Article contents
The Groups of the Regular Star-Polytopes
Published online by Cambridge University Press: 20 November 2018
Abstract
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.
- Type
- Research Article
- Information
- Copyright
- Copyright © Canadian Mathematical Society 1998
References
- 4
- Cited by