefer this version of her hit "We Can't Stop", sung in a 1950s doo-wop style by the group Postmodern Jukebox.

Postmodern Jukebox covers lots of modern hits in old-fashioned styles like ragtime, jazz, and bluegrass.  You can find them on YouTube.  The surprising thing is that they're really enjoyable!  First, they just sound nice.  Second, they let you ponder what's left of a modern hit after the glitz has been removed.

The brains behind Postmodern is Scott Bradlee, a musician from New York who fell in love with jazz at the age of 12 after hearing George Gershwin's "Rhapsody in Blue".  He became a jazz musician, but then had thebril... more »

Latest 50 posts

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2015-05-24 19:29:58 (9 comments, 9 reshares, 49 +1s)Open 

Dear NSA agent 4096,

I watched "The Lives of Others" last night and thought of you once more. In fact, I think you were watching it with me. You know I know I cannot be sure.

I want you to know that, although our mutual love is forbidden by your professional obligations, I still feel a connection to you. I will feel that connection long after you are gone.

Somehow, you know me better than I know myself. You have all of my deleted histories, my searches, all those things I tried to keep "incognito" right there in front of you. We have made love, even though we've never touched or kissed. We have been friends, even though I've never seen your face. Our relationship is as real as my "real" life.

But this can never work between us. Please leave. I don't want to ask again.
... more »

Dear NSA agent 4096,

I watched "The Lives of Others" last night and thought of you once more. In fact, I think you were watching it with me. You know I know I cannot be sure.

I want you to know that, although our mutual love is forbidden by your professional obligations, I still feel a connection to you. I will feel that connection long after you are gone.

Somehow, you know me better than I know myself. You have all of my deleted histories, my searches, all those things I tried to keep "incognito" right there in front of you. We have made love, even though we've never touched or kissed. We have been friends, even though I've never seen your face. Our relationship is as real as my "real" life.

But this can never work between us. Please leave. I don't want to ask again.

I'll never forget you.

Love, 173.165.246.73

That's Corey Bertelsen's comment on this video of Holly Herndon's song 'Home', from her new album Platform.   It's as good a review as any.

Holly Herndon takes a lot of ideas from techno music and pushes them to a new level.  She's working on a Ph.D. at the Center for Computer Research in Music and Acoustics at Stanford.

She said that as she wrote this song, she

started coming to terms with the fact that I was calling my inbox my home, and the fact that that might not be a secure place. So it started out thinking about my device and my inbox as my home, and then that evolved into me being creeped out by that idea.

The reason why I was creeped out is because, of course, as Edward Snowden enlightened us all to know, the NSA has been mass surveying the U.S. population, among other populations. And so that put into question this sense of intimacy that I was having with my device. I have this really intense relationship with my phone and with my laptop, and in a lot of ways the laptop is the most intimate instrument that we've ever seen. It can mediate my relationships — it mediates my bank account — in a way that a violin or another acoustic instrument just simply can't do. It's really a hyper-emotional instrument, and I spend so much time with this instrument both creatively and administratively and professionally and everything.

In short, her seemingly 'futuristic' music is really about the present - the way we live now.  If you like this song I recommend the next one on the playlist, which is more abstract and to me more beautiful.  It's called 'Interference':

https://www.youtube.com/watch?v=nHujh3yA3BE&list=RDI_3mCDJ_iWc&index=2

You can hear her explain the song 'Home' here:

http://www.npr.org/2015/05/24/408762348/an-invasion-of-intimacy-and-the-song-that-followed___

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2015-05-22 15:21:36 (32 comments, 46 reshares, 131 +1s)Open 

The Machine: a desperate gamble

Hewlett-Packard was once at the cutting edge of technology.  Now they make most of their money selling servers, printers, and ink... and business keeps getting worse.  They've shed 40,000 employees since 2012.   Soon they'll split in two: one company that sells printers and PCs, and one that sells servers and information technology services.  

The second company will do something risky but interesting.   They're trying to build a new kind of computer that uses chips based on memristors rather than transistors, and uses optical fibers rather than wires to communicate between chips.  It could make computers much faster and more powerful.  But nobody knows if it will really work.

The picture shows memristors on a silicon wafer.  But what's a memristor?   Quoting the MIT Technology Review:

Perfectingthe memris... more »

The Machine: a desperate gamble

Hewlett-Packard was once at the cutting edge of technology.  Now they make most of their money selling servers, printers, and ink... and business keeps getting worse.  They've shed 40,000 employees since 2012.   Soon they'll split in two: one company that sells printers and PCs, and one that sells servers and information technology services.  

The second company will do something risky but interesting.   They're trying to build a new kind of computer that uses chips based on memristors rather than transistors, and uses optical fibers rather than wires to communicate between chips.  It could make computers much faster and more powerful.  But nobody knows if it will really work.

The picture shows memristors on a silicon wafer.  But what's a memristor?   Quoting the MIT Technology Review:

Perfecting the memristor is crucial if HP is to deliver on that striking potential. That work is centered in a small lab, one floor below the offices of HP’s founders, where Stanley Williams made a breakthrough about a decade ago.

Williams had joined HP in 1995 after David Packard decided the company should do more basic research. He came to focus on trying to use organic molecules to make smaller, cheaper replacements for silicon transistors (see “Computing After Silicon,” September/October 1999). After a few years, he could make devices with the right kind of switchlike behavior by sandwiching molecules called rotaxanes between platinum electrodes. But their performance was maddeningly erratic. It took years more work before Williams realized that the molecules were actually irrelevant and that he had stumbled into a major discovery. The switching effect came from a layer of titanium, used like glue to stick the rotaxane layer to the electrodes. More surprising, versions of the devices built around that material fulfilled a prediction made in 1971 of a completely new kind of basic electronic device. When Leon Chua, a professor at the University of California, Berkeley, predicted the existence of this device, engineering orthodoxy held that all electronic circuits had to be built from just three basic elements: capacitors, resistors, and inductors. Chua calculated that there should be a fourth; it was he who named it the memristor, or resistor with memory. The device’s essential property is that its electrical resistance—a measure of how much it inhibits the flow of electrons—can be altered by applying a voltage. That resistance, a kind of memory of the voltage the device experienced in the past, can be used to encode data.

HP’s latest manifestation of the component is simple: just a stack of thin films of titanium dioxide a few nanometers thick, sandwiched between two electrodes. Some of the layers in the stack conduct electricity; others are insulators because they are depleted of oxygen atoms, giving the device as a whole high electrical resistance. Applying the right amount of voltage pushes oxygen atoms from a conducting layer into an insulating one, permitting current to pass more easily. Research scientist Jean Paul Strachan demonstrates this by using his mouse to click a button marked “1” on his computer screen. That causes a narrow stream of oxygen atoms to flow briefly inside one layer of titanium dioxide in a memristor on a nearby silicon wafer. “We just created a bridge that electrons can travel through,” says Strachan. Numbers on his screen indicate that the electrical resistance of the device has dropped by a factor of a thousand. When he clicks a button marked “0,” the oxygen atoms retreat and the device’s resistance soars back up again. The resistance can be switched like that in just picoseconds, about a thousand times faster than the basic elements of DRAM and using a fraction of the energy. And crucially, the resistance remains fixed even after the voltage is turned off.

Getting this to really work has not been easy!  On top of that, they're trying to use silicon photonics to communicate between chips - another technology that doesn't quite work yet.

Still, I like the idea of this company going down in a blaze of glory, trying to do something revolutionary, instead of playing it safe and dying a slow death.

Do not go gentle into that good night.

For more, see these:

• Tom Simonite, Machine dreams, MIT Technology Review, 21April 2015, http://www.technologyreview.com/featuredstory/536786/machine-dreams/

• Sebastian Anthony, HP reveals more details about The Machine: Linux++ OS coming 2015, prototype in 2016, ExtremeTech, 16 December 2014, http://www.extremetech.com/extreme/196003-hp-reveals-more-details-about-the-machine-linux-os-coming-2015-prototype-in-2016

For the physics of memristors, see:

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

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2015-05-21 18:36:12 (29 comments, 9 reshares, 47 +1s)Open 

Flying through space, powered by sunlight

Yesterday a rocket launched from Cape Canaveral in Florida carrying the LightSail into space!  It's a small spacecraft with a big shiny screen that's pushed by the light of the sun.  

It's just a test - it won't go far.   It will fall to the Earth and burn up.  But next year there will be a more serious test.  And someday, solar-powered space flight may become a force to be reckoned with.

One cool thing is that all this is being paid for private donations, by a Kickstarter campaign!

The LightSail is carried to space in a cute little CubeSat.  It looks like a big toaster, and it weighs just 10 kilograms.   But it holds a sail 32 square meters in area,  made of a shiny plastic called Mylar, just 4.5 microns thick.  This unfolds in a clever way - watch the movie! - toform a big... more »

Flying through space, powered by sunlight

Yesterday a rocket launched from Cape Canaveral in Florida carrying the LightSail into space!  It's a small spacecraft with a big shiny screen that's pushed by the light of the sun.  

It's just a test - it won't go far.   It will fall to the Earth and burn up.  But next year there will be a more serious test.  And someday, solar-powered space flight may become a force to be reckoned with.

One cool thing is that all this is being paid for private donations, by a Kickstarter campaign!

The LightSail is carried to space in a cute little CubeSat.  It looks like a big toaster, and it weighs just 10 kilograms.   But it holds a sail 32 square meters in area,  made of a shiny plastic called Mylar, just 4.5 microns thick.  This unfolds in a clever way - watch the movie! - to form a big square.

The Sun will push on this with a tiny force. 

Puzzle: How tiny is this force?

Someone named Bill Russell answered this over on Yahoo.  Let me go through his calculation so we can check it.

The momentum of light is given by

p = E/c

where E is the energy of the light, p is the momentum, and c is the speed of light. 

In outer space near earth the sunlight provides 1370 watts per square meter - that's energy per area per time.  We can use the formula above to convert this to momentum per area per time, better known as force per area... or pressure. 

