Hypercubes circumscribed in hyperspheres: a constant growth ratio for volumes at large dimensions

Details Hypercubes circumscribed in hyperspheres: a constant growth ratio for volumes at large dimensions Hypercubes circumscribed in hyperspheres: a constant growth ratio for volumes at large dimensions

2004-11-04

arxiv.org/abs/math/0411106v1

Mathematics

Metric Geometry

2 pages, 1 figure

Scientific paper

This note shows that an interesting property arises when considering the relation between the hypersphere volumes at dimensions $n+1$ and $n$, if the hyperspheres circumscribe unitary hypercubes in $n+1$ and $n$ dimensions, respectively . In the limit of large $n$, the ratio $\frac{V_{n+1}}{V_{n}}$ is constant and equal to the irrational number $\sqrt{\frac{\pi e}{2}}$. Some consequences of this result are explored.

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