List of uniform polyhedra

Uniform polyhedra and tilings form a well studied group. They are listed here for quick comparison of their properties and varied naming schemes and symbols.

This list includes:
* all 75 nonprismatic uniform polyhedra;
* a few representatives of the infinite sets of prisms and antiprisms;
* one special case polyhedron, Skilling's figure with overlapping edges.

Not included are:
* 40 potential uniform polyhedra with degenerate vertex figures which have overlapping edges (not counted by Coxeter);
* 11 uniform tessellations with convex faces;
* 14 uniform tilings with nonconvex faces;
* the infinite set of Uniform tilings in hyperbolic plane.

Table of polyhedra

The convex forms are listed in order of degree of vertex configurations from 3 faces/vertex and up, and in increasing sides per face. This ordering allows topological similarities to be show.

Convex forms (3 faces/vertex)

Nonconvex forms with convex faces

Special case

(*1) : The Great disnub dirhombidodecahedron has 120 edges shared by four faces. If counted as two pairs, then there are a total 360 edges. Because of this edge-degeneracy, it is not always considered a uniform polyhedron.

Column key

* Solid classes
** R = 5 Platonic solids
** R+= 4 Kepler-Poinsot polyhedra
** A = 13 Archimedean solids
** C+= 14 Non-convex polyhedra with only convex faces (all of these uniform polyhedra have faces which intersect each other)
** S+= 39 Non-convex polyhedra with complex (star) faces
** P = Infinite series of Convex Regular Prisms and Antiprisms
** P+= Infinite series of Non-convex uniform prisms and antiprisms (these all contain complex (star) faces)
** T = 11 Planar tessellations
* Bowers style acronym - A unique pronounceable abbreviated name created by mathematician "Jonathan Bowers"
* Uniform indexing: U01-U80 (Tetrahedron first, Prisms at 76+)
* Kaleido indexing: K01-K80 (prisms 1-5, Tetrahedron 6+)
* Magnus Wenninger Polyhedron Models: W001-W119
** 1-18 - 5 convex regular and 13 convex semiregular
** 20-22, 41 - 4 non-convex regular
** 19-66 Special 48 stellations/compounds (Nonregulars not given on this list)
** 67-119 - 53 non-convex uniform
* Chi: the Euler characteristic, χ. Uniform tilings on the plane correspond to a torus topology, with Euler characteristic of zero.
* For the plane tilings, the numbers given of vertices, edges and faces show the ratio of such elements in one period of the pattern, which in each case is a rhombus (sometimes a right-angled rhombus, i.e. a square).
* Note on Vertex figure

** The white polygon lines represent the "vertex figure" polygon. The colored faces are included on the vertex figure images help see their relations. Some of the intersecting faces are drawn visually incorrectly because they are not properly intersected visually to show which portions are in front.

References

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External links

* [http://www.software3d.com/Stella.php Stella: Polyhedron Navigator] - Software able to generate and print nets for all uniform polyhedra. Used to create most images on this page.
* [http://www.software3d.com/Uniform.php Paper models]

* Uniform indexing: U1-U80, (Tetrahedron first)
** [http://local.wasp.uwa.edu.au/~pbourke/geometry/polyhedra Uniform Polyhedra (80), Paul Bourke]
** http://mathworld.wolfram.com/UniformPolyhedron.html
** http://www.mathconsult.ch/showroom/unipoly
*** [http://www.mathconsult.ch/showroom/unipoly/list.html All uniform polyhedra by rotation group]
** http://gratrix.net/polyhedra/uniform/summary
** http://www.it-c.dk/edu/documentation/mathworks/math/math/u/u034.htm
** http://www.buddenbooks.com/jb/uniform/

* Kaleido Indexing: K1-K80 (Triangle prism first)
** http://www.math.technion.ac.il/~rl/kaleido
*** http://www.math.technion.ac.il/~rl/docs/uniform.pdf Uniform Solution for Uniform Polyhedra
** http://bulatov.org/polyhedra/uniform
** http://web.ukonline.co.uk/polyhedra/uniform/uniform.html

* Also
** http://www.polyedergarten.de/polyhedrix/e_klintro.htm