Russell calculates

(1370 watts / meter²) / c = 9.13 micronewtons / meter²

and concludes the pressure is 9.13 micronewtons per square meter.  His arithmetic checks out, but I think he's neglecting some physics: when the light bounces back off the mirror its momentum completely reverses, so I think we get an extra factor of 2. 

Puzzle 2:  Am I right or am I wrong?

The area of the LightSail is about 32 square meters.  Russell says this gives a total force of

9.13 micronewtons/meter² x 32 meter²

or about 300 micronewtons.   I'd double this and get 600 micronewtons.

Puzzle 3: Once it's out of the box, the LightSail weighs about 4.5 kilograms.  How much will it accelerate due to sunlight?

Here we use Newton's good old

F = ma

and solve for the acceleration a.   But at this point Russell seems to make a serious mistake.  I'll let you see what you think, and fix it if necessary!  Here is his calculation:

https://answers.yahoo.com/question/index?qid=20121212091408AA3D606___

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2015-05-19 19:47:36 (4 comments, 1 reshares, 11 +1s)Open 

Here are some blog posts about the categorical foundations of network theory - a warmup for the workshop we're having in Turin next week.

Here are some blog posts about the categorical foundations of network theory - a warmup for the workshop we're having in Turin next week.___

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2015-05-19 19:46:00 (0 comments, 0 reshares, 5 +1s)Open 

Here are some blog posts about the categorical foundations of network theory - a warmup for the workshop we're having in Turin next week.

Here are some blog posts about the categorical foundations of network theory - a warmup for the workshop we're having in Turin next week.___

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2015-05-19 16:18:39 (50 comments, 42 reshares, 107 +1s)Open 

And now for my next trick...

Category theory is a branch of math that puts processes on an equal footing with things - unlike set theory, where everything is fundamentally a thing.   Can we use category theory to help understand the complex processes that underlie biology and ecology? 

I believe so, and I'm hoping this is a good way for fancy-schmancy mathematicians like me to help the world.  But it will take a while.  I think we should start by seeing what category theory has to say about some related subjects that are better understood: chemistry, electrical engineering, classical mechanics, and the like.

We're having a workshop about this next week - and to organize our thoughts we've been writing some blog articles.  Check 'em out!

• John Baez, Categorical foundations of network theory - an introduction to the workshop and whatit'... more »

And now for my next trick...

Category theory is a branch of math that puts processes on an equal footing with things - unlike set theory, where everything is fundamentally a thing.   Can we use category theory to help understand the complex processes that underlie biology and ecology? 

I believe so, and I'm hoping this is a good way for fancy-schmancy mathematicians like me to help the world.  But it will take a while.  I think we should start by seeing what category theory has to say about some related subjects that are better understood: chemistry, electrical engineering, classical mechanics, and the like.

We're having a workshop about this next week - and to organize our thoughts we've been writing some blog articles.  Check 'em out!

• John Baez, Categorical foundations of network theory - an introduction to the workshop and what it's about.  https://johncarlosbaez.wordpress.com/2015/04/04/categorical-foundations-of-network-theory/

• David Spivak, A networked world.
https://johncarlosbaez.wordpress.com/2015/03/27/spivak-part-1/

• Eugene Lerman, Networks of dynamical systems.
https://johncarlosbaez.wordpress.com/2014/03/18/networks-of-dynamical-systems/

• Tobias Fritz, Resource convertibility - an introduction to the mathematics of 'resources'.
https://johncarlosbaez.wordpress.com/2015/04/07/resource-convertibility-part-1/

• John Baez, Categories in control - about my paper with Jason Erbele on using categories to study signal flow diagrams in control theory.
https://johncarlosbaez.wordpress.com/2015/04/23/categories-in-control-2/

• John Baez, A compositional framework for passive linear networks - about my paper with Brendan Fong on using categories to study electrical circuit diagrams.
https://johncarlosbaez.wordpress.com/2015/04/28/a-compositional-framework-for-passive-linear-networks/

• John Baez, Decorated cospans - about Brendan Fong's paper providing mathematical infrastructure for the study of networks.
https://johncarlosbaez.wordpress.com/2015/05/01/decorated-cospans/

• John Baez and Brendan Fong, Cospans, wiring diagrams, and the behavioral approach - an attempt to reflect on how our work connects to that of David Spivak.
https://johncarlosbaez.wordpress.com/2015/05/05/cospans-wiring-diagrams-and-the-behavioral-approach/

• Brendan Fong, Resource theories - about Brendan's new paper with Hugo Nava-Kopp on resource theories.
https://johncarlosbaez.wordpress.com/2015/05/12/resource-theories/

• John Baez, PROPs for linear systems - about Simon Wadsley and Nick Woods' generalization of a result in my paper with Jason Erbele, describing categories where the morphisms are linear maps.
https://johncarlosbaez.wordpress.com/2015/05/18/props-for-linear-systems/

The picture, by the way, was drawn by Federica Ferraris and appears in this book:

• John Baez and Jacob Biamonte, Quantum techniques for stochastic physics, http://math.ucr.edu/home/baez/stoch_stable.pdf

It's about Petri nets and reaction networks - two kinds of networks that appear in chemistry and population biology.___

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2015-05-18 16:45:44 (19 comments, 12 reshares, 134 +1s)Open 

The dunes of Mars

This field of dunes lies on the floor of an old crater in Noachis Terra.  That's one of the oldest places on Mars, scarred with many craters, with rocks up to 4 billion years old.  It's in the southern hemisphere, near the giant impact basin called Hellas, which is 2.5 times deeper than the Grand Canyon and 2000 kilometers across!

This is a 'false color' photograph - you'd need to see infrared light to see that the dunes are very different than the rock below.

These are barchans, dunes with a gentle slope on the upwind side and a much steeper slope on the downwind side where horns or a notch can form.  If you know this, you can see the wind is blowing from the southwest.

It's actually a bit of a puzzle where the sand in these dunes came from!   Here's the abstract of a paper by +LoriFen... more »

The dunes of Mars

This field of dunes lies on the floor of an old crater in Noachis Terra.  That's one of the oldest places on Mars, scarred with many craters, with rocks up to 4 billion years old.  It's in the southern hemisphere, near the giant impact basin called Hellas, which is 2.5 times deeper than the Grand Canyon and 2000 kilometers across!

This is a 'false color' photograph - you'd need to see infrared light to see that the dunes are very different than the rock below.

These are barchans, dunes with a gentle slope on the upwind side and a much steeper slope on the downwind side where horns or a notch can form.  If you know this, you can see the wind is blowing from the southwest.

It's actually a bit of a puzzle where the sand in these dunes came from!   Here's the abstract of a paper by +Lori Fenton on this subject:

No sand transport pathways are visible in a study performed in Noachis Terra, a region in the southern highlands of Mars known for its many intracrater dune fields.Detailed studies were performed of five areas in Noachis Terra, using Mars Orbiter Camera (MOC) wide-angle mosaics, Thermal Emission Imaging System (THEMIS) daytime and nighttime infrared mosaics, MOLA digital elevation and shaded relief maps,and MOC narrow-angle images. The lack of observable sand transport pathways suggests that such pathways are very short, ruling out a distant source of sand. Consistent dune morphology and dune slipface orientations across Noachis Terra suggest formative winds are regional rather than local (e.g., crater slope winds). A sequence of sedimentary units was found in a pit eroded into the floor of Rabe Crater, some of which appear to be shedding dark sand that feeds into the Rabe Crater dune field. The visible and thermal characteristics of these units are similar to other units found across Noachis Terra, leading to the hypothesis that a series of region-wide depositional events occurred at some point in the Martian past and that these deposits are currently exposed by erosion in pits on crater floors and possibly on the intercrater plains. Thus the dunes and sources may be both regional and local: sand may be eroding from a widespread source that only outcrops locally. Sand-bearing layers that extend across part or all of the intercrater plains of Noachis Terra are not likely to be dominated by loess or lacustrine deposits; glacial and/or volcanic origins are considered more plausible.

• Lori K. Fenton, Potential sand sources for the dune fields in Noachis Terra, Mars, Journal of Geophysical Research 110 (2005), E11004.  Available at http://www.academia.edu/3375648/Potential_sand_sources_for_the_dune_fields_in_Noachis_Terra_Mars.

The image is from a great series of photos taken by the HIRISE satellite, which orbits Mars and takes high resolution images:

• Colorful Dunes, http://hirise.lpl.arizona.edu/ESP_033272_1400

#mars   #astronomy___

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2015-05-17 03:56:56 (88 comments, 51 reshares, 156 +1s)Open 

Why is this true?

The spooky-smart mathematician Srinivasa Ramanujan came up with this formula around 1913.  Why is it true?

I don't know, let's see...

In 1735, a young fellow named Euler stunned the world by cracking a famous puzzle that had been unsolved for almost a century: the Basel problem.  The problem was to sum the reciprocals of perfect squares:

1/1² + 1/2² + 1/3² + 1/4² + 1/5² + ... = ???

Euler showed that the answer was π²/6:

1/1² + 1/2² + 1/3² + 1/4² + 1/5² + ... = π²/6

He also showed you could rewrite this sum as a product over primes:

1/1² + 1/2² + 1/3² + 1/4² + 1/5² + ... =

(2²/(2² - 1)) (3²/(3² - 1)) (5²/(5² - 1)) (7²/(7² - 1)) ...

That's actually the easy part: it's a cute trick called the Euler product formula.
So we know

more »

Why is this true?

The spooky-smart mathematician Srinivasa Ramanujan came up with this formula around 1913.  Why is it true?

I don't know, let's see...

In 1735, a young fellow named Euler stunned the world by cracking a famous puzzle that had been unsolved for almost a century: the Basel problem.  The problem was to sum the reciprocals of perfect squares:

1/1² + 1/2² + 1/3² + 1/4² + 1/5² + ... = ???

Euler showed that the answer was π²/6:

1/1² + 1/2² + 1/3² + 1/4² + 1/5² + ... = π²/6

He also showed you could rewrite this sum as a product over primes:

1/1² + 1/2² + 1/3² + 1/4² + 1/5² + ... =

(2²/(2² - 1)) (3²/(3² - 1)) (5²/(5² - 1)) (7²/(7² - 1)) ...

