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.
That's a perfectly valid way to "relieve" the paradox, I certainly agree. Others have noted how it might not be all that paradoxical given that there are uncountably many points in one ball and in two copies of it. There are many more, perhaps much more surprising, examples of equal cardinality in math. For that reason, the result is taken by some simply as a statement on the complexity of the group of translations and rotations. Definitely an interesting discussion for sure! Thanks for watching!
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!
5:28 not clear why second order branches do not contain σ^-1. Why e=σ*σ^-1 is not a child of σ. If it were shown as a child the self-referential structure would be more obvious. I think the proposed tree contains itself as subgraphs infinitely many times, as in every grandchild generation there is {e}
I see what you're getting at. I suppose it's a matter of choosing how you want to define the tree. We definitely couldn't define the tree to "contain itself", since the proof requires that every leaf of the tree corresponds one-to-one with an element of the free group F_2. We could however create a structure where, when 'e' is a child of some node, we let that node point to 'e' at the top of the tree. That is, we add arrows pointing back up the tree. (To be a bit pedantic, letting elements "point back" to e would create cycles, which means the structure wouldn't qualify as a tree. Of course, we could fix that problem by just calling it a 'graph', instead of 'tree'.) You could even consider the tree as such a structure, where "upward arrows" (i.e. arrows A -> B where A is of a later generation than B) are simply deleted / hidden. Considering the above, I think adding the self-reference in the tree/graph wouldn't necessarily have made the proof easier to understand, hence why I chose to present it as I did. But I can definitely appreciate that for some people it makes more sense to think of the tree in a self-referential way. Thanks for watching!
I'm a mathematics major and I really enjoyed this. Thank you.
Great to hear! Thanks for watching.
Fantastico - "mesmerising to look at"!
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.
That's a perfectly valid way to "relieve" the paradox, I certainly agree. Others have noted how it might not be all that paradoxical given that there are uncountably many points in one ball and in two copies of it. There are many more, perhaps much more surprising, examples of equal cardinality in math. For that reason, the result is taken by some simply as a statement on the complexity of the group of translations and rotations. Definitely an interesting discussion for sure!
Thanks for watching!
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!
Also, where could I find the full thesis, I’d love to read it!:)
Thanks for the kind words, I'm glad you enjoyed it. You can download my thesis at filebin.net/lhmmlpfqlssrxsjq
@@teunvanwezel2282Shame that the file is no longer there. I'm going through Banach tarski right now but got confused on the selection of D :(
He used Manim, a Python library for MAthematical ANIMations, it's really nice.
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
Well-spotted! I'm pretty sure that's an error, and it's supposed to be theta = pi/2. Not sure what exactly the cause is.
Really nice, thanks, can you please update the link to the thesis? The one in the comment doesn't work
The thesis is now available on my website here:
teunvanwezel.nl/Wezel-van-Teun-Bacehlor-Thesis.pdf
Thanks for watching!
Marvelous video❤.please go on making videos. Sorry I don't speak English very well. I use google's traslation app.😊
5:28 not clear why second order branches do not contain σ^-1. Why e=σ*σ^-1 is not a child of σ. If it were shown as a child the self-referential structure would be more obvious. I think the proposed tree contains itself as subgraphs infinitely many times, as in every grandchild generation there is {e}
I see what you're getting at. I suppose it's a matter of choosing how you want to define the tree.
We definitely couldn't define the tree to "contain itself", since the proof requires that every leaf of the tree corresponds one-to-one with an element of the free group F_2. We could however create a structure where, when 'e' is a child of some node, we let that node point to 'e' at the top of the tree. That is, we add arrows pointing back up the tree. (To be a bit pedantic, letting elements "point back" to e would create cycles, which means the structure wouldn't qualify as a tree. Of course, we could fix that problem by just calling it a 'graph', instead of 'tree'.) You could even consider the tree as such a structure, where "upward arrows" (i.e. arrows A -> B where A is of a later generation than B) are simply deleted / hidden.
Considering the above, I think adding the self-reference in the tree/graph wouldn't necessarily have made the proof easier to understand, hence why I chose to present it as I did. But I can definitely appreciate that for some people it makes more sense to think of the tree in a self-referential way.
Thanks for watching!
❤
Banach Tarski houd me nu ruim 12 jaar bezig 😂
Ja fascinerend is het zeker. Dank voor het kijken!
Figures lie; and liars figure.😂
Nah im good thanks though
Thanks for stopping by though