# Self-Dual, Self-Petrie Covers of Regular Polyhedra

## Abstract

**:**

## 1. Introduction

## 2. Abstract Polyhedra

- Each flag of $\mathcal{P}$ consists of a vertex, an edge, and a face.
- Each edge is incident on exactly two vertices and two faces.
- If F is a vertex and G is a face such that $F\le G$, then there are exactly two edges that are incident to both F and G.
- $\mathcal{P}$ is strongly flag-connected, meaning that if $\mathsf{\Phi}$ and $\mathsf{\Psi}$ are two flags of $\mathcal{P}$, then there is a sequence of flags $\mathsf{\Phi}={\mathsf{\Phi}}_{0},{\mathsf{\Phi}}_{1},\dots ,{\mathsf{\Phi}}_{k}=\mathsf{\Psi}$ such that for $i=0,\dots ,k-1$, the flags ${\mathsf{\Phi}}_{i}$ and ${\mathsf{\Phi}}_{i+1}$ are adjacent, and each ${\mathsf{\Phi}}_{i}$ contains $\mathsf{\Phi}\cap \mathsf{\Psi}$.

#### 2.1. Duality Operations

## 3. Mixing Polyhedra

**Proposition**

**3.1.**

**Proof.**

**Corollary**

**3.2.**

**Proof.**

**Proposition**

**3.3.**

**Proof.**

**Theorem**

**3.4.**

**Proof.**

**Corollary**

**3.5.**

**Proof.**

**Theorem**

**3.6.**

**Proof.**

## 4. The Covers of Universal Polyhedra

## References

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$\mathcal{P}$ | $\left|\mathsf{\Gamma}\right(\mathcal{P}\left)\right|$ | $\left|\mathsf{\Gamma}\right({\mathcal{P}}^{\ast}\left)\right|$ | Method |
---|---|---|---|

${\{2,2k+1\}}_{4k+2}$ | $4(2k+1)$ | $8{(2k+1)}^{3}$ | Hand |

${\{2,4k\}}_{4k}$ | $16k$ | $32{k}^{3}$ | Hand |

${\{2,4k+2\}}_{4k+2}$ | $8(2k+1)$ | $8{(2k+1)}^{3}$ | Hand |

${\{3,3\}}_{4}$ | 24 | ${\left(24\right)}^{3}$ | Thm. 3.4 |

${\{3,4\}}_{6}$ | 48 | ${\left(48\right)}^{3}/8$ | GAP |

${\{3,5\}}_{5}$ | 60 | ${\left(60\right)}^{3}$ | Thm. 3.4 |

${\{3,5\}}_{10}$ | 120 | ${\left(120\right)}^{3}$ | GAP |

${\{3,6\}}_{6}$ | 108 | ${\left(108\right)}^{3}/216$ | GAP |

${\{3,7\}}_{8}$ | 336 | ${\left(336\right)}^{6}/8$ | Cor. 3.5 |

${\{3,7\}}_{9}$ | 504 | ${\left(504\right)}^{6}$ | Cor. 3.5 |

${\{3,7\}}_{13}$ | $1,092$ | ${(1,092)}^{6}$ | Cor. 3.5 |

${\{3,7\}}_{15}$ | $12,180$ | ${(12,180)}^{6}$ | Cor. 3.5 |

${\{3,7\}}_{16}$ | $21,504$ | ${(21,504)}^{6}/8$ | Cor. 3.5 |

${\{3,8\}}_{8}$ | 672 | ${\left(672\right)}^{3}/8$ | Thm. 3.6 |

${\{3,8\}}_{11}$ | $12,144$ | ${(12,144)}^{6}/8$ | Cor. 3.5 |

${\{3,9\}}_{9}$ | $3,420$ | ${(3,420)}^{3}$ | GAP |

${\{3,9\}}_{10}$ | $20,520$ | ${(20,520)}^{6}/216$ | Cor. 3.5 |

${\{4,4\}}_{4k}$ | $64{k}^{2}$ | $64{k}^{6}$ | Hand |

${\{4,4\}}_{4k+2}$ | $16{(2k+1)}^{2}$ | $32{(2k+1)}^{6}$ | Hand |

${\{4,5\}}_{5}$ | 160 | ${\left(160\right)}^{3}$ | Thm. 3.4 |

${\{4,5\}}_{9}$ | $6,840$ | ${(6,840)}^{6}/8$ | Cor. 3.5 |

© 2012 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Cunningham, G. Self-Dual, Self-Petrie Covers of Regular Polyhedra. *Symmetry* **2012**, *4*, 208-218.
https://doi.org/10.3390/sym4010208

**AMA Style**

Cunningham G. Self-Dual, Self-Petrie Covers of Regular Polyhedra. *Symmetry*. 2012; 4(1):208-218.
https://doi.org/10.3390/sym4010208

**Chicago/Turabian Style**

Cunningham, Gabe. 2012. "Self-Dual, Self-Petrie Covers of Regular Polyhedra" *Symmetry* 4, no. 1: 208-218.
https://doi.org/10.3390/sym4010208