Uniform polyhedron

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Wythoffian construction diagram.png
Example forms from the cube and octahedron

A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, 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 (if also face and edge transitive), quasi-regular (if edge transitive but not face transitive) or semi-regular (if neither edge nor face transitive). 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).

There are also two infinite sets of uniform prisms and antiprisms, including convex and star forms.

Dual polyhedra to uniform polyhedra are face-transitive (isohedral) and have regular vertex figures, and are generally classified in parallel with their dual (uniform) polyhedron. The dual of a regular polyhedron is regular, while the dual of an Archimedean solid is a Catalan solid.

The concept of uniform polyhedron is a special case of the concept of uniform polytope, which also applies to shapes in higher-dimensional (or lower-dimensional) space.

History[edit]

Regular star polyhedra:

Other 53 nonregular star polyhedra:

  • Of the remaining 53, Albert Badoureau (1881) discovered 36. Edmund Hess (1878) discovered 2 more and Pitsch (1881) independently discovered 18, of which 15 had not previously been discovered.
  • The geometer H.S.M. 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.
  • Coxeter, Longuet-Higgins & Miller (1954) published the list of uniform polyhedra.
  • Sopov (1970) proved their conjecture that the list was complete.
  • In 1974, Magnus Wenninger published his book Polyhedron models, which lists all 75 nonprismatic uniform polyhedra, with many previously unpublished names given to them by Norman Johnson.
  • Skilling (1975) 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 1987, Edmond Bonan draw all the uniform polyhedra and their duals in 3D, with a Turbo Pascal program called Polyca : almost of them were shown during the International Stereoscopic Union Congress held at the Congress Theatre, Eastbourne, United Kingdom.
  • 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.
  • In 2002 Peter W. Messer discovered a minimal set of closed-form expressions for determining the main combinatorial and metrical quantities of any uniform polyhedron (and its dual) given only its Wythoff symbol.[1]

Uniform star polyhedra[edit]

The 57 nonprismatic nonconvex forms are compiled by Wythoff constructions within Schwarz triangles.

Convex forms by Wythoff construction[edit]

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.

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:

  1. Hosohedrons H2... (Only as spherical tilings)
  2. Dihedrons D2... (Only as spherical tilings)
  3. Prisms P3... (Truncated hosohedrons)
  4. Antiprisms A3... (Snub prisms)

Summary tables[edit]

Johnson name Parent Truncated Rectified Bitruncated
(tr. dual)
Birectified
(dual)
Cantellated Omnitruncated
(Cantitruncated)
Snub
Extended
Schläfli symbol
\begin{Bmatrix} p , q \end{Bmatrix} t\begin{Bmatrix} p , q \end{Bmatrix} \begin{Bmatrix} p \\ q \end{Bmatrix} t\begin{Bmatrix} q , p \end{Bmatrix} \begin{Bmatrix} q , p \end{Bmatrix} r\begin{Bmatrix} p \\ q \end{Bmatrix} t\begin{Bmatrix} p \\ q \end{Bmatrix} s\begin{Bmatrix} p \\ q \end{Bmatrix}
{p,q} t{p,q} r{p,q} 2t{p,q} 2r{p,q} rr{p,q} tr{p,q} sr{p,q}
t0{p,q} t0,1{p,q} t1{p,q} t1,2{p,q} t2{p,q} t0,2{p,q} t0,1,2{p,q} ht0,1,2{p,q}
Wythoff symbol
(p q 2)
q | p 2 2 q | p 2 | p q 2 p | q p | q 2 p q | 2 p q 2 | | p q 2
Coxeter diagram CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png
CDel node 1.pngCDel split1-pq.pngCDel nodes.png
CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png
CDel node.pngCDel split1-pq.pngCDel nodes 11.png
CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png
CDel node 1.pngCDel split1-pq.pngCDel nodes 11.png
CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png
CDel node h.pngCDel split1-pq.pngCDel nodes hh.png
Vertex figure pq (q.2p.2p) (p.q)2 (p.2q.2q) qp (p.4.q.4) (4.2p.2q) (3.3.p.3.q)
Tetrahedral
(3 3 2)
Uniform polyhedron-33-t0.png
{3,3}
Uniform polyhedron-33-t01.png
(3.6.6)
Uniform polyhedron-33-t1.png
(3.3.3.3)
Uniform polyhedron-33-t12.png
(3.6.6)
Uniform polyhedron-33-t2.png
{3,3}
Uniform polyhedron-33-t02.png
(3.4.3.4)
Uniform polyhedron-33-t012.png
(4.6.6)
Uniform polyhedron-33-s012.svg
(3.3.3.3.3)
Octahedral
(4 3 2)
Uniform polyhedron-43-t0.svg
{4,3}
Uniform polyhedron-43-t01.svg
(3.8.8)
Uniform polyhedron-43-t1.svg
(3.4.3.4)
Uniform polyhedron-43-t12.svg
(4.6.6)
Uniform polyhedron-43-t2.svg
{3,4}
Uniform polyhedron-43-t02.png
(3.4.4.4)
Uniform polyhedron-43-t012.png
(4.6.8)
Uniform polyhedron-43-s012.png
(3.3.3.3.4)
Icosahedral
(5 3 2)
Uniform polyhedron-53-t0.png
{5,3}
Uniform polyhedron-53-t01.png
(3.10.10)
Uniform polyhedron-53-t1.png
(3.5.3.5)
Uniform polyhedron-53-t12.png
(5.6.6)
Uniform polyhedron-53-t2.png
{3,5}
Uniform polyhedron-53-t02.png
(3.4.5.4)
Uniform polyhedron-53-t012.png
(4.6.10)
Uniform polyhedron-53-s012.png
(3.3.3.3.5)

