List Of Uniform Polyhedra

COLUMN KEY

Solid classes

--- R = 5 Platonic Solid s

--- R+= 4 Kepler-Poinsot Solid s

--- A = 13 Archimedean Solid s

--- 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 Antiprism s (these all contain complex (star) faces)

--- T = 11 Planar Tessellation s

Bowers Style Acronym

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 polyhedron pictures:

--- Pictures were generated from VRML model viewing software. Some Star Polygon faced polyhedra were drawn using a binary filling algorithm, so the center of a pentagram is not filled. Although the definition of "interior" is ambiguous for self-intersecting faces, Polyhedronists usually consider the multileveled interior as solid faced.

Note on Vertex figure images:

--- 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.

TABLE OF POLYHEDRA

Regular and convex forms (3 faces/vertex)

Regular and convex forms (4 faces/vertex)

Regular and convex forms (5 faces/vertex)

Regular and convex forms (6 faces/vertex)

Nonconvex forms with convex faces

Nonconvex prismatic forms

Nonconvex forms with nonconvex 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.

REFERENCE

EXTERNAL LINKS

Stella: Polyhedron Navigator - Software for generating and printing nets for all uniform polyhedra

Paper models

Uniform indexing: U1-U80, (Tetrahedron first)

--- http://mathworld.wolfram.com/UniformPolyhedron.html

--- http://www.mathconsult.ch/showroom/unipoly

-- 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://home.earthlink.net/~jbuddenh/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://www.physics.orst.edu/~bulatov/polyhedra/uniform

--- http://web.ukonline.co.uk/polyhedra/uniform/uniform.html

Also

--- http://www.polyhedrix.de/e_klintro.htm

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List Of Uniform Polyhedra