List of uniform polyhedra by Wythoff symbol

Short description: none

Template:Polyhedron types There are many relations among the uniform polyhedra.

Here they are grouped by the Wythoff symbol.

## Contents

* 1 Key
* 2 Regular
* 2.1 Convex
* 2.2 Non-convex
* 3 Quasi-regular
* 4 Wythoff p q|r
* 4.1 Truncated regular forms
* 4.2 Hemipolyhedra
* 4.3 Rhombic quasi-regular
* 5 Even-sided forms
* 5.1 Wythoff p q r|
* 5.2 Wythoff p q (r s)|
* 6 Snub polyhedra
* 6.1 Wythoff |p q r
* 6.2 Wythoff |p q r s

## Key

Image
Name
Bowers pet name
V Number of vertices,E Number of edges,F Number of faces=Face configuration
?=Euler characteristic, group=Symmetry group
Wythoff symbol – Vertex figure
W – Wenninger number, U – Uniform number, K- Kaleido number, C -Coxeter number
alternative name
second alternative name

## Regular

All the faces are identical, each edge is identical and each vertex is identical. They all have a Wythoff symbol of the form p|q 2.

### Convex

The Platonic solids.

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### Non-convex

The Kepler-Poinsot solids.

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## Quasi-regular

Each edge is identical and each vertex is identical. There are two types of faces which appear in an alternating fashion around each vertex. The first row are semi-regular with 4 faces around each vertex. They have Wythoff symbol 2|p q. The second row are ditrigonal with 6 faces around each vertex. They have Wythoff symbol 3|p q or 3/2|p q.

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## Wythoff p q|r

### Truncated regular forms

Each vertex has three faces surrounding it, two of which are identical. These all have Wythoff symbols 2 p|q, some are constructed by truncating the regular solids.

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### Hemipolyhedra

The hemipolyhedra all have faces which pass through the origin. Their Wythoff symbols are of the form p p/m|q or p/m p/n|q. With the exception of the tetrahemihexahedron they occur in pairs, and are closely related to the semi-regular polyhedra, like the cuboctohedron.

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### Rhombic quasi-regular

Four faces around the vertex in the pattern p.q.r.q. The name rhombic stems from inserting a square in the cuboctahedron and icosidodecahedron. The Wythoff symbol is of the form p q|r.

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## Even-sided forms

### Wythoff p q r|

These have three different faces around each vertex, and the vertices do not lie on any plane of symmetry. They have Wythoff symbol p q r|, and vertex figures 2p.2q.2r.

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### Wythoff p q (r s)|

Vertex figure p.q.-p.-q. Wythoff p q (r s)|, mixing pqr| and pqs|.

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## Snub polyhedra

These have Wythoff symbol |p q r, and one non-Wythoffian construction is given |p q r s.

### Wythoff |p q r

Symmetry group  |  |  |

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O  |

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Ih |

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I  |

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I  |

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I  |

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### Wythoff |p q r s

Symmetry group

Ih  |

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Original source: https://en.wikipedia.org/wiki/List of uniform polyhedra by Wythoff symbol. Read more

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