The Banach Tarski Paradox: A Visual Proof

This video gives an animated proof of the Banach-Tarski paradox, based on my Bachelor's Thesis titled 'Non-measurable Sets and the Banach-Tarski Paradox'. This video and the written thesis were written at Utrecht University under the supervision of Karma Dajani.

Full thesis available here:

[StephenStruble-kr]

I'm a mathematics major and I really enjoyed this. Thank you.

[JannesBeckeringh]

Fantastico - "mesmerising to look at"!

[scarlettx]

What software/programs did you use to create this?
Fantastic work! I’m currently working on an undergrad project on Banach-Tarski and looking for ways to visualise it, thank you for the inspiration!

[michaelbarker]

It seems to me the paradox just comes from us trying to add a narrative to whats going on by using concepts we are familiar with. Some people might say that we are "cutting up" the sphere but thats simply not at all what we are doing when compared to taking a physical sphere and cutting it up. Anything that we could do to a physical object is measurable which is precisely the thing we are not doing in this paradox. When cutting up the ball using sets the thing that we would have is something where there is no physical correlate which is the non measurable set.

[pitxinuno]

Marvelous video❤.please go on making videos. Sorry I don't speak English very well. I use google's traslation app.😊

[Mebasically]

Really nice, thanks, can you please update the link to the thesis? The one in the comment doesn't work

[savirmaskara]

Hi! Could you explain what you’re doing at 2:30? I’m seeing it as a 17/3 rotation would land you at (1, 0) so I’m definitely misinterpreting it

[BoyKhongklai]

Banach Tarski houd me nu ruim 12 jaar bezig 😂