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AnarchoEngineer@lemmy.dbzer0.com 2 weeks ago
Also isn’t an infinite dimensional sphere practically hollow?
(If you were to integrate the sphere to calculate volume like you do for lower dimensional ones, you would sum the volume of shells—which is just their surface area times a thickness—making it up. With infinite dimensions, each shell becomes infinitely larger than the preceding shell no matter how fine you make the slices. This means the largest shell contains basically all the volume.)
pankuleczkapl@lemmy.dbzer0.com 2 weeks ago
This reasoning is pretty weird, but the conclusion is basically right. That is, there is absolutely no way to extend the conventional notion of volume to R^infinity, which is basically what most people would imagine is the infinite equivalent of our dimensional space.
ComicalMayhem@lemmy.world 2 weeks ago
Wait, I thought the volume of a sphere approaches 0 as dimensions go to infinity, no? Thr general formula for the volume of any nth dimensional sphere has the gamma function in the denominator, which rises faster than whatever is in the numerator. At some point (5 dimensions, iirc) the volume starts decreasing
pankuleczkapl@lemmy.dbzer0.com 2 weeks ago
Yes, but we aren’t talking about the limit of the volume. We are talking about volume in actually infinite dimensional space.
phoenixz@lemmy.ca 2 weeks ago
I imagine that most people don’t have the faintest clue what you’re talking about, though