# Convex-Faced Combinatorially Regular Polyhedra of Small Genus

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. The Eight Maps and Their Polyhedra

#### 2.1. Klein’s Map

#### 2.2. Dyck’s Map

#### 2.3. Coxeter’s Geometric Skew Polyhedra

#### 2.4. Coxeter’s Topological Analogues of Skew Polyhedra

#### 2.5. The Klein–Fricke Map

#### 2.6. The Map ${\{3,10\}}_{6}$

## 3. Completeness of the List

**Lemma**

**3.1.**

**Proof.**

**Conjecture**

**3.2.**

## 4. Open Problems

**Genus 7 or 8.**The maps $R7.1$ of genus 7, as well as $R8.1$ and $R8.2$ of genus 8, are the only regular maps of genus 7 or 8 that could possibly admit a realization as a convex-faced polyhedron in ${\mathbb{E}}^{3}$ (all other maps violate the condition that ${m}_{V}={m}_{F}=1$). The map $R7.1$ of type $\{3,7\}$ has Petrie polygons of length 18 and a group of order 1008; however, most likely, $R7.1$ itself is not the universal map ${\{3,7\}}_{18}$. By contrast, $R8.1$ is the universal map ${\{3,8\}}_{8}$; its automorphism group is a semidirect product of the unitary reflection group ${\left[1\phantom{\rule{0.166667em}{0ex}}1\phantom{\rule{0.166667em}{0ex}}{1}^{4}\right]}^{4}$ in complex 3-space by ${C}_{2}$, and has order 672 (see pp. 296, 399 of Reference [6]). The map $R8.2$ of type $\{3,8\}$ has Petrie polygons of length 14 and again a group of order 672; however, $R8.2$ itself is not the universal map ${\{3,8\}}_{14}$, which is known to be infinite (see p. 399 of Reference [6]). Thus all three maps have much larger automorphism groups than the maps in Table 1. It would be desirable to decide whether or not $R7.1$, $R8.1$, and $R8.2$ admit a polyhedral embedding in ${\mathbb{E}}^{3}$, and if so, to explicitly construct a polyhedron with maximum possible geometric symmetry.

**Leonardo polyhedra.**Among the eight combinatorially regular maps of Table 1, exactly four admit realizations as Leonardo polyhedra in the sense of [29], meaning that the symmetry group of the polyhedron is either the full symmetry group of a Platonic solid or its rotation subgroup. Very few Leonardo polyhedra with large combinatorial automorphism groups seem to be known, and it would be a worthwhile task to find more, particularly Leonardo polyhedra which are combinatorially regular.

**Icosahedral symmetry.**Finally, among all combinatorially regular polyhedra in ${\mathbb{E}}^{3}$ discovered so far, none except the icosahedron has full icosahedral symmetry or icosahedral rotation symmetry. Any combinatorially regular polyhedra with icosahedral symmetry would likely be very interesting. As mentioned earlier, when self-intersections are permitted, full icosahedral symmetry can be obtained, for example, for Gordan’s maps.

## Acknowledgment

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Genus g | Name or Type | Group Order | Face Vector $({\mathit{f}}_{0},{\mathit{f}}_{1},{\mathit{f}}_{2})$ | References | Map of [11] | Symmetry Group | Max. |
---|---|---|---|---|---|---|---|

3 | ${\{3,7\}}_{8}$ | 336 | (24,84,56) | Klein [17,18] | $R3.1$ | ${[3,3]}^{+}$ | |

3 | ${\{3,8\}}_{6}$ | 192 | (12,48,32) | Dyck [15,16] | $R3.2$ | ${D}_{3}$ | ∗ |

5 | $\{3,8\}$ | 384 | (24,96,64) | Klein and Fricke [19] | $R5.1$ | ${[3,4]}^{+}$ | ∗ |

5 | $\{4,5|4\}$ | 320 | (32,80,40) | Coxeter [10] | $R5.3$ | ${C}_{2}^{3}$ | |

5 | $\{5,4|4\}$ | 320 | (40,80,32) | Coxeter [10] | $R{5.3}^{\ast}$ | ${C}_{2}^{3}$ | |

6 | ${\{3,10\}}_{6}$ | 300 | (15,75,50) | Coxeter and Moser [5] | $R6.1$ | ${D}_{3}$ | ∗ |

6 | $\{4,6|3\}$ | 240 | (20,60,30) | Coxeter [10], | $R6.2$ | $[3,3]$ | ∗ |

Boole Stott [23] | |||||||

6 | $\{6,4|3\}$ | 240 | (30,60,20) | Coxeter [10], | $R{6.2}^{\ast}$ | $[3,3]$ | ∗ |

Boole Stott [23] |

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Schulte, E.; Wills, J.M.
Convex-Faced Combinatorially Regular Polyhedra of Small Genus. *Symmetry* **2012**, *4*, 1-14.
https://doi.org/10.3390/sym4010001

**AMA Style**

Schulte E, Wills JM.
Convex-Faced Combinatorially Regular Polyhedra of Small Genus. *Symmetry*. 2012; 4(1):1-14.
https://doi.org/10.3390/sym4010001

**Chicago/Turabian Style**

Schulte, Egon, and Jörg M. Wills.
2012. "Convex-Faced Combinatorially Regular Polyhedra of Small Genus" *Symmetry* 4, no. 1: 1-14.
https://doi.org/10.3390/sym4010001