And a sampling of Dihedral symmetries:

(p 2 2) Parent Truncated Rectified Bitruncated
(tr. dual)
Birectified
(dual)
Cantellated Omnitruncated
(Cantitruncated)
Snub
Extended
Schläfli symbol
\begin{Bmatrix} p , 2 \end{Bmatrix} t\begin{Bmatrix} p , 2 \end{Bmatrix} \begin{Bmatrix} p \\ 2 \end{Bmatrix} t\begin{Bmatrix} 2 , p \end{Bmatrix} \begin{Bmatrix} 2 , p \end{Bmatrix} r\begin{Bmatrix} p \\ 2 \end{Bmatrix} t\begin{Bmatrix} p \\ 2 \end{Bmatrix} s\begin{Bmatrix} p \\ 2 \end{Bmatrix}
{p,2} t{p,2} r{p,2} 2t{p,2} 2r{p,2} rr{p,2} tr{p,2} sr{p,2}
t0{p,2} t0,1{p,2} t1{p,2} t1,2{p,2} t2{p,2} t0,2{p,2} t0,1,2{p,2} ht0,1,2{p,2}
Wythoff symbol 2 | p 2 2 2 | p 2 | p 2 2 p | 2 p | 2 2 p 2 | 2 p 2 2 | | p 2 2
Coxeter-Dynkin diagram CDel node 1.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel 2.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel 2.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel 2.pngCDel node 1.png CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel p.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel 2.pngCDel node 1.png CDel node h.pngCDel p.pngCDel node h.pngCDel 2x.pngCDel node h.png
Vertex figure p2 (2.2p.2p) (p. 2.p. 2) (p. 4.4) 2p (p. 4.2.4) (4.2p.4) (3.3.p. 3.2)
Dihedral
(2 2 2)
Digonal dihedron.png
{2,2}
2.4.4 Digonal dihedron.png
2.2.2.2

4.4.2
Digonal dihedron.png
{2,2}
2.4.2.4 Spherical square prism2.png
4.4.4
Spherical digonal antiprism.png
3.3.3.2
Dihedral
(3 2 2)
Trigonal dihedron.png
{3,2}
Hexagonal dihedron.png
2.6.6
2.3.2.3 Spherical triangular prism.png
4.4.3
Spherical trigonal hosohedron.png
{2,3}
Spherical triangular prism.png
2.4.3.4
Spherical hexagonal prism2.png
4.4.6
Spherical trigonal antiprism.png
3.3.3.3
Dihedral
(4 2 2)
{4,2} 2.8.8 2.4.2.4 Spherical square prism.png
4.4.4
Spherical square hosohedron.png
{2,4}
Spherical square prism.png
2.4.4.4
Spherical octagonal prism2.png
4.4.8
Spherical square antiprism.png
3.3.3.4
Dihedral
(5 2 2)
{5,2} 2.10.10 2.5.2.5 Spherical pentagonal prism.png
4.4.5
Spherical pentagonal hosohedron.png
{2,5}
Spherical pentagonal prism.png
2.4.5.4
Spherical decagonal prism2.png
4.4.10
Spherical pentagonal antiprism.png
3.3.3.5
Dihedral
(6 2 2)
Hexagonal dihedron.png
{6,2}
Dodecagonal dihedron.png
2.12.12
Hexagonal dihedron.png
2.6.2.6
Spherical hexagonal prism.png
4.4.6
Spherical hexagonal hosohedron.png
{2,6}
Spherical hexagonal prism.png
2.4.6.4
Spherical dodecagonal prism2.png
4.4.12
Spherical hexagonal antiprism.png
3.3.3.6

Wythoff construction operators[edit]

