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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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Volume formulae for regular hyperbolic cubes
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by T. H. Marshall PDF
Conform. Geom. Dyn. 2 (1998), 25-28 Request permission

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
  • N.G. De Bruijn, Asymptotic Methods in Analysis, North-Holland, 1970.
  • Uffe Haagerup and Hans J. Munkholm, Simplices of maximal volume in hyperbolic $n$-space, Acta Math. 147 (1981), no. 1-2, 1–11. MR 631085, DOI 10.1007/BF02392865
  • J. W. Milnor, ‘How to Compute Volume in Hyperbolic Space’ in Collected Papers Vol. 1, Geometry (Publish or Perish, 1994).
  • John G. Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, vol. 149, Springer-Verlag, New York, 1994. MR 1299730, DOI 10.1007/978-1-4757-4013-4
  • 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
  • Received by editor(s): August 15, 1997
  • Received by editor(s) in revised form: November 26, 1997
  • Published electronically: February 11, 1998
  • © Copyright 1998 American Mathematical Society
  • 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
  • MathSciNet review: 1600384