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The Groups of the Regular Star-Polytopes

Published online by Cambridge University Press:  20 November 2018

Peter McMullen
Peter McMullen*
Affiliation:
University College London Gower Street London WC1E 6BT England, e-mail: p.mcmullen@ucl.ac.uk
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Abstract

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The regular star-polyhedron $\left\{ 5,\,\frac{5}{2} \right\}$ is isomorphic to the abstract polyhedron $\left\{ 5,\,5|3 \right\}$, where the last entry “3” in its symbol denotes the size of a hole, given by the imposition of a certain extra relation on the group of the hyperbolic honeycomb $\left\{ 5,\,5 \right\}$. Here, analogous formulations are found for the groups of the regular 4-dimensional star-polytopes, and for those of the non-discrete regular 4-dimensional honeycombs. In all cases, the extra group relations to be imposed on the corresponding Coxeter groups are those arising from “deep holes”; thus the abstract description of $\left\{ 5,\,{{3}^{k}},\,\frac{5}{2} \right\}\,\text{is}\,\left\{ 5,\,{{3}^{k}},\,5|3 \right\}\,\text{for}\,k\,=\,1\,\text{or}\,\text{2}$. The non-discrete quasi-regular honeycombs in ${{\mathbb{E}}^{3}}$, on the other hand, are not determined in an analogous way.

Keywords

51M2052C99
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 50 , Issue 2 , 01 April 1998 , pp. 426 - 448
Copyright
Copyright © Canadian Mathematical Society 1998

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