Hyperspheres and Hypertori

An enlightening discussion about pi and tau.

Hyperspheres and Hypertori

by 1=0 » Tue Mar 21, 2017 6:36 pm

For the surface area and volume of a unit n-dimensional hypersphere, we have the formulas:
S=τ^floor(n/2)/(n-2)!! ×2 if n is odd
V=τ^floor(n/2)/n!! ×2 if n is odd
These are simple and easy to calculate and don't require a gamma function.
Additionally, tau simplifies the formulas for hypertori as well, and I'm surprised no one ever seems to talk about them. For an n-torus (here, n is the dimension of the surface, not the interior), the formulas are
S=τ^n times all the radii
V=1/2*τ^n times all the radii (with the last one squared)
These are extremely simple in terms of tau!
τ>π
1=0
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