Asymptotic Volume Formulae and Hyperbolic Ball Packing

T.H. Marshall


We prove that the volume of an n-dimensional regular spherical
simplex of edge length $r< pi/2$ is asymptotically ${rm
e}^{-x} x^{n/2} {(n+1)}^{1/2}/n!$, as $n rightarrow infty$, where $x=sec r
-1$. The same is true for hyperbolic simplices if we set $x=1-{rm
sech} r$ and replace
${rm e}^{-x}$ by
${rm e}^x$.

We obtain error bounds for this asymptotic, and apply it to
find an upper bound for the density of ball packings of balls of a
given radius in hyperbolic n-space, for all sufficiently large n.

Hyperbolic simplex, Volume, Ball packing

Math Review Classification
Primary 30F40, 57N10 ; Secondary 20F55

Last Updated

14 pages

This article is available in: