Conformal Geometry and Dynamics

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Volume formulae for regular hyperbolic cubes

Author(s): T. H. Marshall
Journal: Conform. Geom. Dyn. 2 (1998), 25-28.
MSC (1991): Primary 51M10, 51M25, 52A35, 52A38; Secondary 05B45, 51M20, 52A40
Posted: February 11, 1998
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Abstract: We express the volume of a regular cube in hyperbolic $n$ -space as an integral on $[0, \infty)$ , and derive from this an asymptotic volume formula for the regular ideal hyperbolic $n$ -cube. This in turn is applied to finding an asymptotic lower bound for the least number of simplices into which a Euclidean $n$ -cube can be triangulated.

References:

1.: N.G. De Bruijn, Asymptotic Methods in Analysis, North-Holland, 1970.
2.: U. Haagerup and H. J. Munkholm, Simplices of Maximal Volume in Hyperbolic $n$ -Space, Acta Math. 147 (1981), 1-11. MR 82j:53116
3.: J. W. Milnor, `How to Compute Volume in Hyperbolic Space' in Collected Papers Vol. 1, Geometry (Publish or Perish, 1994).
4.: J. G. Ratcliffe, Foundations of Hyperbolic Manifolds, Springer-Verlag, 1994. MR 95j:57011
5.: W. D. Smith, Studies in Computational Geometry Motivated by Mesh Generation, Thesis, Princeton, 1989.


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