- Uniform polyhedron
A

**uniform polyhedron**is apolyhedron which hasregular polygon s as faces and istransitive on its vertices (i.e. there is anisometry mapping any vertex onto any other). It follows that all vertices are congruent, and the polyhedron has a high degree of reflectional and rotationalsymmetry .Uniform polyhedra may be regular, quasi-regular or semi-regular. The faces and vertices need not be convex, so many of the uniform polyhedra are also star polyhedra.

Excluding the

infinite set s, there are 75 uniform polyhedra (or 76 if edges are allowed to coincide).Categories include:

* Infinite sets of uniform prisms and antiprisms (including star forms)

* 5Platonic solid s - regular convex polyhedra

* 4Kepler-Poinsot polyhedra - regular nonconvex polyhedra

* 13Archimedean solid s - quasiregular and semiregular convex polyhedra

* 14 nonconvex polyhedra with convex faces

* 39 nonconvex polyhedra with nonconvex faces

* 1 polyhedron found byJohn Skilling with pairs of edges that coincide, calledGreat disnub dirhombidodecahedron (Skilling's figure).They can also be grouped by their

symmetry group , which is done below.**History*** The

Platonic solid s date back to the classical Greeks and were studied byPlato , Theaetetus andEuclid .

*Johannes Kepler (1571-1630) was the first to publish the complete list ofArchimedean solid s after the original work ofArchimedes was lost.

* Kepler (1619) discovered two of the regularKepler-Poinsot polyhedra andLouis Poinsot (1809) discovered the other two.

* Of the remaining 66, Albert Badoureau (1881) discovered 37.Edmund Hess (1878) discovered 2 more and Pitsch (1881) independently discovered 18, of which 15 had not previously been discovered.

* The famous geometer Donald Coxeter discovered the remaining twelve in collaboration withJ.C.P. Miller (1930-1932) but did not publish. M.S. and H.C. Longuet-Higgins and independently discovered 11 of these.

* In 1954 H.S.M. Coxeter, M.S. Longuet-Higgins, J.C.P. Miller published the list of uniform polyhedra.

* In 1970 S. P. Sopov proved their conjecture that the list was complete.

* In 1974,Magnus Wenninger published his book "Polyhedron models", which is the first published list of all 75 nonprismatic uniform polyhedra, with many previously unpublished names given to them by Norman Johnson.

* In 1975,John Skilling independently proved the completeness, and showed that if the definition of uniform polyhedron is relaxed to allow edges to coincide then there is just one extra possibility.

* In 1993, Zvi Har'El produced a complete kaleidoscopic construction of the uniform polyhedra and duals with a computer program called**Kaleido**, and summarized in a paper "Uniform Solution for Uniform Polyhedra", counting figures 1-80.

* Also in 1993, R. Mäder ported this Kaleido solution toMathematica with a slightly different indexing system.**Indexing**Four numbering schemes for the uniform polyhedra are in common use, distinguished by letters:

* [

**C**] Coxeter et al., 1954, showed the convex forms as figures 15 through 32; three prismatic forms, figures 33–35; and the nonconvex forms, figures 36–92.

* [**W**] Wenninger, 1974, has 119 figures: 1-5 for the Platonic solids, 6-18 for the Archimedean solids, 19-66 for stellated forms including the 4 regular nonconvex polyhedra, and ended with 67-119 for the nonconvex uniform polyhedra.

* [**K**] Kaleido, 1993: The 80 figures were grouped by symmetry: 1-5 as representatives of the infinite families of prismatic forms with dihedral symmetry, 6-9 withtetrahedral symmetry , 10-26 withOctahedral symmetry , 46-80 withicosahedral symmetry .

