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Platonic solid: Tetrahedron
Uniform star polyhedron: Snub dodecadodecahedron

In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent.

Uniform polyhedra may be regular (if also face- and edge-transitive), quasi-regular (if also 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.

There are two infinite classes of uniform polyhedra, together with 75 other polyhedra:

Hence 5 + 13 + 4 + 53 = 75.

There are also many degenerate uniform polyhedra with pairs of edges that coincide, including one found by John Skilling called the great disnub dirhombidodecahedron (Skilling's figure).

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.

Definition

The Original Sin in the theory of polyhedra goes back to Euclid, and through Kepler, Poinsot, Cauchy and many others continues to afflict all the work on this topic (including that of the present author). It arises from the fact that the traditional usage of the term “regular polyhedra” was, and is, contrary to syntax and to logic: the words seem to imply that we are dealing, among the objects we call “polyhedra”, with those special ones that deserve to be called “regular”. But at each stage— Euclid, Kepler, Poinsot, Hess, Brückner,…—the writers failed to define what are the “polyhedra” among which they are finding the “regular” ones.

(Branko Grünbaum 1994)

Coxeter, Longuet-Higgins & Miller (1954) define uniform polyhedra to be vertex-transitive polyhedra with regular faces. They define a polyhedron to be a finite set of polygons such that each side of a polygon is a side of just one other polygon, such that no non-empty proper subset of the polygons has the same property. By a polygon they implicitly mean a polygon in 3-dimensional Euclidean space; these are allowed to be non-convex and to intersect each other.

There are some generalizations of the concept of a uniform polyhedron. If the connectedness assumption is dropped, then we get uniform compounds, which can be split as a union of polyhedra, such as the compound of 5 cubes. If we drop the condition that the realization of the polyhedron is non-degenerate, then we get the so-called degenerate uniform polyhedra. These require a more general definition of polyhedra. Grünbaum (1994) gave a rather complicated definition of a polyhedron, while McMullen & Schulte (2002) gave a simpler and more general definition of a polyhedron: in their terminology, a polyhedron is a 2-dimensional abstract polytope with a non-degenerate 3-dimensional realization. Here an abstract polytope is a poset of its "faces" satisfying various condition, a realization is a function from its vertices to some space, and the realization is called non-degenerate if any two distinct faces of the abstract polytope have distinct realizations. Some of the ways they can be degenerate are as follows:

History

Regular convex polyhedra

Nonregular uniform convex polyhedra

Regular star polyhedra

Other 53 nonregular star polyhedra

Uniform star polyhedra

Main article: Uniform star polyhedron

The great dirhombicosidodecahedron, the only non-Wythoffian uniform polyhedron
The great dirhombicosidodecahedron, the only non-Wythoffian uniform polyhedron

The 57 nonprismatic nonconvex forms, with exception of the great dirhombicosidodecahedron, are compiled by Wythoff constructions within Schwarz triangles.

Convex forms by Wythoff construction

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

The convex uniform polyhedra can be named by Wythoff construction operations on 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 the 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.

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

  1. Hosohedra H2... (only as spherical tilings)
  2. Dihedra D2... (only as spherical tilings)
  3. Prisms P3... (truncated hosohedra)
  4. Antiprisms A3... (snub prisms)

Summary tables

Johnson name Parent Truncated Rectified Bitruncated
(tr. dual)
Birectified
(dual)
Cantellated Omnitruncated
(cantitruncated)
Snub
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
Extended
Schläfli symbol
{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
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

3.3.3
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

3.3.3
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

4.4.4
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

3.3.3.3
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.svg

5.5.5
Uniform polyhedron-53-t01.svg

3.10.10
Uniform polyhedron-53-t1.svg

3.5.3.5
Uniform polyhedron-53-t12.svg

5.6.6
Uniform polyhedron-53-t2.svg

3.3.3.3.3
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:

(The sphere is not cut, only the tiling is cut.) (On a sphere, an edge is the arc of the great circle, the shortest way, between its two vertices. Hence, a digon whose vertices are not polar-opposite is flat: it looks like an edge.)