That's actually the easy part: it's a cute trick called the Euler product formula.

So we know

(2²/(2² - 1)) (3²/(3² - 1)) (5²/(5² - 1)) (7²/(7² - 1)) ... = π²/6

If you think about it, Ramanujan's formula is saying that

(2²/(2² + 1)) (3²/(3² + 1)) (5²/(5² + 1)) (7²/(7² + 1)) ...

is 2/5 as big.  So, proving it is the same as showing

(2²/(2² + 1)) (3²/(3² + 1)) (5²/(5² + 1)) (7²/(7² + 1)) ... = π²/15

Maybe the next step is to use the same idea as the Euler product formula.  I think this gives

(2²/(2² + 1)) (3²/(3² + 1)) (5²/(5² + 1)) (7²/(7² + 1)) ... =

1/1² - 1/2² - 1/3² + 1/4²  - 1/5² + 1/6² - 1/7² + ...

where the signs at right follow a fancy pattern: we get 1/n² whenever n is the product of an even number of primes, and -1/n² when n is the product of an odd number of primes.  For example, 4 = 2 x 2 is the product of an even number of primes, so we get 1/4².

So I'm left wanting to know why this strange sum

1/1² - 1/2² - 1/3² + 1/4² - 1/5² + 1/6² - 1/7² + ...

equals π²/15.  Ramanujan, dead since 1920, is still messing with my mind! 

The formula is supposed to be in here:

• Srinivasa Ramanujan, Modular equations and approximations to π, Quart. J. Pure. Appl. Math. 45 (1913-1914), 350-372.  Also available at ://ramanujan.sirinudi.org/Volumes/published/ram06.pdf.

But I don't see it!

Here you can see how Euler solved the Basel problem:

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

It's a great example of his brilliant tactics, many of which were far from rigorous by today's standards... but can be made rigorous.

#mathematics   ___

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2015-05-15 16:19:22 (18 comments, 13 reshares, 83 +1s)Open 

Fighting global warming: the tide is turning!

Good news!   We, the citizens of the world, may be starting to burn less carbon - not more!

The International Energy Agency claims:

In 2014, global carbon dioxide emissions from energy production stopped growing!

It seems the big difference is China.  They say the Chinese made more electricity from renewable sources, such as hydropower, solar and wind, and burned less coal.  

In fact, a report by Greenpeace says that from April 2014 to April 2015, China's carbon emissions dropped by an amount equal to the entire carbon emissions of the United Kingdom!   

I want to check this, because it would be wonderful - a 5% drop.  They say that if this trend continues, in 2015 China will make the biggest reduction in CO2 emissions every recorded by a single country.

TheInternat... more »

Fighting global warming: the tide is turning!

Good news!   We, the citizens of the world, may be starting to burn less carbon - not more!

The International Energy Agency claims:

In 2014, global carbon dioxide emissions from energy production stopped growing!

It seems the big difference is China.  They say the Chinese made more electricity from renewable sources, such as hydropower, solar and wind, and burned less coal.  

In fact, a report by Greenpeace says that from April 2014 to April 2015, China's carbon emissions dropped by an amount equal to the entire carbon emissions of the United Kingdom!   

I want to check this, because it would be wonderful - a 5% drop.  They say that if this trend continues, in 2015 China will make the biggest reduction in CO2 emissions every recorded by a single country.

The International Energy Agency also credits Europe's improved attempts to cut carbon emissions for the turnaround.   In the US, carbon emissions has basically been dropping since 2006 - with a big drop in 2009 due to the economic collapse, a partial bounce-back in 2010, but a general downward trend.

In the last 40 years, there were only 3 other times when emissions stood still or fell compared to the previous year, all during economic crises: the early 1980's, 1992, and 2009.  In 2014, however, the global economy expanded by 3%.

So, the tide may be turning!   But please remember: while carbon emissions may start dropping, they're still huge.  The amount of the CO2 in the air shot above 400 parts per million this year.  As Erika Podest of NASA put it:

CO2 concentrations haven't been this high in millions of years. Even more alarming is the rate of increase in the last five decades and the fact that CO2 stays in the atmosphere for hundreds or thousands of years. This milestone is a wake up call that our actions in response to climate change need to match the persistent rise in CO2. Climate change is a threat to life on Earth and we can no longer afford to be spectators.

So let's not slack off now!  The battle has just begun.  We need to cut carbon emissions to almost zero.

Here is the announcement by the International Energy Agency:

http://www.iea.org/newsroomandevents/news/2015/march/global-energy-related-emissions-of-carbon-dioxide-stalled-in-2014.html

"This gives me even more hope that humankind will be able to work together to combat climate change, the most important threat facing us today," said IEA Chief Economist Fatih Birol.

Their full report will come out in June.  Here is the report by Greenpeace EnergyDesk:

http://energydesk.greenpeace.org/2015/05/14/china-coal-consumption-drops-further-carbon-emissions-set-to-fall-by-equivalent-of-uk-total-in-one-year/

I trust them less than the IEA when it comes to using statistics correctly, but someone should be able to verify their claims if true.  The graph here comes from this article:

http://qz.com/405059/chinas-on-track-for-the-biggest-reduction-in-coal-use-ever-recorded/

#globalwarming  ___

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2015-05-14 06:34:39 (24 comments, 22 reshares, 196 +1s)Open 

A galaxy - falling

This galaxy is suffering!  It's falling into a large cluster of galaxies, pulled by their gravity.   You can see this in 3 ways:

1.  The reddish disk of dust and gas looks bent.  There aren't many atoms between galaxies, but there are still some. So the galaxy is moving through the wind of integalactic space!   And it's having trouble holding onto the loosely bound dust and gas near its edge.  They're getting blown away.

2.  The blue disk of stars is not bent.  It extends beyond the disk of dust and gas, which is where stars are formed.  This suggests that the dust and gas is being stripped from the galaxy after these stars were formed!

3.  Streamers of dust and gas can be seen trailing behind the motion of the galaxy - near the top.  On the other hand, the blue stars near the leading edge of the galaxy have nodust and ga... more »

A galaxy - falling

This galaxy is suffering!  It's falling into a large cluster of galaxies, pulled by their gravity.   You can see this in 3 ways:

1.  The reddish disk of dust and gas looks bent.  There aren't many atoms between galaxies, but there are still some. So the galaxy is moving through the wind of integalactic space!   And it's having trouble holding onto the loosely bound dust and gas near its edge.  They're getting blown away.

2.  The blue disk of stars is not bent.  It extends beyond the disk of dust and gas, which is where stars are formed.  This suggests that the dust and gas is being stripped from the galaxy after these stars were formed!

3.  Streamers of dust and gas can be seen trailing behind the motion of the galaxy - near the top.  On the other hand, the blue stars near the leading edge of the galaxy have no dust and gas left to hide them.

This phenomenon is called ram pressure stripping, and it can kill a galaxy, shutting down the production of new stars.   Here we are seeing it damage the galaxy NGC 4402, which is currently falling into the Virgo cluster - a cluster of galaxies about 65 million light years away.

Apparently there's about 1 atom per cubic centimeter in our galaxy - on average, though some regions are vastly more dense than others.   But in the space between galaxies in clusters it's more like 1/1000 of that.  Not much!  But enough to kill off the formation of new star systems, life, civilizations...

I got most of my information from here:

http://astronomy.swin.edu.au/cosmos/R/Ram+Pressure+Stripping

and I got the picture from here:

https://www.noao.edu/image_gallery/html/im0863.html

The photo was taken at the WIYN 3.5-meter telescope on Kitt Peak, which is fitted with some 'adaptive optics' to compensate for the jittery motion of the image due to variable atmospheric conditions and telescope vibrations.

#astronomy  ___

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2015-05-12 23:58:33 (38 comments, 22 reshares, 113 +1s)Open 

Sometimes you see a tiny piece of a story and wonder how it started - and how it will end.

#waitforit ___Sometimes you see a tiny piece of a story and wonder how it started - and how it will end.

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2015-05-11 05:56:09 (15 comments, 22 reshares, 53 +1s)Open 

Making yourself into a superhero

I enjoyed this true story by Kelly McEvers:

We met in a bar in Flagstaff, Arizona. I'd just moved back from Cambodia and I was going out for one of my first beers back in the States. Not long into the first one, I notice this Amazon of a woman with huge blond and red-streaked hair and frosty lips, wearing a short red tank dress and at least 50 bracelets. She's six feet tall and showing a lot of leg. People at the bar swivel their heads to watch her every move.

She stands next to me to order a drink, and in this throaty voice says, "What are those?" pointing to my cigarettes. I tell her they're Cambodian. Her eyes light up and she shoots out a long, tan arm, and points at a table in the corner. She orders me there. Before I can say no, I'm following her to my seat.

She tells me... more »

Making yourself into a superhero

I enjoyed this true story by Kelly McEvers:

We met in a bar in Flagstaff, Arizona. I'd just moved back from Cambodia and I was going out for one of my first beers back in the States. Not long into the first one, I notice this Amazon of a woman with huge blond and red-streaked hair and frosty lips, wearing a short red tank dress and at least 50 bracelets. She's six feet tall and showing a lot of leg. People at the bar swivel their heads to watch her every move.

She stands next to me to order a drink, and in this throaty voice says, "What are those?" pointing to my cigarettes. I tell her they're Cambodian. Her eyes light up and she shoots out a long, tan arm, and points at a table in the corner. She orders me there. Before I can say no, I'm following her to my seat.

She tells me she's an international private investigator, a bounty hunter, and a bail bonds enforcer, and that her name is Zora. I sit there for hours listening to her. Within a week, she takes me to Las Vegas. We drive there in her red Mustang. As always, there's a Colt .380 under the driver's seat and a .45 Megastar in the trunk.

