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I was playing around, and stumbled upon something quite beautiful..
If you sum over the volumes of all 2k-dimensional unit spheres, that sum converges to e^π.
I have no idea why this should be true..pic.twitter.com/QvsQJkAuc2
Hmm. Not immediately! I will try to understand the recursion you are hinting at: a formula for V_{2k+2} in terms of V_k. If I could find an intuitive explanation of that, I'd consider my work done. I think the sum is a bit of a red herring.
could be! but if I see x^k / k! , I cant resist the temptation.. especially when there's a π !
Part of me is thinks a geometric interpretation might be tricky because you're adding together things with different units? (unless you imagine this as a sum of various infinite-dimensional cylinder/extruded-sphere things)
But we can normalize the parameter to be dimensionless
How does geometric intuition deal with the first term of the sum, a zero-dimensional sphere? What is that anyway, and why is its volume equal to 1?
A zero-dimensional sphere is a point and it's 0-dimensional volume is 1. That makes some sense, since it's *one* point.
(I'm not sure the 'perhaps' will prove srictly necessary there ;) ).
I tried to think geometrically about this sum and didn't get very far, since he was adding thing with different units. Now I feel pressured to continue. 
I think it'd be better to put in the radius r of the spheres instead of setting it to 1. Then the sum is exp(r pi). 1/n
I feel like if someone must know anything about a geometric interpretation of that, it's @3blue1brown...?
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