# The complete set of uniform polyhedra

@article{Skilling1975TheCS, title={The complete set of uniform polyhedra}, author={James Skilling}, journal={Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences}, year={1975}, volume={278}, pages={111 - 135} }

A definitive enumeration of all uniform polyhedra is presented (a uniform polyhedron has all vertices equivalent and all its faces regular polygons). It is shown that the set of uniform polyhedra presented by Coxeter, Longuet-Higgins & Miller (1953) is essentially complete. However, after a natural extension of their definition of a polyhedron to allow the coincidence of two or more edges, one extra polyhedron is found, namely the great disnub dirhombidodecahedron.

## 35 Citations

### " New " uniform polyhedra

- Mathematics, Art
- 2003

Definitions of polygons and polyhedra, more general than the one traditionally accepted, allow the construction of "new" uniform polyhedra. Although regular polyhedra, as well as some other classes…

### Uniform solution for uniform polyhedra

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An arbitrary precision solution of uniform polyhedra and their duals is presented. The solution is uniform for all polyhedra given by their kaleidoscopic construction, with no need to ‘examine’ each…

### Closed-Form Expressions for Uniform Polyhedra and Their Duals

- MathematicsDiscret. Comput. Geom.
- 2002

The main combinatorial and metrical quantities of uniform polyhedra and their duals are presented as closed-form expressions derived from the Wythoff symbols.

### Uniform compounds of uniform polyhedra

- ArtMathematical Proceedings of the Cambridge Philosophical Society
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Abstract The work of Coxeter, Longuet-Higgins and Miller (1953) and of Skilling (1975) is extended to give a complete list of uniform compounds of uniform polyhedra. Symmetry relationships between…

### Wythoffian Skeletal Polyhedra in Ordinary Space, I

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The blueprint for the construction is described and the Wythoffians for distinguished classes of regular polyhedra are treated, which are vertex-transitive and often feature vertex configurations with an attractive mix of different face shapes.

### Definitions in Polytopes

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Multi-shell clusters represent complex structures, the study of which needs rigorous definitions in graph theory, geometry, set theory, etc. Within this chapter, main definitions for polyhedra,…

### Vertex-Transitive Polyhedra of Higher Genus, I

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- 2017

It is shown that the symmetry group must also act regularly on the vertices, and the case of rotational tetrahedral symmetry is settled completely: the unique example is already known.

### Tiling with Regular Star Polygons

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The Archimedean tilings (Figure 1) and polyhedra will be familiar to many readers. They have the property that the tiles of the tiling, or the faces of the polyhedron, are regular polygons, and that…

### Isogonal Prismatoids

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- 1997

One of the aims is to point out the extent of inadequacies in the study of the polyhedra in which faces may be self-intersecting polygons, and distinct faces may intersect in various ways, by a discussion of isogonal prismatoids.

### STELLATION OF POLYHEDRA, AND COMPUTER IMPLEMENTATION

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The construction and display of graphic representations of stel- lated polyhedra requires a painstaking and close analysis of the geometrical objects generated by extending the facial planes of the…