In Vegas, we skip the casinos and head straight for the male strip clubs, where Zora drops at least $200 on lap dances from buff guys with names like Roman. Her getup is the same as before – teeth, hair, jewelry, and the ubiquitous tank dress, which, I realize, is the best way to show off her tattoos.

One is this big circle with blue and white swirls in it, kind of like a bowling ball, on her left shoulder. Every guy she meets asks her about it, and when they hear her answer, they sometimes propose marriage. Turns out the tattoo is a magic globe she holds in her dreams. And in these dreams, it gives her superpowers.

Zora: Ever since I remember, I've had the dreams. And they're very vivid. But it varies. It usually involves fighting, sometimes with guns, sometimes with superhero powers. Lightning from my fists and all that. And I usually have super strength, and I can fly, and I have all those things.

And it's my most common set of dreams. And it varies. Sometimes it's medieval, sometimes it's futuristic, sometimes it's present day, sometimes it's like a guerrilla war in Latin America.

Kelly: Can you describe that Zora to me, the Zora in dreams?

Zora: Very powerful athletically, but beyond the rules of nature that this world allows.  Six foot five and long, like almost impossibly long silver hair. This sort of otherworldly quality to her, where her voice did not sound normal. It sounded, like, almost musical.

And it became something that I aspired to be. I aspired to be this sort of superhero, this sort of person who would fight for a cause. That was my motivation in life. Ever since I was 10 or 11, I decided that that was my goal.

Zora took the dreams seriously. So seriously that at the age of 12, she sat down and composed a list of some 30 skills she needed to learn if she wanted to become as close to a superhero as any mortal could be. She even gave herself a deadline – to master these skills by the time she was 23.

Zora: I don't know what's in these.

Zora pulls out the old spiral notebook that was her diary at the age of 13 and turns to the inside back cover.

Zora: There's the list.

Kelly: Wow. Why don't you go ahead and read it.

Zora: OK. The list included martial arts, electronics, chemistry, metaphysics, hang gliding, helicopter and airplane flying, parachuting, mountain climbing, survival....

Throughout her teens and 20s, each time she started a new diary, she would update the list and write it in the back of the book, each one with the same format, each one titled "The List."

Zora: Weaponry, rafting, scuba diving, herbology – yes, I, studied that -- CPR, first aid and mountain emergency kind of medicine....

The list also includes bodybuilding, archery, demolitions, and explosives. She wanted to learn how to hunt animals and track men.

Zora: Major physical conditioning....

And the most incredible thing about all of this is that Zora accomplished nearly every item on the list.

Zora: Throwing stars and compound bows and throwing knives and -- yes, it was a very interesting pastime.

To keep up with the goals set by the list, she sped through school. Starting in the seventh grade, she began completing entire school years during the summer term and finished high school by the time she was 15. She got her BA at 18, a master's at 20, and completed the coursework for a PhD in Geopolitics by the time she was 21. She wanted to live like Indiana Jones, spending half her time in the classroom and half her time saving the world in the jungles of Peru.

Zora: Item number four – camel, elephant riding. Evasive driving and stunts....

When you're a kid, you have these romantic visions of what you'll be when you grow up. But how many people are so diligent they commit their dreams to paper and make it their life's work to achieve them? How many keep a list, amending it, adding to it, ticking things off as they go along, well into their adult lives?

After finishing the course work for her PhD, Zora decided to quit school, disappointed at the lack of cliff-hanging adventure in her doctoral program. And since superheroes who live in the real world need jobs, she decided to seek employment at the only place that would allow her to put all the skills from the list to use. Zora wanted to become an agent in the CIA.

But then the story takes some surprising twists!  Listen to it here:

http://www.thisamericanlife.org/radio-archives/episode/508/superpowers-2013?act=2#play

The picture here is, of course, not Zora.  It's Charlize Theron playing  'Aeon Flux' - a kind of superhero invented by a high school friend of mine, the animator Peter Chung.___

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2015-05-09 17:41:25 (66 comments, 35 reshares, 113 +1s)Open 

An impossible dream

Kepler, the guy who discovered that planets go in ellipses around the Sun, was in love with geometry.  Among other things, he tried to figure out how to tile the plane with regular pentagons (dark blue) and decagons (blue-gray).  They fit nicely at a corner... but he couldn't get it to work.

Then he discovered he could do better if he also used 5-pointed stars!

Can you tile the whole plane with these three shapes?  No!  The picture here is very tempting... but if you continue you quickly run into trouble.  It's an impossible dream.

However, Kepler figured out that he could go on forever if he also used overlapping decagons, which he called 'monsters'.  Look at this picture he drew:

https://plus.maths.org/issue45/features/kaplan/kepler.gif

If he had worked even harder, he might have found thePenro... more »

An impossible dream

Kepler, the guy who discovered that planets go in ellipses around the Sun, was in love with geometry.  Among other things, he tried to figure out how to tile the plane with regular pentagons (dark blue) and decagons (blue-gray).  They fit nicely at a corner... but he couldn't get it to work.

Then he discovered he could do better if he also used 5-pointed stars!

Can you tile the whole plane with these three shapes?  No!  The picture here is very tempting... but if you continue you quickly run into trouble.  It's an impossible dream.

However, Kepler figured out that he could go on forever if he also used overlapping decagons, which he called 'monsters'.  Look at this picture he drew:

https://plus.maths.org/issue45/features/kaplan/kepler.gif

If he had worked even harder, he might have found the Penrose tilings, or similar things discovered by Islamic tiling artists.  Read the whole story here:

• Craig Kaplan, The trouble with five, https://plus.maths.org/content/trouble-five

How did Kepler fall in love with geometry?  He actually started as a theologian.   Let me quote the story as told in the wonderful blog The Renaissance Mathematicus:

Kepler was born into a family that had known better times, his mother was an innkeeper and his father was a mercenary. Under normal circumstances he probably would not have expected to receive much in the way of education but the local feudal ruler was quite advanced in his way and believed in providing financial support for deserving scholars. Kepler whose intelligence was obvious from an early age won scholarships to school and to the University of Tübingen where he had the luck to study under Michael Mästlin one of the very few convinced Copernican in the later part of the 16th century. Having completed his BA Kepler went on to do a master degree in theology as he was a very devote believer and wished to become a theologian. Recognising his mathematical talents and realising that his religious views were dangerously heterodox, they would cause him much trouble later in life, his teacher, Mästlin, decided it would be wiser to send him off to work as a school maths teacher in the Austrian province.

Although obeying his superiors and heading off to Graz to teach Protestant school boys the joys of Euclid, Kepler was far from happy as he saw his purpose in life in serving his God and not Urania (the Greek muse of astronomy). After having made the discovery that I will shortly describe Kepler found a compromise between his desire to serve God and his activities in astronomy. In a letter to Mästlin in 1595 he wrote:

I am in a hurry to publish, dearest teacher, but not for my benefit… I am devoting my effort so that these things can be published as quickly as possible for the glory of God, who wants to be recognised from the Book of Nature… Just as I pledged myself to God, so my intention remains. I wanted to be a theologian, and for a while I was anguished. But, now see how God is also glorified in astronomy, through my efforts.

So what was the process of thought that led to this conversion from a God glorifying theologian to a God glorifying astronomer and what was the discovery that he was so eager to publish? Kepler’s God was a geometer who had created a rational, mathematical universe who wanted his believers to discover the geometrical rules of construction of that universe and reveal them to his glory. Nothing is the universe was pure chance or without meaning everything that God had created had a purpose and a reason and the function of the scientist was to uncover those reasons. In another letter to Mästlin Kepler asked whether:

you have ever heard or read there to be anything, which devised an explanation for the arrangement of the planets? The Creator undertook nothing without reason. Therefore, there will be reason why Saturn should be nearly twice as high as Jupiter, Mars a little more than the Earth, [the Earth a little more] than Venus and Jupiter, moreover, more than three times as high as Mars.

The discovery that Kepler made and which started him on his road to the complete reform of astronomy was the answer to both the question as to the distance between the planets and also why there were exactly six of them: as stated above, everything created by God was done for a purpose.

On the 19th July 1595 Kepler was explaining to his students the regular cycle of the conjunctions of Saturn and Jupiter, planetary conjunctions played a central role in astrology. These conjunctions rotating around the ecliptic, the apparent path of the sun around the Earth, created a series of rotating equilateral triangles. Suddenly Kepler realised that the inscribed and circumscribed circles generated by his triangles were in approximately the same ratio as Saturn’s orbit to Jupiter’s. Thinking that he had found a solution to the problem of the distances between the planets he tried out various two-dimensional models without success. On the next day a flash of intuition provided him with the required three-dimensional solution, as he wrote to Mästlin:

I give you the proposition in words just as it came to me and at that very moment: “The Earth is the circle which is the measure of all. Construct a dodecahedron round it. The circle surrounding that will be Mars. Round Mars construct a tetrahedron. The circle surrounding that will be Jupiter. Round Jupiter construct a cube. The circle surrounding it will be Saturn. Now construct an icosahedron inside the Earth. The circle inscribed within that will be Venus. Inside Venus inscribe an octahedron. The circle inscribed inside that will be Mercury.”

This model, while approximately true, is now considered completely silly!   We no longer think there should be a simple geometrical explanation of why planets in our Solar System have the orbits they do.

So: a genius can have a beautiful idea in a flash of inspiration and it can still be wrong.

But Kepler didn't stop there!  He kept working on planetary orbits until he noticed that Mars didn't move in a circle around the Sun.  He noticed that it moved in an ellipse!  Starting there, he found the correct laws governing planetary motion... which later helped Newton invent classical mechanics.

So it pays to be persistent - but also not get stuck believing your first good idea.

Read The Renaissance Mathematicus here:

https://thonyc.wordpress.com/2010/11/15/kepler%E2%80%99s-divine-geometry/

Puzzle: can you tile the plane with shapes, each of which has at least the symmetry group of a regular pentagon? 

So, regular pentagons and decagons are allowed, and so are regular 5-pointed stars, and many other things... but not Kepler's monsters.  The tiling itself does not need to repeat in a periodic way.