(p 2 2) Parent Truncated Rectified Bitruncated
(tr. dual)
Birectified
(dual)
Cantellated Omnitruncated
(cantitruncated)
Snub
Coxeter 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
Extended
Schläfli symbol
{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
Vertex figure p2 2.2p.2p p.2.p.2 p.4.4 2p p.4.2.4 4.2p.4 3.3.3.p
Dihedral
(2 2 2)
Digonal dihedron.png

{2,2}
Tetragonal dihedron.png

2.4.4
Digonal dihedron.png

2.2.2.2
Tetragonal dihedron.png

4.4.2
Digonal dihedron.png

2.2
Tetragonal dihedron.png

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

3.3
Hexagonal dihedron.png

2.6.6
Trigonal dihedron.png

2.3.2.3
Spherical triangular prism.png

4.4.3
Spherical trigonal hosohedron.png

2.2.2
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)
Tetragonal dihedron.png

4.4
2.8.8
Tetragonal dihedron.png

2.4.2.4
Spherical square prism.png

4.4.4
Spherical square hosohedron.png

2.2.2.2
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)
Pentagonal dihedron.png

5.5
2.10.10
Pentagonal dihedron.png

2.5.2.5
Spherical pentagonal prism.png

4.4.5
Spherical pentagonal hosohedron.png

2.2.2.2.2
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

6.6
Dodecagonal dihedron.png

2.12.12
Hexagonal dihedron.png

2.6.2.6
Spherical hexagonal prism.png

4.4.6
Spherical hexagonal hosohedron.png

2.2.2.2.2.2
Spherical hexagonal prism.png

2.4.6.4
Spherical dodecagonal prism2.png

4.4.12
Spherical hexagonal antiprism.png

3.3.3.6

(3 3 2) Td tetrahedral symmetry

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 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 in the alternately colored triangles on a sphere:

Tetrakishexahedron.jpg
Tetrahedral reflection domains.png
Sphere symmetry group td.png
# Name Graph
A3
Graph
A2
Picture Tiling Vertex
figure
Coxeter
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
Tetratetrahedron
(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 Cantellated tetrahedron
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 Omnitruncated tetrahedron
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.svg
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.svg
Regular polygon 3.svg

2 {3}
Regular polygon 3.svg

{3}
20 30 12

(4 3 2) Oh octahedral symmetry

The octahedral symmetry of the sphere generates 7 uniform polyhedra, and a 7 more by alternation. Six 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 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 in the alternately colored triangles on a sphere:

Disdyakisdodecahedron.jpg
Octahedral reflection domains.png
Sphere symmetry group oh.png
# Name Graph
B3
Graph
B2
Picture Tiling Vertex
figure
Coxeter
and Schläfli
symbols
Face counts by position Element counts
Pos. 2
CDel node.pngCDel 4.pngCDel node.pngCDel 2.png
[4]
(6)
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]
(8)
Faces Edges Vertices
7 Cube
3-cube t0.svg
3-cube t0 B2.svg
Uniform polyhedron-43-t0.svg
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.svg
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.svg
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.svg
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.svg
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 4.svg

{4}
Regular polygon 4.svg

{4}
Regular polygon 3.svg

{3}
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] Snub octahedron
(same as Icosahedron)
3-cube h01.svg
3-cube h01 B2.svg
Uniform polyhedron-43-h01.svg
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
[4] (same as Cuboctahedron)
3-simplex t02 A2.svg
3-simplex t02.svg
Uniform polyhedron-33-t02.png
Uniform tiling 332-t02.png
Cuboctahedron vertfig.png
CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel nodes 11.pngCDel split2.pngCDel node.png
rr{3,3}
14 24 12
[5] (same as Truncated octahedron)
3-simplex t012 A2.svg
3-simplex t012.svg
Uniform polyhedron-33-t012.png
Uniform tiling 332-t012.png
Truncated octahedron vertfig.png
CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
= CDel nodes 11.pngCDel split2.pngCDel node 1.png
tr{3,3}
14 36 24
[9] Cantic snub octahedron
(same as Rhombicuboctahedron)
3-cube t02.svg
3-cube t02 B2.svg
Rhombicuboctahedron uniform edge coloring.png
Uniform tiling 432-t02.png
Small rhombicuboctahedron vertfig.png
CDel node 1.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png
s2{3,4}=rr{3,4}
26 48 24
11 Snub cuboctahedron
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.svg
Regular polygon 3.svg

2 {3}
Regular polygon 3.svg

{3}
38 60 24

(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 Coxeter 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 in the alternately colored triangles on a sphere:

Disdyakistriacontahedron.jpg
Icosahedral reflection domains.png
Sphere symmetry group ih.png

# Name Graph
(A2)
[6]
Graph
(H3)
[10]
Picture Tiling Vertex
figure
Coxeter
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 A2 projection.svg
Dodecahedron H3 projection.svg
Uniform polyhedron-53-t0.svg
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 A2 projection.svg
Icosahedron H3 projection.svg
Uniform polyhedron-53-t2.svg
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.svg
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.svg
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 10.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.svg
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 icosidodecahedron
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.svg
Regular 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)

Main article: Prismatic uniform polyhedron

The dihedral symmetry of the sphere generates two infinite sets of uniform polyhedra, prisms and antiprisms, and two more infinite set of degenerate polyhedra, the hosohedra and dihedra which exist 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 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

