Support Vsauce, your brain, Alzheimer's research, and other YouTube educators by joining THE CURIOSITY BOX: a seasonal delivery of viral science toys made by Vsauce! A portion of all proceeds goes to Alzheimer's research and our Inquisitive Fellowship, a program that gives money and resources directly to growing STEM channels here on YouTube! https://www.curiositybox.com Q: "What's an anagram of Banach-Tarski?" A: "Banach-Tarski Banach-Tarski." twitter: https://www.twitter.com/tweetsauce Instagram: http://www.instagram.com/electricpants Kevin’s Field Day video: https://www.youtube.com/watch?v=1zARMZ08ums Field Day: https://www.youtube.com/channel/UCRPktNf5vnBR1J4e7t1RUVg Deep dream animation by http://instagram.com/NaderMakki/ If you like it, you'll love this video also by Nader: https://www.youtube.com/watch?v=fJ9j_z2kXI0 Chocolate illusion: http://mathandmultimedia.com/2014/07/22/explanation-infinite-chocolate-bars/ Chocolate illusion video: https://www.youtube.com/watch?v=dmBsPgPu0Wc related Numberphile videos: sizes of infinity (includes diagonal argument): https://www.youtube.com/watch?v=elvOZm0d4H0 infinity paradoxes: https://www.youtube.com/watch?v=dDl7g_2x74Q Vi Hart on types of infinity: https://www.youtube.com/watch?v=23I5GS4JiDg Countable & uncountable definitions: http://mathinsight.org/definition/uncountable https://en.wikipedia.org/wiki/Countable_set https://en.wikipedia.org/wiki/Uncountable_set Banach-Tarski on wikipedia: https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox Banach-Tarski proofs: http://math.uchicago.edu/~may/REU2014/REUPapers/Robinson.pdf https://www.math.hmc.edu/~su/papers.dir/banachtarski.pdf http://people.math.umass.edu/~weston/oldpapers/banach.pdf Banach-Tarski explinations online: http://www.irregularwebcomic.net/2339.html http://www.kuro5hin.org/comments/2003/5/23/134430/275?pid=5#10 http://skepticsplay.blogspot.co.uk/2010/05/doubling-sphere.html http://austinrochford.com/posts/2014-05-14-banach-tarski-paradox.html http://www.math.cornell.edu/~mec/Summer2009/Whieldon/Math_Explorers_Club%3A__Lesson_Links/Entries/2009/7/28_Lesson_6%3A__Whats_an_Anagram_of_Banach-Tarski.html http://rachellevanger.com/index_files/BT%20Animated%20Presentation%20Web.pdf http://quibb.blogspot.co.uk/2013_03_01_archive.html http://blog.computationalcomplexity.org/2011/04/what-did-banachs-wife-think-of-banach.html https://geopolicraticus.wordpress.com/tag/banach-tarski-paradox/ http://dgleahy.com/p47.html https://www.math.hmc.edu/funfacts/ffiles/30001.1-3-8.shtml Cayley graph animated gif: https://twitter.com/GIFsofWikipedia/status/624202342259240960 Hilbert’s hotel on wikipedia: https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel types of infinity: http://www.xamuel.com/levels-of-infinity/ set theory and quantum physics: http://link.springer.com/article/10.1007/BF02213427#page-1 LHC gif: http://cms.web.cern.ch/news/lhc-data-be-made-public-open-access-initiative Zermelo-Fraenkel axioms of mathematics: http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html Is math invented or discovered? http://phys.org/news/2013-09-mathematics-effective-world.html http://www.scientificamerican.com/article/is-the-universe-made-of-math-excerpt/ more deep dream images: https://www.reddit.com/r/deepdream/ BOOKS: The Pea and the Sun: http://www.amazon.com/The-Pea-Sun-Mathematical-Paradox/dp/1568813279 The Outer Limits of Reason: http://www.amazon.com/Outer-Limits-Reason-Science-Mathematics/dp/0262019353 Why Beliefs Matter: http://www.amazon.com/Why-Beliefs-Matter-Reflections-Science/dp/0198704992 Things to Make and do in the Fourth Dimension: http://www.amazon.com/Things-Make-Fourth-Dimension-Mathematicians/dp/0374275653 Music by http://www.youtube.com/JakeChudnow and http://www.audionetwork.com