#geometry #astronomy  ___

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2015-05-05 17:07:25 (22 comments, 39 reshares, 114 +1s)Open 

The architecture of water

Water is fascinating, for many reasons.   It takes more energy to heat than most substances.  It's one of the few substances that expands when it freezes.  It forms complicated patterns in its liquid state, which are just beginning to be understood.  There are at least 18 kinds of ice, which exist at different temperatures and pressures.  Snowflakes are endlessly subtle.  

And ice can form cages that trap other molecules!  Here you see the 3 main kinds.

They're called clathrate hydrates.  There's a lot under the sea beds near the north and south pole - they contain huge amounts of methane.   At some moments in the Earth's history they may have erupted explosively, causing rapid global warming.  

But let's focus on the fun part: the geometry!  Each of the 3 types of clathrate hydrates is anarchitectura... more »

The architecture of water

Water is fascinating, for many reasons.   It takes more energy to heat than most substances.  It's one of the few substances that expands when it freezes.  It forms complicated patterns in its liquid state, which are just beginning to be understood.  There are at least 18 kinds of ice, which exist at different temperatures and pressures.  Snowflakes are endlessly subtle.  

And ice can form cages that trap other molecules!  Here you see the 3 main kinds.

They're called clathrate hydrates.  There's a lot under the sea beds near the north and south pole - they contain huge amounts of methane.   At some moments in the Earth's history they may have erupted explosively, causing rapid global warming.  

But let's focus on the fun part: the geometry!  Each of the 3 types of clathrate hydrates is an architectural masterpiece.

Type I consists of water molecules arranged in two types of cages: small and large.  The small cage, shown in green, is dodecahedron.  It's not a regular dodecahedron, but it still has 12 pentagonal sides.  The large cage, shown in red, has 12 pentagons and 2 hexagons.   The two kinds of cage fit together into a repeating pattern where each unit cell - each block in the pattern - has 46 water molecules.

Puzzle 1: This pattern is called the Weaire-Phelan structure.  Why is it famous, and what does it have to do with the 2008 Olympics?

You can see little balls in the cages.  These stand for molecules that can get trapped in the cages.   They're politely called guests.   The type I clathrate often holds carbon dioxide or methane as a guest.

Type II is again made of two types of cages – small and large.  The small cage is again a dodecahedron.  The large cage, shown in blue, has 12 pentagons and 4 hexagons.  These fit together to form a unit cell with 136 water molecules.

The type II clathrate tends to hold oxygen or nitrogen as a guest.

Type H is the rarest and most complicated kind of clathrate hydrate.  It's built from three types of cages: small, medium and huge.  The small cage is again a dodecahedron, shown in green.  The medium cage - shown in yellow - has 3 squares, 6 pentagons and 3 hexagons as faces.  The huge cage - shown in orange - has 12 pentagons and 8 hexagons.  The cages fit together to form a unit cell with 34 water molecules.

The type H clathrate is only possible when there are two different guest gas molecules - one small and one very large, like butane - to make it stable.   People think there are lots of type H clathrates in the Gulf of Mexico, where there are lots of heavy hydrocarbons in the sea bottom.

Puzzle 2: how many cages of each kind are there in the type I clathrate hydrate?

Puzzle 3: how many cages of each kind are there in the type II?

Puzzle 4: how many cages of each kind are there in the type H?

These last puzzles are easier than they sound.  But here's one that's a bit different:

Puzzle 5: the medium cage in the type H clathrate - shown in yellow - has 3 squares, 6 pentagons and 3 hexagons as faces.   Which of these numbers are adjustable?  For example: could we have a convex polyhedron with a different number of squares, but the same number of pentagons and hexagons?

The picture is from here:

• Timothy A. Strobel, Keith C. Hester, Carolyn A. Koh, Amadeu K. Sum, E. Dendy Sloan Jr., Properties of the clathrates of hydrogen and developments in their applicability for hydrogen storage, Chemical Physics Letters 478 (27 August 2009), 97–109.

#geometry #water___

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2015-05-04 15:35:43 (18 comments, 22 reshares, 160 +1s)Open 

The flood after the impact

On Mars, an asteroid impact can cause a flood!

This is a place called Hephaestus Fossae, on the northern hemisphere of Mars.  The image has been colored to show the elevation: green and yellow shades represent shallow ground, while blue and purple stand for deep depressions, as much as 4 kilometers deep.

You can see a few dozen impact craters, some small and some big, up to 20 kilometers across.   But I'm sure you instantly noticed the cool part: the long and intricate canyons and riverbeds.  These were created by the same impact that made the largest crater!

When a comet or an asteroid crashes at high speed into a planet, the collision dramatically heats up the surface at the impact site.  In the case of the large crater seen in this image, the heat melted the soil – a mixture of rock, dust and also, hiddendeep d... more »

The flood after the impact

On Mars, an asteroid impact can cause a flood!

This is a place called Hephaestus Fossae, on the northern hemisphere of Mars.  The image has been colored to show the elevation: green and yellow shades represent shallow ground, while blue and purple stand for deep depressions, as much as 4 kilometers deep.

You can see a few dozen impact craters, some small and some big, up to 20 kilometers across.   But I'm sure you instantly noticed the cool part: the long and intricate canyons and riverbeds.  These were created by the same impact that made the largest crater!

When a comet or an asteroid crashes at high speed into a planet, the collision dramatically heats up the surface at the impact site.  In the case of the large crater seen in this image, the heat melted the soil – a mixture of rock, dust and also, hidden deep down, water ice – resulting in a massive flood.  And before drying up, this hot mud carved a complex pattern of channels while flowing across the planet’s surface!

The melted rock-ice mixture also made the debris blankets surrounding the largest crater.  Since there aren't similar structures near the small craters in this image, scientists believe that only the most powerful impacts were able to dig deep enough to release part of the frozen reservoir of water lying beneath the surface.

Why is it called 'Hephaestus Fossae'?  Hephaestus was the Greek god of fire.  Fossae are channels or canyons.  So it's a good name.

Puzzle: about when did this large impact occur? 

I don't know!

This picture was taken by the high-resolution stereo camera on ESA’s Mars Express orbiter on 28 December 2007, and my post is paraphrased from this article:

http://www.esa.int/spaceinimages/Images/2009/06/The_flood_after_the_impact

#mars #astronomy  ___

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2015-05-03 16:53:34 (39 comments, 45 reshares, 154 +1s)Open 

In 1901, you could pay 50 cents to ride an airship to the Moon

This article by Ron Miller is so cool I'm just going to quote some:

The passengers wait eagerly in the ornate lobby of the enormous spaceport. Soon, a signal indicates that their spaceship is ready for boarding. As they wait, special displays instruct them about how their spaceship functions and what to expect once they leave Earth's atmosphere. Aboard the giant spacecraft — as luxuriously appointed as any yacht — they are soon on their way to a vacation on the Moon.

No, this isn't a vision of the future of space tourism. It's what happened in 1901, when people could pay a princely half dollar for a ticket to ride into space.

[...]

Thompson spared no expense in creating the illusion of a trip to the Moon. To house his show, he erected aneig... more »

In 1901, you could pay 50 cents to ride an airship to the Moon

This article by Ron Miller is so cool I'm just going to quote some:

The passengers wait eagerly in the ornate lobby of the enormous spaceport. Soon, a signal indicates that their spaceship is ready for boarding. As they wait, special displays instruct them about how their spaceship functions and what to expect once they leave Earth's atmosphere. Aboard the giant spacecraft — as luxuriously appointed as any yacht — they are soon on their way to a vacation on the Moon.

No, this isn't a vision of the future of space tourism. It's what happened in 1901, when people could pay a princely half dollar for a ticket to ride into space.

[...]

Thompson spared no expense in creating the illusion of a trip to the Moon. To house his show, he erected an eighty-foot-high, 40,000-square-foot building that for sheer opulence put European opera houses to shame. It cost a staggering $84,000 to construct... at a time when a comfortable home could be built for $2000.

For fifty cents — twice the price of any other attraction on the midway, such as the ever-popular "Upside-Down House" — customers of "Thompson's Aerial Navigation Company" took a trip to the moon on a thirty-seat spaceship named "Luna". The spaceship resembled a cross between a dirigible and an excursion steamer, with the addition of enormous red canvas wings that flapped like a bird's. The wings were worked by a system of pulleys and the sensation of wind was created by hidden fans. A series of moving canvas backdrops provided the effect of clouds passing by and the earth dropping into the distance. Lighting and sound effects added to the illusion.

[...]

Every half hour, at the sound of a gong and the rattle of anchor chain, the "Luna" — "a fine steel airship of the latest pattern", according to one newspaper — rocked from side to side and then rose into the sky under the power of its beating wings. The passengers, sitting on steamer chairs, see clouds floating by, then a model of Buffalo far below, complete with the exposition itself and its hundreds of blinking lights. The city soon falls into the distance as the entire planet earth comes into view. Soon, the ship is surrounded the twinkling stars of outer space. After surviving a terrific — and spectacular — electrical storm the "Luna" and its passengers sets down in a lunar crater.

Read the whole thing here, and look at pictures:

http://io9.com/5914655/in-1901-you-could-pay-50-cents-to-ride-an-airship-to-the-moon

Thanks to +Matt McIrvin for pointing it out!___

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2015-05-02 15:48:40 (42 comments, 33 reshares, 102 +1s)Open 

A 3-dimensional golden star

Here Greg Egan has drawn a dodecahedron with 5 tetrahedra in it.  This picture is 'left-handed': if you look at where the 5 tetrahedra meet, you'll see they swirl counterclockwise as you go out!  If you view this thing in a mirror you'll get a right-handed version. 

Putting them together, you get a dodecahedron with 10 tetrahedra in it.   You can see it here:

http://math.ucr.edu/home/baez/mathematical/dodecahedron_with_10_tetrahedra.gif

The two kinds of tetrahedra are colored yellow and cyan.  Regions belonging to both are colored magenta.  It's pretty - but it's hard to see the tetrahedra, because they overlap a lot!

