The Banach–Tarski Paradox

Captions provided by CCTubes – Captioning the Internet! 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