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: https://www.twitter.com/tweetsauce Instagram: http://www.instagram.com/electricpants Kevin’s Field Day video: https://www.youtube.com/watchv=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/watchv=fJ9j_z2kXI0 Chocolate illusion: http://mathandmultimedia.com/2014/07/22/explanation-infinite-chocolate-bars/ Chocolate illusion video: https://www.youtube.com/watchv=dmBsPgPu0Wc related Numberphile videos: sizes of infinity (includes diagonal argument): https://www.youtube.com/watchv=elvOZm0d4H0 infinity paradoxes: https://www.youtube.com/watchv=dDl7g_2x74Q Vi Hart on types of infinity: https://www.youtube.com/watchv=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/2003/5/23/134430/275pid=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/Summer20