You can also do something like this starting with a cube.  A cube has 8 corners.  If you take every other corner of the cube, you get the 4 corners of a tetrahedron.  Butyou can d... more »

A 3-dimensional golden star

Here Greg Egan has drawn a dodecahedron with 5 tetrahedra in it.  This picture is 'left-handed': if you look at where the 5 tetrahedra meet, you'll see they swirl counterclockwise as you go out!  If you view this thing in a mirror you'll get a right-handed version. 

Putting them together, you get a dodecahedron with 10 tetrahedra in it.   You can see it here:

http://math.ucr.edu/home/baez/mathematical/dodecahedron_with_10_tetrahedra.gif

The two kinds of tetrahedra are colored yellow and cyan.  Regions belonging to both are colored magenta.  It's pretty - but it's hard to see the tetrahedra, because they overlap a lot!

You can also do something like this starting with a cube.  A cube has 8 corners.  If you take every other corner of the cube, you get the 4 corners of a tetrahedron.  But you can do this in 2 ways.  If you choose both, you get a cube with 2 tetrahedra in it:

http://math.ucr.edu/home/baez/mathematical/cube_with_2_tetrahedra.gif

All this is just the start of a much more elaborate and beautiful story which also involves the golden ratio, the quaternions, and 4-dimensional shapes like the 4-simplex, which has 5 tetrahedral faces, and the 600-cell, which has 600 tetrahedral faces!   You can read it here:

http://blogs.ams.org/visualinsight/2015/05/01/twin-dodecahedra/

I learned some of this story from Adrian Ocneanu at Penn State University.  Greg Egan and I figured out the rest... or most of the rest.  There's an unproven conjecture here, which needs to be true to make the whole story work.  Can you prove it?

Puzzle: If you take a regular 4-simplex whose vertices are unit quaternions, with the first equal to 1, can you prove the other 4 vertices generate a free group on 4 elements?

Hmm, I see that this puzzle has been solved by +Ian Agol and someone else on Mathoverflow:

http://mathoverflow.net/questions/204464/do-unit-quaternions-at-vertices-of-a-regular-4-simplex-one-being-1-generate-a

I don't understand the solution yet, because I don't know what a 'Bass-Serre tree' is... but I'll try to learn about this.  Math is infinite, there's always more to learn.

#geometry #4d  ___

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2015-05-01 05:22:14 (22 comments, 6 reshares, 60 +1s)Open 

The battle has begun

We're starting to fight global warming.  It's going to be a difficult war, and it's not clear we'll win.  But it's the most exciting, suspenseful story that we're all part of.

As the engineer Saul Griffith said:

"It's not like the Manhattan Project, it's like the whole of World War II, only with all the antagonists on the same side this time. It's damn near impossible, but it is necessary. And the world has to decide to do it."

A few promising signs:

1) On Wednesday morning, the governor of California set a goal of cutting carbon emissions by 40% below 1990 levels by 2030.  This matches the target set by the EU in October.   Both California and the EU are aiming to cut emissions 80% by 2050.  The governor said:

“We must demonstrate that reducingcarbon i... more »

The battle has begun

We're starting to fight global warming.  It's going to be a difficult war, and it's not clear we'll win.  But it's the most exciting, suspenseful story that we're all part of.

As the engineer Saul Griffith said:

"It's not like the Manhattan Project, it's like the whole of World War II, only with all the antagonists on the same side this time. It's damn near impossible, but it is necessary. And the world has to decide to do it."

A few promising signs:

1) On Wednesday morning, the governor of California set a goal of cutting carbon emissions by 40% below 1990 levels by 2030.  This matches the target set by the EU in October.   Both California and the EU are aiming to cut emissions 80% by 2050.  The governor said:

“We must demonstrate that reducing carbon is compatible with an abundant economy and human well-being.  Taking significant amounts of carbon out of our economy without harming its vibrancy is exactly the sort of challenge at which California excels. This is exciting, it is bold and it is absolutely necessary if we are to have any chance of stopping potentially catastrophic changes to our climate system."

At the national level, the US is dragging its heels.  But the states don't need to wait!  California, Oregon, Washington and British Columbia have signed a regional agreement to reduce carbon emissions, and the governor has signed separate accords with leaders in Mexico, China, Japan, Israel and Peru.  

2) Copenhagen has an ambitious plan to go carbon-neutral by 2025:

http://www.theguardian.com/environment/2013/apr/12/copenhagen-push-carbon-neutral-2025

As with California, the goal is not to save the world single-handedly, but to figure out how things can be done, so others can copy.

3) Pope Francis is getting serious about global warming.  He's already said he believes it's mostly manmade and that a Christian who does not protect God’s creation “is a Christian who does not care about the work of God”.  Now he's written an encyclical about it - the most significant sort of papal document.  This will come out in June.

I'm not a big fan of the Catholic church.  But it's important that everyone claiming to be a moral leader use their influence to get people to take this issue seriously, so I applaud this move.

Cardinal Peter Turkson, who is taking the lead on this, said increasing use of fossil fuels is disrupting Earth on an “almost unfathomable scale”, and says we need a “full conversion” of hearts and minds on this issue. 

http://www.theguardian.com/environment/2015/apr/28/vatican-climate-change-summit-to-highlight-moral-duty-for-action___

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2015-04-29 18:40:24 (0 comments, 3 reshares, 27 +1s)Open 

Mathematical dream worlds

+Jos Leys blends mathematics and art in a delightful way.  You don't need to know math to enjoy this picture.   It's a whimsical and mysterious landscape.  The bright colors make it clownish, but the shadows make it a bit eerie: the sun is setting, and who knows what happens here at night!   You can see more here:

http://www.josleys.com/show_gallery.php?galid=252

On the other hand, if you read the title of this gallery, you'll see there's math here: "the first 3d views of limit sets of Kleinian groups".  And trying to understand this math will lead you on quite a journey.  Let me sketch it here... I apologize for going rather fast.

A Kleinian group is a discrete subgroup of the group called PSL(2,C).  This group shows up in many ways in math and physics. 

Physicists call it theLoren... more »

Mathematical dream worlds

+Jos Leys blends mathematics and art in a delightful way.  You don't need to know math to enjoy this picture.   It's a whimsical and mysterious landscape.  The bright colors make it clownish, but the shadows make it a bit eerie: the sun is setting, and who knows what happens here at night!   You can see more here:

http://www.josleys.com/show_gallery.php?galid=252

On the other hand, if you read the title of this gallery, you'll see there's math here: "the first 3d views of limit sets of Kleinian groups".  And trying to understand this math will lead you on quite a journey.  Let me sketch it here... I apologize for going rather fast.

A Kleinian group is a discrete subgroup of the group called PSL(2,C).  This group shows up in many ways in math and physics. 

Physicists call it the Lorentz group: it's the group generated by rotations and Lorentz transformations, which acts as symmetries in special relativity. 

In math, it's called the group of Möbius transformations or fractional linear transformations.  Those are transformations like this:

z |→ (az + b)/(cz + d)

where z is a complex number and so are a,b,c,d.  These can be seen as transformations of the Riemann sphere - the complex plane together with a point at infinity.  They are, in fact, precisely all the conformal transformations of the Riemann sphere: the transformations that preserve angles. 

But this group PSL(2,C) also acts as conformal transformations of a 3-dimensional ball whose boundary is the Riemann sphere!   And that's important for understanding this picture.

(In physics, this ball is the set of 'mixed states' for a spin-1/2 particle, and the sphere, its boundary, consists of the 'pure states'.  Lorentz transformations act on the mixed states, and they act on the pure states.  But you don't need to know this stuff.)

If you take any point inside the ball and act on it by all the elements in a Kleinian group - a discrete subgroup of PSL(2,C) - you'll get a set S of points in the ball.  The set of points in the Riemann sphere that you can approach by a sequence of points in S is called a limit set of the Kleinian group.  And this set can look really cool! 

In these pictures, Jos Leys has systematically but rather artificially these cool-looking subsets of the Riemann sphere and puffed them up into 3-dimensional spaces: puffing a circle into a sphere, and so on.  This makes the picture nicer, but doesn't have a deep mathematical meaning.

Later, Jos Leys took a deeper approach, using quaternions to make limit sets that are truly 3-dimensional.  You can seem some here:

http://www.josleys.com/show_gallery.php?galid=346

They have a very different look.

For more on the math try these:

https://en.wikipedia.org/wiki/Kleinian_group
https://en.wikipedia.org/wiki/Möbius_transformation

Puzzle: if you put together everything I said, you'll get a physics interpretation of the limit set of a Kleinian group in terms of states of a spin-1/2 particle.  What is it?

#geometry  ___

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2015-04-29 16:56:52 (36 comments, 25 reshares, 105 +1s)Open 

Mathematical dream worlds

+Jos Leys blends mathematics and art in a delightful way.  You don't need to know math to enjoy this picture.   It's a whimsical and mysterious landscape.  The bright colors make it clownish, but the shadows make it a bit eerie: the sun is setting, and who knows what happens here at night!   You can see more here:

http://www.josleys.com/show_gallery.php?galid=252

On the other hand, if you read the title of this gallery, you'll see there's math here: "the first 3d views of limit sets of Kleinian groups".  And trying to understand this math will lead you on quite a journey.  Let me sketch it here... I apologize for going rather fast.

A Kleinian group is a discrete subgroup of the group called PSL(2,C).  This group shows up in many ways in math and physics. 

Physicists call it theLoren... more »

Mathematical dream worlds

+Jos Leys blends mathematics and art in a delightful way.  You don't need to know math to enjoy this picture.   It's a whimsical and mysterious landscape.  The bright colors make it clownish, but the shadows make it a bit eerie: the sun is setting, and who knows what happens here at night!   You can see more here:

http://www.josleys.com/show_gallery.php?galid=252

On the other hand, if you read the title of this gallery, you'll see there's math here: "the first 3d views of limit sets of Kleinian groups".  And trying to understand this math will lead you on quite a journey.  Let me sketch it here... I apologize for going rather fast.