* [**U**] Mathematica, 1993, follows the Kaleido series with the 5 prismatic forms moved to last, so that the nonprismatic forms become 1–75.**Nonconvex uniform polyhedra**The 57 nonprismatic nonconvex forms are compiled by Wythoff constructions within

Schwarz triangle s.:"Main article:

Nonconvex uniform polyhedron "**Convex forms by Wythoff construction**The convex uniform polyhedra can be named by

Wythoff construction operations and can be named in relation to the regular form.In more detail the convex uniform polyhedron are given below by their

Wythoff construction within each symmetry group.Within the Wythoff construction, there are repetitions created by lower symmetry forms. The cube is a regular polyhedron, and a square prism. The

octahedron is a regular polyhedron, and a triangular antiprism. Theoctahedron is also a "rectified tetrahedron". Many polyhedra are repeated from different construction sources and are colored differently.The Wythoff construction applies equally to uniform polyhedra and uniform tilings on the surface of a sphere, so images of both are given. The spherical tilings including the set of

hosohedron s anddihedron s which are degenerate polyhedra.These symmetry groups are formed from the reflectional

point groups in three dimensions , each represented by a fundamental triangle (p q r), where p>1, q>1, r>1 and 1/p+1/q+1/r<1.*

Tetrahedral symmetry (3 3 2) - order 24

*Octahedral symmetry (4 3 2) - order 48

*Icosahedral symmetry (5 3 2) - order 60

*Dihedral symmetry (n 2 2), for all n=3,4,5,... - order "4n"The remaining nonreflective forms are constructed by alternation operations applied to the polyhedra with an even number of sides.

Along with the prisms and their

dihedral symmetry , the spherical Wythoff construction process adds two "regular" classes which become degenerate as polyhedra - the "dihedra" and hosohedra, the first having only two faces, and the second only two vertices. The truncation of the regular "hosohedra" creates the prisms.Below the convex uniform polyhedra are indexed 1-18 for the nonprismatic forms as they are presented in the tables by symmetry form. Repeated forms are in brackets.

For the infinite set of prismatic forms, they are indexed in four families:

#Hosohedron s H_{2...}(Only as spherical tilings)

#Dihedron s D_{2...}(Only as spherical tilings)

# Prisms P_{3...}(Truncated hosohedrons)

#Antiprism s A_{3...}(Snub prisms)**Summary tables****(5 3 2) I**_{h}Icosahedral symmetryThe

icosahedral symmetry of the sphere generates 7 uniform polyhedra, and a 1 more by alternation. Only one is repeated from the tetrahedral and octahedral symmetry table above.The icosahedral symmetry is represented by a fundamental triangle (5 3 2) counting the mirrors at each vertex. It can also be represented by the

Coxeter group G_{2}or [5,3] , as well as a .There are 60 triangles, visible in the faces of the

disdyakis triacontahedron and alternately colored triangles on a sphere::**(4 2 2) D**_{4h}dihedral symmetryThere are 16 fundamental triangles, visible in the faces of the

octagonal bipyramid and alternately colored triangles on a sphere::**See also***

Polyhedron

**Regular polyhedron

**Quasiregular polyhedron

**Semiregular polyhedron

*List of uniform polyhedra

*List of Wenninger polyhedron models

*Polyhedron model

*List of uniform polyhedra by vertex figure

*List of uniform polyhedra by Wythoff symbol **References***Brückner, M. "Vielecke und vielflache. Theorie und geschichte.". Leipzig, Germany: Teubner, 1900. [

*http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=ABN8316.0001.001*]

*H.S.M. Coxeter, M.S. Longuet-Higgins, J.C.P. Miller, "Uniform polyhedra",**Phil. Trans.**1954, 246 A, 401-50 [*http://links.jstor.org/sici?sici=0080-4614%2819540513%29246%3A916%3C401%3AUP%3E2.0.CO%3B2-4*]

*S. P. Sopov "A proof of the completeness on the list of elementary homogeneous polyhedra." (Russian) Ukrain. Geometr. Sb. No. 8, (1970), 139-156

*

*John Skilling , "The complete set of uniform polyhedra.", Philos. Trans. Roy. Soc. London Ser. A 278 (1975), 111-135 [*http://links.jstor.org/sici?sici=0080-4614%2819750306%29278%3A1278%3C111%3ATCSOUP%3E2.0.CO%3B2-F*]

* Har'El, Z. [*http://www.math.technion.ac.il/~rl/docs/uniform.pdf "Uniform Solution for Uniform Polyhedra."*] , Geometriae Dedicata 47, 57-110, 1993. [*http://www.math.technion.ac.il/~rl Zvi Har’El*] , [*http://www.math.technion.ac.il/~rl/kaleido Kaleido software*] , [*http://www.math.technion.ac.il/~rl/kaleido/poly.html Images*] , [*http://www.math.technion.ac.il/~rl/kaleido/dual.html dual images*]

* [*http://www.mathconsult.ch/showroom/unipoly Mäder, R. E.*] "Uniform Polyhedra." Mathematica J. 3, 48-57, 1993. [*http://library.wolfram.com/infocenter/Articles/2254*]**External links***

* [*http://www.math.technion.ac.il/~rl/docs/uniform.pdf Uniform Solution for Uniform Polyhedra*]

* [*http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra*]

* [*http://www.georgehart.com/virtual-polyhedra/uniform-info.html Virtual Polyhedra*] Uniform Polyhedra

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

* Paper models:

** [*http://www.software3d.com/Uniform.php Uniform/Dual Polyhedra*]

** [*http://www.polyedergarten.de/ Paper Models of Uniform (and other) Polyhedra*]

* [*http://www.tfh-berlin.de/~s17299/applets/polyh2.html uniform polyhedra in 3d*]

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Uniform polyhedron compound**— A uniform polyhedron compound is a polyhedral compound whose constituents are identical (although possibly enantiomorphous) uniform polyhedra, in an arrangement that is also uniform: the symmetry group of the compound acts transitively on the… … Wikipedia**Nonconvex uniform polyhedron**— In geometry, a nonconvex uniform polyhedron, or uniform star polyhedron, is a self intersecting uniform polyhedron. Each can contain either star polygon faces, star polygon vertex figures or both.The complete set of 57 nonprismatic uniform star… … Wikipedia**Prismatic uniform polyhedron**— A pentagrammic antiprism is made of two regular pentagrams and 10 equilateral triangles. In geometry, a prismatic uniform polyhedron is a uniform polyhedron with dihedral symmetry. They exist in two infinite families, the uniform prisms and the… … Wikipedia**Polyhedron**— Polyhedra redirects here. For the relational database system, see Polyhedra DBMS. For the game magazine, see Polyhedron (magazine). For the scientific journal, see Polyhedron (journal). Some Polyhedra Dodecahedron (Regular polyhedron) … Wikipedia**Uniform star polyhedron**— A display of uniform polyhedra at the Science Museum in London … Wikipedia**Uniform polytope**— A uniform polytope is a vertex transitive polytope made from uniform polytope facets. A uniform polytope must also have only regular polygon faces.Uniformity is a generalization of the older category semiregular, but also includes the regular… … Wikipedia**Uniform coloring**— In geometry, a uniform coloring is a property of a uniform figure (uniform tiling or uniform polyhedron) that is colored to be vertex transitive. Different symmetries can be expressed on the same geometric figure with the faces following… … Wikipedia**Uniform great rhombicosidodecahedron**— In geometry, the uniform great rhombicosidodecahedron is a nonconvex uniform polyhedron, indexed as U67. It is also called the quasirhombicosidodecahedron .This model shares the name with the convex great rhombicosidodecahedron, which is also… … Wikipedia**Uniform great rhombicuboctahedron**— In geometry, the Uniform great rhombicuboctahedron is a nonconvex uniform polyhedron, indexed as U17.It shares the vertex arrangement with the convex truncated cube.This model shares the name with the convex great rhombicuboctahedron, which is… … Wikipedia**Polyhedron model**— A polyhedron model is a physical construction of a polyhedron, constructed from cardboard, plastic board, wood board or other panel material, or, less commonly, solid material.Since there are 75 uniform polyhedra, including the five regular… … Wikipedia