Operation Symbol Coxeter
diagram
Description
Parent {p,q}
t0{p,q}
CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png Any regular polyhedron or tiling
Rectified (r) r{p,q}
t1{p,q}
CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png The edges are fully truncated into single points. The polyhedron now has the combined faces of the parent and dual.
Birectified (2r)
(also dual)
2r{p,q}
t2{p,q}
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png
Dual Cube-Octahedron.svg
The birectified (dual) is a further truncation so that the original faces are reduced to points. New faces are formed under each parent vertex. The number of edges is unchanged and are rotated 90 degrees. The dual of the regular polyhedron {p, q} is also a regular polyhedron {q, p}.
Truncated (t) t{p,q}
t0,1{p,q}
CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png Each original vertex is cut off, with a new face filling the gap. Truncation has a degree of freedom, which has one solution that creates a uniform truncated polyhedron. The polyhedron has its original faces doubled in sides, and contains the faces of the dual.
Cube truncation sequence.svg
Bitruncated (2t)
(also truncated dual)
2t{p,q}
t1,2{p,q}
CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png Same as truncated dual.
Cantellated (rr)
(Also expanded)
rr{p,q} CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is half way between both the parent and dual forms.
Cube cantellation sequence.svg
Cantitruncated (tr)
(Also omnitruncated)
tr{p,q}
t0,1,2{p,q}
CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png The truncation and cantellation operations are applied together to create an omnitruncated form which has the parent's faces doubled in sides, the dual's faces doubled in sides, and squares where the original edges existed.
Alternation operations
Operation Symbol Coxeter
diagram
Description
Snub rectified (sr) sr{p,q} CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png The alternated cantitruncated. All the original faces end up with half as many sides, and the squares degenerate into edges. Since the omnitruncated forms have 3 faces/vertex, new triangles are formed. Usually these alternated faceting forms are slightly deformed thereafter in order to end again as uniform polyhedra. The possibility of the latter variation depends on the degree of freedom.
Snubcubes in grCO.svg
Snub (s) s{p,2q} CDel node h.pngCDel p.pngCDel node h.pngCDel 2x.pngCDel q.pngCDel node.png Alternated truncation
Cantic snub (s2) s2{p,2q} CDel node h.pngCDel p.pngCDel node h.pngCDel 2x.pngCDel q.pngCDel node 1.png
Alternated cantellation (hrr) hrr{2p,2q} CDel node h.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 2x.pngCDel q.pngCDel node h.png Only possible in uniform tilings (infinite polyhedra), alternation of CDel node 1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 2x.pngCDel q.pngCDel node 1.png
For example, CDel node h.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h.png
Half (h) h{2p,q} CDel node h1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png Alternation of CDel node 1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png, same as CDel labelp.pngCDel branch 10ru.pngCDel split2-qq.pngCDel node.png
Cantic (h2) h2{2p,q} CDel node h1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png Same as CDel labelp.pngCDel branch 10ru.pngCDel split2-qq.pngCDel node 1.png
Half rectified (hr) hr{2p,2q} CDel node.pngCDel 2x.pngCDel p.pngCDel node h1.pngCDel 2x.pngCDel q.pngCDel node.png Only possible in uniform tilings (infinite polyhedra), alternation of CDel node.pngCDel 2x.pngCDel p.pngCDel node 1.pngCDel 2x.pngCDel q.pngCDel node.png, same as CDel labelp.pngCDel branch 10ru.pngCDel 2a2b-cross.pngCDel branch 10lu.pngCDel labelq.png or CDel labelp.pngCDel branch 10r.pngCDel iaib.pngCDel branch 01l.pngCDel labelq.png
For example, CDel node.pngCDel 4.pngCDel node h1.pngCDel 4.pngCDel node.png = CDel nodes 10ru.pngCDel 2a2b-cross.pngCDel nodes 10lu.png or CDel nodes 11.pngCDel iaib.pngCDel nodes.png
Quarter (q) q{2p,2q} CDel node h1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 2x.pngCDel q.pngCDel node h1.png Only possible in uniform tilings (infinite polyhedra), same as CDel labelq.pngCDel branch 11.pngCDel papb-cross.pngCDel branch 10l.pngCDel labelq.png
For example, CDel node h1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h1.png = CDel nodes 11.pngCDel 2a2b-cross.pngCDel nodes 10lu.png or CDel nodes 11.pngCDel iaib.pngCDel nodes 10l.png

(3 3 2) Td Tetrahedral symmetry[edit]

The tetrahedral symmetry of the sphere generates 5 uniform polyhedra, and a 6th form by a snub operation.

The tetrahedral symmetry is represented by a fundamental triangle with one vertex with two mirrors, and two vertices with three mirrors, represented by the symbol (3 3 2). It can also be represented by the Coxeter group A2 or [3,3], as well as a Coxeter-Dynkin diagram: CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png.