A Kleinian group is a discrete subgroup of the group called PSL(2,C).  This group shows up in many ways in math and physics. 

Physicists call it the Lorentz group: it's the group generated by rotations and Lorentz transformations, which acts as symmetries in special relativity. 

In math, it's called the group of Möbius transformations or fractional linear transformations.  Those are transformations like this:

z |→ (az + b)/(cz + d)

where z is a complex number and so are a,b,c,d.  These can be seen as transformations of the Riemann sphere - the complex plane together with a point at infinity.  They are, in fact, precisely all the conformal transformations of the Riemann sphere: the transformations that preserve angles. 

But this group PSL(2,C) also acts as conformal transformations of a 3-dimensional ball whose boundary is the Riemann sphere!   And that's important for understanding this picture.

(In physics, this ball is the set of 'mixed states' for a spin-1/2 particle, and the sphere, its boundary, consists of the 'pure states'.  Lorentz transformations act on the mixed states, and they act on the pure states.  But you don't need to know this stuff.)

If you take any point inside the ball and act on it by all the elements in a Kleinian group - a discrete subgroup of PSL(2,C) - you'll get a set S of points in the ball.  The set of points in the Riemann sphere that you can approach by a sequence of points in S is called a limit set of the Kleinian group.  And this set can look really cool! 

In these pictures, Jos Leys has systematically but rather artificially taken these cool-looking subsets of the Riemann sphere and puffed them up into 3-dimensional spaces: puffing a circle into a sphere, and so on.  This makes the picture nicer, but doesn't have a deep mathematical meaning.

Later, Jos Leys took a deeper approach, using quaternions to make limit sets that are truly 3-dimensional.  You can seem some here:

http://www.josleys.com/show_gallery.php?galid=346

They have a very different look.

For more on the math try these:

https://en.wikipedia.org/wiki/Kleinian_group
https://en.wikipedia.org/wiki/Möbius_transformation

Puzzle: if you put together everything I said, you'll get a physics interpretation of the limit set of a Kleinian group in terms of states of a spin-1/2 particle.  What is it?

#geometry  ___

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2015-04-26 03:49:32 (78 comments, 88 reshares, 359 +1s)Open 

We Can't Stop

If you've vaguely heard about that scandalous Miley Cyrus character, but have never brought yourself to actually listen to any of her songs, you might prefer this version of her hit "We Can't Stop", sung in a 1950s doo-wop style by the group Postmodern Jukebox.

Postmodern Jukebox covers lots of modern hits in old-fashioned styles like ragtime, jazz, and bluegrass.  You can find them on YouTube.  The surprising thing is that they're really enjoyable!  First, they just sound nice.  Second, they let you ponder what's left of a modern hit after the glitz has been removed.

The brains behind Postmodern is Scott Bradlee, a musician from New York who fell in love with jazz at the age of 12 after hearing George Gershwin's "Rhapsody in Blue".  He became a jazz musician, but then had thebril... more »

We Can't Stop

If you've vaguely heard about that scandalous Miley Cyrus character, but have never brought yourself to actually listen to any of her songs, you might prefer this version of her hit "We Can't Stop", sung in a 1950s doo-wop style by the group Postmodern Jukebox.

Postmodern Jukebox covers lots of modern hits in old-fashioned styles like ragtime, jazz, and bluegrass.  You can find them on YouTube.  The surprising thing is that they're really enjoyable!  First, they just sound nice.  Second, they let you ponder what's left of a modern hit after the glitz has been removed.

The brains behind Postmodern is Scott Bradlee, a musician from New York who fell in love with jazz at the age of 12 after hearing George Gershwin's "Rhapsody in Blue".  He became a jazz musician, but then had the brilliant idea of giving modern songs old-fashioned arrangements.  In 2009  he released "Hello My Ragtime '80s", which combined popular music from the 1980s with ragtime-style piano.  In 2013 he formed Postmodern Jukebox, and they first became famous with this song... probably sung in his living room.  The lead singer is Robyn Adele Anderson.

Some other good ones:

"All About That Bass" - https://www.youtube.com/watch?v=aLnZ1NQm2uk

"Creep" - https://www.youtube.com/watch?v=m3lF2qEA2cw

"Blurred Lines" - https://www.youtube.com/watch?v=Nz-OMn1o22Y

"Call Me Maybe" - https://www.youtube.com/watch?v=5meWI3iX1sE

If you don't know the originals of these songs, you have been living under a rock - which is not necessarily a bad thing.  Now you can catch up without ever entering the modern world!  Go straight to the postmodern world.

What's interesting, of course, is how well these songs do with old-fashioned arrangements.  At a certain basic level, like the chord progressions, American popular music is remarkably slow to change.___

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2015-04-24 16:00:05 (35 comments, 69 reshares, 178 +1s)Open 

The toughest animal on the planet

A rotifer is an animal that lives in water and sweeps food into its mouth with small hairs.  There are many kinds, most less than a millimeter in length.  They can eat anything smaller than their head.

The toughest are the bdelloid rotifers.  These can survive being completely dried out for up to 9 years!  When they dry out, they sometimes crack.  Even their DNA can crack... but when they get wet, they come back to life!

Thanks to this strange lifestyle, their DNA gets mixed with other DNA.   Up to 10% of their active genes come from bacteria, fungi and algae!!! 

Scientists have found DNA from 500 different species in the genes of a rotifer from Australia.  "It's a genetic mosaic. It takes pieces of DNA from all over the place," said one of the study's authors. "Its biochemistryis a mos... more »

The toughest animal on the planet

A rotifer is an animal that lives in water and sweeps food into its mouth with small hairs.  There are many kinds, most less than a millimeter in length.  They can eat anything smaller than their head.

The toughest are the bdelloid rotifers.  These can survive being completely dried out for up to 9 years!  When they dry out, they sometimes crack.  Even their DNA can crack... but when they get wet, they come back to life!

Thanks to this strange lifestyle, their DNA gets mixed with other DNA.   Up to 10% of their active genes come from bacteria, fungi and algae!!! 

Scientists have found DNA from 500 different species in the genes of a rotifer from Australia.  "It's a genetic mosaic. It takes pieces of DNA from all over the place," said one of the study's authors. "Its biochemistry is a mosaic in the same way. It's a real mishmash of activities."

Perhaps because of this, bdelloid rotifers don't bother to have sex. 

Their ability to survive dry conditions makes them great at living in desert lakes and mud puddles that dry up.  But they also use this ability to beat some parasites.  When they dry out, the parasites die... but the rotifers survive!

On top of all this, bdelloid rotifers can survive high doses of radiation.  My guess is that this is just a side-effect of having really good genetic repair mechanisms.

Puzzle 1: what does 'bdelloid' mean?

Puzzle 2: what other words begin with 'bd'... and why?

Here's the paper that found 10% of active genes and 40% of all enzyme activity in bdelloid rotifers involve foreign DNA:

• Chiara Boschetti, Adrian Carr, Alastair Crisp, Isobel Eyres, Yuan Wang-Koh, Esther Lubzens, Timothy G. Barraclough, Gos Micklem and Alan Tunnacliffe, Biochemical diversification through foreign gene expression in bdelloid rotifers, PLOS Genetics, 15 November 2012, http://journals.plos.org/plosgenetics/article?id=10.1371/journal.pgen.1003035.

The  animated gif is from here:

http://merismo.tumblr.com/post/43868329996/gif-rotifer-with-cilia-on-corona-present-mastax

#spnetwork #bdelloid #rotifer doi:10.1371/journal.pgen.1003035___

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2015-04-22 17:12:00 (38 comments, 25 reshares, 82 +1s)Open 

The insanity of infinite reflections

This picture by +John Valentine shows a ball inside a mirrored spheroid... together with all its reflections! The real ball is at lower right.  The rest are reflections.  They form crazy patterns - the kind of thing mathematicians think about when they can't sleep at night.

This is like a picture I showed you earlier, made by +Refurio Anachro. But now the ball is lit from three directions with soft red, green, and blue lights, so we can see things more clearly.  The view simulates an ultra-wide-angle camera. 

This is just a low-resolution closeup of a much bigger and more detailed picture!  You can get other views here, along with a good discussion:

https://plus.google.com/u/0/114187364719055671781/posts/RzdjJwARTu6

You can get a really big version here:
ht... more »

The insanity of infinite reflections

This picture by +John Valentine shows a ball inside a mirrored spheroid... together with all its reflections! The real ball is at lower right.  The rest are reflections.  They form crazy patterns - the kind of thing mathematicians think about when they can't sleep at night.

This is like a picture I showed you earlier, made by +Refurio Anachro. But now the ball is lit from three directions with soft red, green, and blue lights, so we can see things more clearly.  The view simulates an ultra-wide-angle camera. 

This is just a low-resolution closeup of a much bigger and more detailed picture!  You can get other views here, along with a good discussion:

https://plus.google.com/u/0/114187364719055671781/posts/RzdjJwARTu6

You can get a really big version here:

https://sites.google.com/site/csjohnv/Ball001d-16k-low.jpg?attredirects=0&d=1

This is 16384 × 16384 pixels and about 16 megabytes.  If you know a nice way to display such a big image online, which makes it easy to zoom in on pieces, please try it!

Puzzle 1: what creates the black 'zone of invisibility', and the fractal hexagonal patterns near the zone of invisibility?

I don't really know the answer in detail - this could be a great math project.

I've watched a number of movies where the climactic final scene involves people fighting inside a hall of mirrors, where it's hard to tell who is real and who is a reflection.  Orson Welles' 1947 classic Lady from Shanghai may be the first - if you haven't seen that, you should definitely watch it.  Another that stands out is Bruce Lee's Enter the Dragon. 

Puzzle 2: what other movies or stories do you know involving this theme?___

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2015-04-20 15:35:47 (84 comments, 30 reshares, 109 +1s)Open 

Twin dodecahedra

Here Greg Egan has drawn two regular dodecahedra, in red and blue.  They share some corners - and these are the corners of a cube, shown in green! 

