The Groups of the Regular Star-Polytopes

McMullen, Peter

Canadian journal of mathematics, Tome 50 (1998) no. 2, pp. 426-448

Résumé

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.

Détail

DOI : 10.4153/CJM-1998-023-7

Mots-clés : 51M20, 52C99

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