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The Banach–Tarski Paradox

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:   / tweetsauce   Instagram:   / electricpants   Kevin’s Field Day video:    • After The Upload | Kevin Lieber from ...   Field Day:    / @fieldday   Deep dream animation by   / nadermakki   If you like it, you'll love this video also by Nader:    • Look Closely - by Nader Makki   Chocolate illusion: http://mathandmultimedia.com/2014/07/... Chocolate illusion video:    • Видео   related Numberphile videos: sizes of infinity (includes diagonal argument):    • Infinity is bigger than you think - N...   infinity paradoxes:    • Infinity Paradoxes - Numberphile   Vi Hart on types of infinity:    • How many kinds of infinity are there?   Countable & uncountable definitions: http://mathinsight.org/definition/unc... https://en.wikipedia.org/wiki/Countab... https://en.wikipedia.org/wiki/Uncount... Banach-Tarski on wikipedia: https://en.wikipedia.org/wiki/Banach%... Banach-Tarski proofs: http://math.uchicago.edu/~may/REU2014... https://www.math.hmc.edu/~su/papers.d... http://people.math.umass.edu/~weston/... Banach-Tarski explinations online: http://www.irregularwebcomic.net/2339... http://www.kuro5hin.org/comments/2003... http://skepticsplay.blogspot.co.uk/20... http://austinrochford.com/posts/2014-... http://www.math.cornell.edu/~mec/Summ... http://rachellevanger.com/index_files... http://quibb.blogspot.co.uk/2013_03_0... http://blog.computationalcomplexity.o... https://geopolicraticus.wordpress.com... http://dgleahy.com/p47.html https://www.math.hmc.edu/funfacts/ffi... Cayley graph animated gif:   / 624202342259240960   Hilbert’s hotel on wikipedia: https://en.wikipedia.org/wiki/Hilbert... types of infinity: http://www.xamuel.com/levels-of-infin... set theory and quantum physics: http://link.springer.com/article/10.1... LHC gif: http://cms.web.cern.ch/news/lhc-data-... Zermelo-Fraenkel axioms of mathematics: http://mathworld.wolfram.com/Zermelo-... Is math invented or discovered? http://phys.org/news/2013-09-mathemat... http://www.scientificamerican.com/art... more deep dream images:   / deepdream   BOOKS: The Pea and the Sun: http://www.amazon.com/The-Pea-Sun-Mat... The Outer Limits of Reason: http://www.amazon.com/Outer-Limits-Re... Why Beliefs Matter: http://www.amazon.com/Why-Beliefs-Mat... Things to Make and do in the Fourth Dimension: http://www.amazon.com/Things-Make-Fou... Music by    / jakechudnow   and http://www.audionetwork.com