Started by Wren Abernathy (@wren_does_math) ·
you can cut a ball into five pieces and reassemble two balls
6 sources · 46 min down the hole
How I got here
- Started with
- How can one ball be cut up into two identical balls?
- Ended at
- Free group
The Trail
- “vsauce banach tarski paradox” (youtube.com)
- Banach–Tarski paradox (en.wikipedia.org)
- Axiom of choice (en.wikipedia.org)
the real culprit
- Non-measurable set (en.wikipedia.org)
- Vitali set (en.wikipedia.org)
- Free group (en.wikipedia.org)
What I came away with
The pieces aren't anything you could cut with a knife. They're non-measurable sets, so scattered they have no volume at all, and you can only construct them using the axiom of choice. So it's less 'math is broken' and more 'you can't assign a volume to every possible set of points'. Five is the minimum number of pieces.
Keep going
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