Comment on Under what circumstances or axioms do spheres (the shape) have infinite surface area?
thebestaquaman@lemmy.world 10 months agoI believe the surface area of an n-dimensional hypersphere is (n - 1) pi r^{n - 1}. In that case (I may have some factors wrong here, just going off memory), an infinite-dimensional hypersphere has infinite surface area as long as it has non-zero radius.
loppy@fedia.io 10 months ago
It does indeed scale with r^(n-1), but your factors are not close at all. It involves the gamma function, which in this case can be expanded into various factorials and also a factor of sqrt(pi) when n is odd. According to Wikipedia, the expression is 2pi^(n/2)r^(n-1)/Gamma(n/2).
thebestaquaman@lemmy.world 10 months ago
While I’m completely open that my factor is likely wrong here, the expression you provided is definitely wrong in both the 2D and 3D case (I’m assuming the r superscript on the pi was a typo), since it gives neither n = 2 => A = 2 pi r nor n = 3 => A = 4 pi r^2.