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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
The general case of the sum over the volumes of all 2k-dimensional spheres of radius R
I have defined the dimensionless radius r to be the radius R measured in units of some "base length" R₀pic.twitter.com/SJjKy0o6Mr
So, if I get this thing right, the infinite sum of 2k-spheres where the volume of each goes to 0 as k goes to infinity, is not infinite?
We're not adding up an infinite number of *infinite* dimensions spheres (k-> infinity), we're adding up all k dimensional spheres.
How do you write math symbols like that?
Hi Dillon - as a non-mathematician outsider, does this describe in some way what happens w the volume of a Schwarzchild black hole (given arbitrary n of dimensions)? What happens if you define the r as 2GM/c^2? Again, sorry if this is stupid, my interest exceeds my ability!
@ risk of compounding my stupidity, is there any reason why a BH would not have an arbitrary n of dimensions within the event horizon?
a fact like that, sounds like it might come in very useful for someone, some day, for something, affording even greater insight, into something else ...
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