Volume formulae for regular hyperbolic cubes
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- by T. H. Marshall
- Conform. Geom. Dyn. 2 (1998), 25-28
- DOI: https://doi.org/10.1090/S1088-4173-98-00025-3
- Published electronically: 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
- 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.
Bibliographic 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