There are 24 triangles, visible in the faces of the tetrakis hexahedron and alternately colored triangles on a sphere:

Tetrakishexahedron.jpg Tetrahedral reflection domains.pngSphere symmetry group td.png
# Name Graph
A3
Graph
A2
Picture Tiling Vertex
figure
Coxeter-Dynkin
and Schläfli
symbols
Face counts by position Element counts
Pos. 2
CDel node.pngCDel 3.pngCDel node.png
[3]
(4)
Pos. 1
CDel node.pngCDel 2.pngCDel node.png
[2]
(6)
Pos. 0
CDel 2.pngCDel node.pngCDel 3.pngCDel node.png
[3]
(4)
Faces Edges Vertices
1 Tetrahedron 3-simplex t0.svg 3-simplex t0 A2.svg Uniform polyhedron-33-t0.png Uniform tiling 332-t0-1-.png Tetrahedron vertfig.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{3,3}
Regular polygon 3.svg
{3}
4 6 4
[1] Birectified tetrahedron
(Same as tetrahedron)
3-simplex t0.svg 3-simplex t0 A2.svg Uniform polyhedron-33-t2.png Uniform tiling 332-t2.png Tetrahedron vertfig.png CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t2{3,3}={3,3}
Regular polygon 3.svg
{3}
4 6 4
2 Rectified tetrahedron
(Same as octahedron)
3-simplex t1.svg 3-simplex t1 A2.svg Uniform polyhedron-33-t1.png Uniform tiling 332-t1-1-.png Octahedron vertfig.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t1{3,3}=r{3,3}
Regular polygon 3.svg
{3}
Regular polygon 3.svg
{3}
8 12 6
3 Truncated tetrahedron 3-simplex t01.svg 3-simplex t01 A2.svg Uniform polyhedron-33-t01.png Uniform tiling 332-t01-1-.png Truncated tetrahedron vertfig.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t0,1{3,3}=t{3,3}
Regular polygon 6.svg
{6}
Regular polygon 3.svg
{3}
8 18 12
[3] Bitruncated tetrahedron
(Same as truncated tetrahedron)
3-simplex t01.svg 3-simplex t01 A2.svg Uniform polyhedron-33-t12.png Uniform tiling 332-t12.png Truncated tetrahedron vertfig.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t1,2{3,3}=t{3,3}
Regular polygon 3.svg
{3}
Regular polygon 6.svg
{6}
8 18 12
4 Rhombitetratetrahedron
(Same as cuboctahedron)
3-simplex t02.svg 3-simplex t02 A2.svg Uniform polyhedron-33-t02.png Uniform tiling 332-t02.png Cuboctahedron vertfig.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,2{3,3}=rr{3,3}
Regular polygon 3.svg
{3}
Regular polygon 4.svg
{4}
Regular polygon 3.svg
{3}
14 24 12
5 Truncated tetratetrahedron
(Same as truncated octahedron)
3-simplex t012.svg 3-simplex t012 A2.svg Uniform polyhedron-33-t012.png Uniform tiling 332-t012.png Truncated octahedron vertfig.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2{3,3}=tr{3,3}
Regular polygon 6.svg
{6}
Regular polygon 4.svg
{4}
Regular polygon 6.svg
{6}
14 36 24
6 Snub tetratetrahedron
(Same as icosahedron)
Icosahedron graph A3.png Icosahedron graph A2.png Uniform polyhedron-33-s012.png Spherical snub tetrahedron.png Icosahedron vertfig.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{3,3}
Regular polygon 3.svg
{3}
Regular polygon 3.svgRegular polygon 3.svg
2 {3}
Regular polygon 3.svg
{3}
20 30 12

(4 3 2) Oh Octahedral symmetry[edit]

The octahedral symmetry of the sphere generates 7 uniform polyhedra, and a 3 more by alternation. Four of these forms are repeated from the tetrahedral symmetry table above.

The octahedral symmetry is represented by a fundamental triangle (4 3 2) counting the mirrors at each vertex. It can also be represented by the Coxeter group B2 or [4,3], as well as a Coxeter-Dynkin diagram: CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png.