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

Octahedron.jpg
Sphere symmetry group d2h.png
# Name Picture Tiling Vertex
figure
Coxeter
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)
Tetragonal dihedron.png
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
Spherical square prism2.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
Spherical digonal antiprism.png
Tetrahedron vertfig.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.png
sr{2,2}
Regular polygon 3.svg
Regular polygon 3.svg

2 {3}
  4 6 4

(3 2 2) D3h dihedral symmetry

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
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
Spherical 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
Spherical hexagonal prism2.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
Spherical 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.svg
Regular polygon 3.svg

2 {3}
  8 12 6
P3 Cantic snub trigonal dihedron
(Triangular prism)
Triangular prism.png
Spherical triangular prism.png
Triangular prism vertfig.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node 1.png
s2{2,3}=t{2,3}
5 9 6

(4 2 2) D4h dihedral symmetry

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
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
Tetragonal dihedron.png
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
Spherical square hosohedron.png
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
Spherical square 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
Spherical octagonal prism2.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
Spherical 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.svg
Regular polygon 3.svg

2 {3}
  10 16 8
P4
[7]
Cantic snub square dihedron
(Cube)
Tetragonal prism.png
Spherical square prism.png
Cube vertfig.png
CDel node h.pngCDel 4.pngCDel node h.pngCDel 2x.pngCDel node 1.png
s2{4,2}=t{2,4}
6 12 8
A2
[1]
Snub square hosohedron
(Digonal antiprism)
(Tetrahedron)
Uniform polyhedron-33-t2.png
Spherical digonal antiprism.png
Tetrahedron vertfig.png
CDel node.pngCDel 4.pngCDel node h.pngCDel 2x.pngCDel node h.png
s{2,4}=sr{2,2}
4 6 4

(5 2 2) D5h dihedral symmetry

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
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
Pentagonal dihedron.png
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
Spherical pentagonal hosohedron.png
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
Spherical 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
Spherical decagonal prism2.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
Spherical 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.svg
Regular polygon 3.svg

2 {3}
  12 20 10
P5 Cantic snub pentagonal dihedron
(Pentagonal prism)
Pentagonal prism.png
Spherical pentagonal prism.png
Pentagonal prism vertfig.png
CDel node h.pngCDel 5.pngCDel node h.pngCDel 2x.pngCDel node 1.png
s2{5,2}=t{2,5}
7 15 10

(6 2 2) D6h dihedral symmetry

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
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.svg
Regular polygon 3.svg

2 {3}
  14 24 12
P3 Cantic hexagonal dihedron
(Triangular prism)
Triangular prism.png
Spherical triangular prism.png
Triangular prism vertfig.png
CDel node h1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node 1.png = CDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
h2{6,2}=t{2,3}
5 9 6
P6 Cantic snub hexagonal dihedron
(Hexagonal prism)
Hexagonal prism.png
Spherical hexagonal prism.png
Hexagonal prism vertfig.png
CDel node h.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node 1.png
s2{6,2}=t{2,6}
8 18 12
A3
[2]
Snub hexagonal hosohedron
(same as Triangular antiprism)
(same as octahedron)
Trigonal antiprism.png
Spherical trigonal antiprism.png
Octahedron vertfig.png
CDel node.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.png
s{2,6}=sr{2,3}
8 12 6

Wythoff construction operators

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. Polyhedra are named by the number of sides of the two regular forms: {p,q} and {q,p}, like cuboctahedron for r{4,3} between a cube and octahedron.
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.jpg
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. A birectification can be seen as the dual.
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 A bitruncation can be seen as the truncation of the dual. A bitruncated cube is a truncated octahedron.
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. A cantellated polyhedron is named as a rhombi-r{p,q}, like rhombicuboctahedron for rr{4,3}.
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

See also

Notes

  1. ^ Regular Polytopes, p.13
  2. ^ Piero della Francesca's Polyhedra
  3. ^ "Stéréo-Club Français - Galerie : Polyedres".
  4. ^ Dr. Zvi Har’El (December 14, 1949 – February 2, 2008) and International Jules Verne Studies - A Tribute
  5. ^ Har'El, Z. Uniform Solution for Uniform Polyhedra., Geometriae Dedicata 47, 57-110, 1993. Zvi Har’El, Kaleido software, Images, dual images
  6. ^ Mäder, R. E. Uniform Polyhedra. Mathematica J. 3, 48-57, 1993. [1]
  7. ^ Messer, Peter W. (2002). "Closed-Form Expressions for Uniform Polyhedra and Their Duals". Discrete & Computational Geometry. 27: 353–375. doi:10.1007/s00454-001-0078-2.

References

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds