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. |
Keywords
Hyperbolic simplex, Volume, Ball packing
Math Review Classification
Primary 30F40, 57N10
; Secondary 20F55
Last Updated
21/3/97
Length
14 pages
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