There are 48 triangles, visible in the faces of the disdyakis dodecahedron and alternately colored triangles on a sphere:

Disdyakisdodecahedron.jpg Octahedral reflection domains.pngSphere symmetry group oh.png
# Name Graph
B3
Graph
B2
Picture Tiling Vertex
figure
Coxeter-Dynkin
and Schläfli
symbols
Face counts by position Element counts
Pos. 2
CDel node.pngCDel 4.pngCDel node.pngCDel 2.png
[4]
(8)
Pos. 1
CDel node.pngCDel 2.pngCDel 2.pngCDel node.png
[2]
(12)
Pos. 0
CDel 2.pngCDel node.pngCDel 3.pngCDel node.png
[3]
(6)
Faces Edges Vertices
7 Cube 3-cube t0.svg 3-cube t0 B2.svg Uniform polyhedron-43-t0.png Uniform tiling 432-t0.png Cube vertfig.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
{4,3}
Regular polygon 4.svg
{4}
6 12 8
[2] Octahedron 3-cube t2.svg 3-cube t2 B2.svg Uniform polyhedron-43-t2.png Uniform tiling 432-t2.png Octahedron vertfig.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
{3,4}
Regular polygon 3.svg
{3}
8 12 6
[4] rectified cube
rectified octahedron
(Cuboctahedron)
3-cube t1.svg 3-cube t1 B2.svg Uniform polyhedron-43-t1.png Uniform tiling 432-t1.png Cuboctahedron vertfig.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
{4,3}
Regular polygon 4.svg
{4}
Regular polygon 3.svg
{3}
14 24 12
8 Truncated cube 3-cube t01.svg 3-cube t01 B2.svg Uniform polyhedron-43-t01.png Uniform tiling 432-t01.png Truncated cube vertfig.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
t0,1{4,3}=t{4,3}
Regular polygon 8.svg
{8}
Regular polygon 3.svg
{3}
14 36 24
[5] Truncated octahedron 3-cube t12.svg 3-cube t12 B2.svg Uniform polyhedron-43-t12.png Uniform tiling 432-t12.png Truncated octahedron vertfig.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1{3,4}=t{3,4}
Regular polygon 4.svg
{4}
Regular polygon 6.svg
{6}
14 36 24
9 Cantellated cube
cantellated octahedron
Rhombicuboctahedron
3-cube t02.svg 3-cube t02 B2.svg Uniform polyhedron-43-t02.png Uniform tiling 432-t02.png Small rhombicuboctahedron vertfig.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,2{4,3}=rr{4,3}
Regular polygon 8.svg
{8}
Regular polygon 4.svg
{4}
Regular polygon 6.svg
{6}
26 48 24
10 Omnitruncated cube
omnitruncated octahedron
Truncated cuboctahedron
3-cube t012.svg 3-cube t012 B2.svg Uniform polyhedron-43-t012.png Uniform tiling 432-t012.png Great rhombicuboctahedron vertfig.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2{4,3}=tr{4,3}
Regular polygon 8.svg
{8}
Regular polygon 4.svg
{4}
Regular polygon 6.svg
{6}
26 72 48
[6] Alternated truncated octahedron
(Same as Icosahedron)
3-cube h01.svg 3-cube h01 B2.svg Uniform polyhedron-43-h01.png Spherical alternated truncated octahedron.png Icosahedron vertfig.png CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png
= CDel nodes hh.pngCDel split2.pngCDel node h.png
s{3,4}=sr{3,3}
Regular polygon 3.svg
{3}
Regular polygon 3.svg
{3}
20 30 12
[1] Half cube
(Same as tetrahedron)
3-simplex t0 A2.svg 3-simplex t0.svg Uniform polyhedron-33-t2.png Uniform tiling 332-t2.png Tetrahedron vertfig.png CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
= CDel nodes 10ru.pngCDel split2.pngCDel node.png
h{4,3}={3,3}
Regular polygon 3.svg
1/2 {3}
4 6 4
[2] Cantic cube
(Same as Truncated tetrahedron)
3-simplex t01 A2.svg 3-simplex t01.svg Uniform polyhedron-33-t12.png Uniform tiling 332-t12.png Truncated tetrahedron vertfig.png CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
= CDel nodes 10ru.pngCDel split2.pngCDel node 1.png
h2{4,3}=t{3,3}
Regular polygon 6.svg
1/2 {6}
Regular polygon 3.svg
1/2 {3}
8 18 12
11 Snub cube Snub cube A2.png Snub cube B2.png Uniform polyhedron-43-s012.png Spherical snub cube.png Snub cube vertfig.png CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{4,3}
Regular polygon 4.svg
{4}
Regular polygon 3.svgRegular polygon 3.svg
2 {3}
Regular polygon 3.svg
{3}
38 60 24

(5 3 2) Ih Icosahedral symmetry[edit]

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 Coxeter-Dynkin diagram: CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png.