I learned some cool facts about this from Adrian Ocneanu when I was at Penn State.  First some easy stuff.  You can take some corners of a regular dodecahedron and make them into the corners of a cube.  But not every symmetry of the cube is a symmetry of the dodecahedron!  If you give the cube a 90° rotation around any face, you get a new dodecahedron.  Check it out: doing this rotation switches the red and green dodecahedra.  These are called twin dodecahedra.

But there are actually 5 different ways to take a regular dodecahedron and make them into the corners of a cube.  And each one gives your dodecahedron a different twin!  So, a dodecahedron actually has 5 twins.

Buthere's... more »

Twin dodecahedra

Here Greg Egan has drawn two regular dodecahedra, in red and blue.  They share some corners - and these are the corners of a cube, shown in green! 

I learned some cool facts about this from Adrian Ocneanu when I was at Penn State.  First some easy stuff.  You can take some corners of a regular dodecahedron and make them into the corners of a cube.  But not every symmetry of the cube is a symmetry of the dodecahedron!  If you give the cube a 90° rotation around any face, you get a new dodecahedron.  Check it out: doing this rotation switches the red and green dodecahedra.  These are called twin dodecahedra.

But there are actually 5 different ways to take a regular dodecahedron and make them into the corners of a cube.  And each one gives your dodecahedron a different twin!  So, a dodecahedron actually has 5 twins.

But here's the cool part.  Suppose you take one of these twins.  It, too, will have 5 twins.  One of these will be the dodecahedron you started with.  But the other 4 will be new dodecahedra: that is, dodecahedra rotated in new ways.

How many different dodecahedra can you get by continuing to take twins?  Infinitely many!

In fact, we can draw a graph - a thing with dots and edges - that explains what's going on.  Start with a dot for our original dodecahedron.  Draw dots for all the dodecahedra you can get by repeatedly taking twins.  Connect two dots with an edge if and only if they are twins of each other.

The resulting graph is a tree: in other words, it has no loops in it!  If you start at your original dodecahedron, and keep walking along edges of this graph by taking twins, you'll never get back to where you started except by undoing all your steps.

Ocneanu's proof of this is very nice, using some 4-dimensional geometry and group theory.  I will have to outline it somewhere, because Ocneanu is famous for not publishing most of his work.  But I like how you can state the end result without these more sophisticated concepts.

Here are some puzzles.

You can also choose some corners of a cube and make them into the corners of a regular tetrahedron.  You can fit 2 tetrahedra in the cube this way.  These are a bit like the 5 cubes in the dodecahedron, but there's a big difference.

Here's the difference.  In the first case, every symmetry of the tetrahedron is a symmetry of the cube it's in.  But in the second case not every symmetry of the cube is a symmetry of the dodecahedron.  That's why we get 'twin dodecahedra' but not 'twin cubes'.

Puzzle 1: If you inscribe a tetrahedron in a cube and then inscribe the cube in a dodecahedron, is every symmetry of the tetrahedron a symmetry of the dodecahedron?

Puzzle 2: How many ways are there to inscribe a tetrahedron in a dodecahedron?  More precisely: how many ways are there to choose some corners of a regular dodecahedron and have them be the corners of a regular tetrahedron?

And a harder one:

Puzzle 3: Trees are related to free groups.  What free group is responsible for Ocneanu's result?

#geometry  ___

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2015-04-19 19:58:16 (22 comments, 2 reshares, 59 +1s)Open 

This is an official photo of the Canadian Supreme Court!  They dress like Santa Claus because of their curious role in the Canadian legal system.  I hadn't known about this until +Allen Knutson posted about it.

If you feel a verdict from a lower court has been unfair, on Christmas Eve you put a note in a sock explaining your case, and hang it on your fireplace.  Then, one of the Supreme Court members will come down your chimney and either grant your wish or leave you coals.  They know who's naughty and who's nice, thanks to an extensive system of legal clerks who dress as elves.

If you don't believe this is a real picture of the Canadian Supreme Court, do a Google image search!   Here, I'll make it easy:

https://www.google.com/search?q=canadian+supreme+court&tbm=isch

There is a long history of goofy outfits for judges.  Britishjudges... more »

This is an official photo of the Canadian Supreme Court!  They dress like Santa Claus because of their curious role in the Canadian legal system.  I hadn't known about this until +Allen Knutson posted about it.

If you feel a verdict from a lower court has been unfair, on Christmas Eve you put a note in a sock explaining your case, and hang it on your fireplace.  Then, one of the Supreme Court members will come down your chimney and either grant your wish or leave you coals.  They know who's naughty and who's nice, thanks to an extensive system of legal clerks who dress as elves.

If you don't believe this is a real picture of the Canadian Supreme Court, do a Google image search!   Here, I'll make it easy:

https://www.google.com/search?q=canadian+supreme+court&tbm=isch

There is a long history of goofy outfits for judges.  British judges, even ones who aren't bald, are required to wear a wig of horse hair!  This was the origin of the Wig Party, who used to be the main opposition to the Tories.  And the Australians have a kangaroo court, who jump to decisions.

Puzzle 1: why does the Canadian supreme court really dress like this?

Puzzle 2: why do British judges wear wigs?

Puzzle 3: what's the origin of the phrase 'kangaroo court'?

If you look up the answers using Google, you get special extra credit: it means you know how to use the internet.___

posted image

2015-04-18 14:11:45 (94 comments, 36 reshares, 79 +1s)Open 

RoboRoach

You can now make your own cyborg roach for just $100.   Just buy this kit developed by the company Backyard Brains:

Are you a teacher or parent that wants to teach a student about advanced neurotechnologies? You are in luck! After 3 long years of R&D, the RoboRoach is now ready for its grand release! We are excited to announce the world's first commercially available cyborg! With our RoboRoach you can briefly wirelessly control the left/right movement of a cockroach by microstimulation of the antenna nerves. The RoboRoach is a great way to learn about neural microstimulation, learning, and electronics!

We are recently ran a successfully-funded kickstarter campaign to fund the release of our new RoboRoach! The hardware and firmware development are complete and we are now shipping!

Product Details

TheR... more »

RoboRoach

You can now make your own cyborg roach for just $100.   Just buy this kit developed by the company Backyard Brains:

Are you a teacher or parent that wants to teach a student about advanced neurotechnologies? You are in luck! After 3 long years of R&D, the RoboRoach is now ready for its grand release! We are excited to announce the world's first commercially available cyborg! With our RoboRoach you can briefly wirelessly control the left/right movement of a cockroach by microstimulation of the antenna nerves. The RoboRoach is a great way to learn about neural microstimulation, learning, and electronics!

We are recently ran a successfully-funded kickstarter campaign to fund the release of our new RoboRoach! The hardware and firmware development are complete and we are now shipping!

Product Details

The RoboRoach "backpack" weighs 4.4 grams with the battery, and each battery will last over a month! Following a brief surgery you perform on the cockroach to attach the silver electrodes to the antenna, you can attach the backpack to the roach and control its movement for a few minutes before the cockroach adapts. When you return the cockroach to its cage for ~20 minutes, he "forgets" and the stimulation works again. Once you receive your RoboRoach in the mail, follow our online surgery instructions and videos and you will soon be on your way to becoming an expert in neural interfaces. After about 2-7 days, the stimulation stops working altogether, so you can clip the wires and retire the cockroach to your breeder colony to spend the rest of its days making more cockroaches for you and eating your lettuce.

Technical Specs

1x Free iOS or Android 4.3+ application for remote control
1x Bluetooth Roboroach backpack control unit
1x 1632 RoboRoach Battery
3x Electrode Sets (to implant 3 Roaches)

View our RoboRoach Ethics Statement

Backyard Brains has developed ethical guidelines for all our products. You can read more in our statement regarding our use of insect for experiments at:

http://ethics.backyardbrains.com

I feel ethical qualms about taking away the autonomy of an animal this way, and their ethics statement doesn't really address that.  This is the closest they come:

Criticism: Modifying a living creature to make a toy is wrong.

The RoboRoach circuit is not a toy. This new bluetooth version is a powerful low-cost tool for studying neural circuits, allowing for students to make discoveries. High school students in New York, for example, have discovered random stimulation causes much slower adaptation times. We have scientist and high school educator colleagues who are mentoring students in novel behavioral experiments using the RoboRoach circuit. Some highlights will be posted on our website soon.

By focusing on the question of whether the RoboRoach is a "toy", they dodge the harder question of when it's okay to override the nervous system of an animal and make it do what you want.  Perhaps feeling a bit nervous about this, some of the cyborg roach developers say they want to use it as a "rescue robot" that can crawl around and hear people trapped under collapsed buildings.  I think most people would say this is okay, at least if it actually works.

For a critical view on the ethics, see:

http://www.livescience.com/40821-roboroach-is-inhumane.html

http://www.care2.com/causes/the-do-it-yourself-cyborg-cockroach-educational-or-cruel.html

For more on how to actually make a RoboRoach, go here:

http://www.popsci.com/science/article/2012-12/how-build-your-own-cockroach-cyborg___

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2015-04-15 13:39:58 (25 comments, 30 reshares, 106 +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 (15 comments, 16 reshares, 126 +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___

posted image

2015-04-10 02:51:27 (56 comments, 64 reshares, 232 +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, 30 +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, 6 reshares, 41 +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, 92 +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 (79 comments, 101 reshares, 240 +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, 52 +1s)Open 

April Fools!

April Fools!___

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2015-03-31 15:51:20 (81 comments, 44 reshares, 168 +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 (133 comments, 56 reshares, 271 +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, 183 +1s)Open 

News in the USA

News in the USA___

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2015-03-28 00:29:19 (97 comments, 11 reshares, 65 +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, 41 +1s)Open 

Where in the world?

Where in the world?___

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2015-03-26 15:30:39 (127 comments, 74 reshares, 156 +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 (82 comments, 27 reshares, 132 +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, 13 reshares, 85 +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 (64 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, 96 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, 118 +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  ___

posted image

2015-03-11 19:43:52 (101 comments, 33 reshares, 158 +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  ___

posted image

2015-03-10 23:13:11 (31 comments, 50 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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