Comment on Under what circumstances or axioms do spheres (the shape) have infinite surface area?
thebestaquaman@lemmy.world 2 days 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 2 days 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 2 days 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.