There are 120 triangles, visible in the faces of the disdyakis triacontahedron and alternately colored triangles on a sphere:

Disdyakistriacontahedron.jpg Icosahedral reflection domains.pngSphere symmetry group ih.png
# Name Graph
(A2)
[6]
Graph
(H3)
[10]
Picture Tiling Vertex
figure
Coxeter-Dynkin
and Schläfli
symbols
Face counts by position Element counts
Pos. 2
CDel node.pngCDel 5.pngCDel node.pngCDel 2.png
[5]
(12)
Pos. 1
CDel node.pngCDel 2.pngCDel node.png
[2]
(30)
Pos. 0
CDel 2.pngCDel node.pngCDel 3.pngCDel node.png
[3]
(20)
Faces Edges Vertices
12 Dodecahedron Dodecahedron t0 A2.png Dodecahedron t0 H3.png Uniform polyhedron-53-t0.png Uniform tiling 532-t0.png Dodecahedron vertfig.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
{5,3}
Regular polygon 5.svg
{5}
12 30 20
[6] Icosahedron Icosahedron t0 A2.png Icosahedron t0 H3.png Uniform polyhedron-53-t2.png Uniform tiling 532-t2.png Icosahedron vertfig.png CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
{3,5}
Regular polygon 3.svg
{3}
20 30 12
13 Rectified dodecahedron
Rectified icosahedron
Icosidodecahedron
Dodecahedron t1 A2.png Dodecahedron t1 H3.png Uniform polyhedron-53-t1.png Uniform tiling 532-t1.png Icosidodecahedron vertfig.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
t1{5,3}=r{5,3}
Regular polygon 5.svg
{5}
Regular polygon 3.svg
{3}
32 60 30
14 Truncated dodecahedron Dodecahedron t01 A2.png Dodecahedron t01 H3.png Uniform polyhedron-53-t01.png Uniform tiling 532-t01.png Truncated dodecahedron vertfig.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
t0,1{5,3}=t{5,3}
Regular polygon 5.svg
{10}
Regular polygon 3.svg
{3}
32 90 60
15 Truncated icosahedron Icosahedron t01 A2.png Icosahedron t01 H3.png Uniform polyhedron-53-t12.png Uniform tiling 532-t12.png Truncated icosahedron vertfig.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1{3,5}=t{3,5}
Regular polygon 5.svg
{5}
Regular polygon 6.svg
{6}
32 90 60
16 Cantellated dodecahedron
Cantellated icosahedron
Rhombicosidodecahedron
Dodecahedron t02 A2.png Dodecahedron t02 H3.png Uniform polyhedron-53-t02.png Uniform tiling 532-t02.png Small rhombicosidodecahedron vertfig.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,2{5,3}=rr{5,3}
Regular polygon 5.svg
{5}
Regular polygon 4.svg
{4}
Regular polygon 3.svg
{3}
62 120 60
17 Omnitruncated dodecahedron
Omnitruncated icosahedron
Truncated icosidodecahedron
Dodecahedron t012 A2.png Dodecahedron t012 H3.png Uniform polyhedron-53-t012.png Uniform tiling 532-t012.png Great rhombicosidodecahedron vertfig.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2{5,3}=tr{5,3}
Regular polygon 10.svg
{10}
Regular polygon 4.svg
{4}
Regular polygon 6.svg
{6}
62 180 120
18 Snub dodecahedron
Snub icosahedron
Snub dodecahedron A2.png Snub dodecahedron H2.png Uniform polyhedron-53-s012.png Spherical snub dodecahedron.png Snub dodecahedron vertfig.png CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{5,3}
Regular polygon 5.svg
{5}
Regular polygon 3.svgRegular polygon 3.svg
2 {3}
Regular polygon 3.svg
{3}
92 150 60

(p 2 2) Prismatic [p,2], I2(p) family (Dph Dihedral symmetry)[edit]

The dihedral symmetry of the sphere generates two infinite sets of uniform polyhedra, prisms and antiprisms, and two more infinite set of degenerate polygons, the hosohedrons and dihedrons which exists as tilings on the sphere.

The dihedral symmetry is represented by a fundamental triangle (p 2 2) counting the mirrors at each vertex. It can also be represented by the Coxeter group I2(p) or [n,2], as well as a prismatic Coxeter-Dynkin diagram: CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png.

Below are the first five dihedral symmetries: D2 ... D6. The dihedral symmetry Dp has order 4n, represented the faces of a bipyramid, and on the sphere as an equator line on the longitude, and n equally-spaced lines of longitude.

