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Conformal Geometry and Dynamics

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Volume formulae for regular hyperbolic cubes


Author: T. H. Marshall
Journal: Conform. Geom. Dyn. 2 (1998), 25-28
MSC (1991): Primary 51M10, 51M25, 52A35, 52A38; Secondary 05B45, 51M20, 52A40
DOI: https://doi.org/10.1090/S1088-4173-98-00025-3
Published electronically: February 11, 1998
MathSciNet review: 1600384
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Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 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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Additional Information

T. H. Marshall
Affiliation: Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand
Email: t_marshall@math.auckland.ac.nz

DOI: https://doi.org/10.1090/S1088-4173-98-00025-3
Keywords: Hyperbolic cube, volume, simplex, triangulation
Received by editor(s): August 15, 1997
Received by editor(s) in revised form: November 26, 1997
Published electronically: February 11, 1998
Article copyright: © Copyright 1998 American Mathematical Society

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