# Uniform polyhedron

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Uniform polyhedron

A uniform polyhedron is a polyhedron which has regular polygons as faces and is transitive on its vertices (i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the polyhedron has a high degree of reflectional and rotational symmetry.

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 sets, 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)
* 5 Platonic solids - regular convex polyhedra
* 4 Kepler-Poinsot polyhedra - regular nonconvex polyhedra
* 13 Archimedean solids - quasiregular and semiregular convex polyhedra
* 14 nonconvex polyhedra with convex faces
* 39 nonconvex polyhedra with nonconvex faces
* 1 polyhedron found by John Skilling with pairs of edges that coincide, called Great disnub dirhombidodecahedron (Skilling's figure).

They can also be grouped by their symmetry group, which is done below.

History

* The Platonic solids date back to the classical Greeks and were studied by Plato, Theaetetus and Euclid.
*Johannes Kepler (1571-1630) was the first to publish the complete list of Archimedean solids after the original work of Archimedes was lost.
* Kepler (1619) discovered two of the regular Kepler-Poinsot polyhedra and Louis 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 with J.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"Polyhedron models", which is the first published list of all 75 nonprismatic uniform polyhedra, with many previously unpublished names given to them by Norman Johnson.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 to Mathematica 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 with tetrahedral symmetry, 10-26 with Octahedral symmetry, 46-80 with icosahedral 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 triangles.

:"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. The octahedron 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 hosohedrons and dihedrons 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:
# Hosohedrons H2... (Only as spherical tilings)
# Dihedrons D2... (Only as spherical tilings)
# Prisms P3... (Truncated hosohedrons)
# Antiprisms A3... (Snub prisms)

Summary tables

(5 3 2) Ih Icosahedral symmetry

The 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 G2 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) D4hdihedral symmetry

There are 16 fundamental triangles, visible in the faces of the octagonal bipyramid and alternately colored triangles on a sphere::

*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]

*
* [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]

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