(2 2 2) dihedral symmetry[edit]

There are 8 fundamental triangles, visible in the faces of the square bipyramid (Octahedron) and alternately colored triangles on a sphere:

Octahedron.svg Sphere symmetry group d2h.png
# Name Picture Tiling Vertex
figure
Coxeter-Dynkin
and Schläfli
symbols
Face counts by position Element counts
Pos. 2
CDel node.pngCDel 2.pngCDel node.pngCDel 2.png
[2]
(2)
Pos. 1
CDel node.pngCDel 2.pngCDel node.png
[2]
(2)
Pos. 0
CDel 2.pngCDel node.pngCDel 2.pngCDel node.png
[2]
(2)
Faces Edges Vertices
D2
H2
digonal dihedron
digonal hosohedron
Digonal dihedron.png CDel node 1.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
{2,2}
Regular digon in spherical geometry-2.svg
{2}
2 2 2
D4 truncated digonal dihedron
(Same as square dihedron)
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node.png
t{2,2}={4,2}
Regular polygon 4.svg
{4}
2 4 4
P4
[7]
omnitruncated digonal dihedron
(Same as cube)
Uniform polyhedron 222-t012.png Cube vertfig.png CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t0,1,2{2,2}=tr{2,2}
Regular polygon 4.svg
{4}
Regular polygon 4.svg
{4}
Regular polygon 4.svg
{4}
6 12 8
A2
[1]
snub digonal dihedron
(Same as tetrahedron)
Uniform polyhedron-33-t2.png Tetrahedron vertfig.png CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.png
sr{2,2}
Regular polygon 3.svgRegular polygon 3.svg
2 {3}
  4 6 4

(3 2 2) D3hdihedral symmetry[edit]

There are 12 fundamental triangles, visible in the faces of the hexagonal bipyramid and alternately colored triangles on a sphere:

Hexagonale bipiramide.png Sphere symmetry group d3h.png
# Name Picture Tiling Vertex
figure
Coxeter-Dynkin
and Schläfli
symbols
Face counts by position Element counts
Pos. 2
CDel node.pngCDel 3.pngCDel node.pngCDel 2.png
[3]
(2)
Pos. 1
CDel node.pngCDel 2.pngCDel node.png
[2]
(3)
Pos. 0
CDel 2.pngCDel node.pngCDel 2.pngCDel node.png
[2]
(3)
Faces Edges Vertices
D3 Trigonal dihedron Trigonal dihedron.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
{3,2}
Regular polygon 3.svg
{3}
2 3 3
H3 Trigonal hosohedron Trigonal hosohedron.png CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
{2,3}
Regular digon in spherical geometry-2.svg
{2}
3 3 2
D6 Truncated trigonal dihedron
(Same as hexagonal dihedron)
Hexagonal dihedron.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node.png
t{3,2}
Regular polygon 6.svg
{6}
2 6 6
P3 Truncated trigonal hosohedron
(Triangular prism)
Triangular prism.png Triangular prism vertfig.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t{2,3}
Regular polygon 3.svg
{3}
Regular polygon 4.svg
{4}
5 9 6
P6 Omnitruncated trigonal dihedron
(Hexagonal prism)
Hexagonal prism.png Hexagonal prism vertfig.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t0,1,2{2,3}=tr{2,3}
Regular polygon 6.svg
{6}
Regular polygon 4.svg
{4}
Regular polygon 4.svg
{4}
8 18 12
A3
[2]
Snub trigonal dihedron
(Same as Triangular antiprism)
(Same as octahedron)
Trigonal antiprism.png Octahedron vertfig.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node h.png
sr{2,3}
Regular polygon 3.svg
{3}
Regular polygon 3.svgRegular polygon 3.svg
2 {3}
  8 12 6

(4 2 2) D4hdihedral symmetry[edit]

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

Octagonal bipyramid.png
# Name Picture Tiling Vertex
figure
Coxeter-Dynkin
and Schläfli
symbols
Face counts by position Element counts
Pos. 2
CDel node.pngCDel 4.pngCDel node.pngCDel 2.png
[4]
(2)
Pos. 1
CDel node.pngCDel 2.pngCDel node.png
[2]
(4)
Pos. 0
CDel 2.pngCDel node.pngCDel 2.pngCDel node.png
[2]
(4)
Faces Edges Vertices
D4 square dihedron CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.png
{4,2}
Regular polygon 4.svg
{4}
2 4 4
H4 square hosohedron CDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.png
{2,4}
Regular digon in spherical geometry-2.svg
{2}
4 4 2
D8 Truncated square dihedron
(Same as octagonal dihedron)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node.png
t{4,2}
Regular polygon 8.svg
{8}
2 8 8
P4
[7]
Truncated square hosohedron
(Cube)
Tetragonal prism.png Cube vertfig.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t{2,4}
Regular polygon 4.svg
{4}
Regular polygon 4.svg
{4}
6 12 8
D8 Omnitruncated square dihedron
(Octagonal prism)
Octagonal prism.png Octagonal prism vertfig.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t0,1,2{2,4}=tr{2,4}
Regular polygon 8.svg
{8}
Regular polygon 4.svg
{4}
Regular polygon 4.svg
{4}
10 24 16
A4 Snub square dihedron
(Square antiprism)
Square antiprism.png Square antiprism vertfig.png CDel node h.pngCDel 4.pngCDel node h.pngCDel 2x.pngCDel node h.png
sr{2,4}
Regular polygon 4.svg
{4}
Regular polygon 3.svgRegular polygon 3.svg
2 {3}
  10 16 8

(5 2 2) D5h dihedral symmetry[edit]

There are 20 fundamental triangles, visible in the faces of the decagonal bipyramid and alternately colored triangles on a sphere:

Decagonal bipyramid.png
# Name Picture Tiling Vertex
figure
Coxeter-Dynkin
and Schläfli
symbols
Face counts by position Element counts
Pos. 2
CDel node.pngCDel 5.pngCDel node.pngCDel 2.png
[5]
(2)
Pos. 1
CDel node.pngCDel 2.pngCDel node.png
[2]
(5)
Pos. 0
CDel 2.pngCDel node.pngCDel 2.pngCDel node.png
[2]
(5)
Faces Edges Vertices
D5 Pentagonal dihedron CDel node 1.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node.png
{5,2}
Regular polygon 5.svg
{5}
2 5 5
H5 Pentagonal hosohedron CDel node.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node 1.png
{2,5}
Regular digon in spherical geometry-2.svg
{2}
5 5 2
D10 Truncated pentagonal dihedron
(Same as decagonal dihedron)
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 2.pngCDel node.png
t{5,2}
Regular polygon 10.svg
{10}
2 10 10
P5 Truncated pentagonal hosohedron
(Same as pentagonal prism)
Pentagonal prism.png Pentagonal prism vertfig.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t{2,5}
Regular polygon 5.svg
{5}
Regular polygon 4.svg
{4}
7 15 10
P10 Omnitruncated pentagonal dihedron
(Decagonal prism)
Decagonal prism.png Decagonal prism vf.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t0,1,2{2,5}=tr{2,5}
Regular polygon 10.svg
{10}
Regular polygon 4.svg
{4}
Regular polygon 4.svg
{4}
12 30 20
A5 Snub pentagonal dihedron
(Pentagonal antiprism)
Pentagonal antiprism.png Pentagonal antiprism vertfig.png CDel node h.pngCDel 5.pngCDel node h.pngCDel 2x.pngCDel node h.png
sr{2,5}
Regular polygon 5.svg
{5}
Regular polygon 3.svgRegular polygon 3.svg
2 {3}
  12 20 10

(6 2 2) D6hdihedral symmetry[edit]

There are 24 fundamental triangles, visible in the faces of the dodecagonal bipyramid and alternately colored triangles on a sphere.

# Name Picture Tiling Vertex
figure
Coxeter-Dynkin
and Schläfli
symbols
Face counts by position Element counts
Pos. 2
CDel node.pngCDel 6.pngCDel node.pngCDel 2.png
[6]
(2)
Pos. 1
CDel node.pngCDel 2.pngCDel node.png
[2]
(6)
Pos. 0
CDel 2.pngCDel node.pngCDel 2.pngCDel node.png
[2]
(6)
Faces Edges Vertices
D6 Hexagonal dihedron Hexagonal dihedron.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.png
{6,2}
Regular polygon 6.svg
{6}
2 6 6
H6 Hexagonal hosohedron Hexagonal hosohedron.png CDel node.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node 1.png
{2,6}
Regular digon in spherical geometry-2.svg
{2}
6 6 2
D12 Truncated hexagonal dihedron
(Same as dodecagonal dihedron)
Dodecagonal dihedron.png CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node.png
t{6,2}
Regular polygon 10.svg
{12}
2 12 12
H6 Truncated hexagonal hosohedron
(Same as hexagonal prism)
Hexagonal prism.png Spherical hexagonal prism.png Hexagonal prism vertfig.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t{2,6}
Regular polygon 6.svg
{6}
Regular polygon 4.svg
{4}
8 18 12
P12 Omnitruncated hexagonal dihedron
(Dodecagonal prism)
Dodecagonal prism.png Spherical truncated hexagonal prism.png Dodecagonal prism vf.png CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t0,1,2{2,6}=tr{2,6}
Regular polygon 10.svg
{12}
Regular polygon 4.svg
{4}
Regular polygon 4.svg
{4}
14 36 24
A6 Snub hexagonal dihedron
(Hexagonal antiprism)
Hexagonal antiprism.png Spherical hexagonal antiprism.png Hexagonal antiprism vertfig.png CDel node h.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.png
sr{2,6}
Regular polygon 6.svg
{6}
Regular polygon 3.svgRegular polygon 3.svg
2 {3}
  14 24 12

See also[edit]

Notes[edit]

References[edit]

External links[edit]