Uniform polyhedron

Last updated
Platonic solid: Tetrahedron Tetrahedron.jpg
Platonic solid: Tetrahedron
Uniform star polyhedron: Snub dodecadodecahedron Snub dodecadodecahedron.png
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.

Contents

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.

(BrankoGrünbaum  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

The great dirhombicosidodecahedron, the only non-Wythoffian uniform polyhedron Great dirhombicosidodecahedron.png
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 Polyhedron truncation example3.png
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 nameParentTruncatedRectifiedBitruncated
(tr. dual)
Birectified
(dual)
CantellatedOmnitruncated
(cantitruncated)
Snub
Coxeter diagram CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel 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.pngCDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel 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 22 q | p2 | p q2 p | qp | q 2p q | 2p q 2 || p q 2
Vertex figure pqq.2p.2p(p.q)2p.2q.2qqpp.4.q.44.2p.2q3.3.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)ParentTruncatedRectifiedBitruncated
(tr. dual)
Birectified
(dual)
CantellatedOmnitruncated
(cantitruncated)
Snub
Coxeter diagram CDel node 1.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel node 1.pngCDel p.pngCDel node 1.pngCDel 2.pngCDel node.pngCDel node.pngCDel p.pngCDel node 1.pngCDel 2.pngCDel node.pngCDel node.pngCDel p.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel node 1.pngCDel p.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel node 1.pngCDel p.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 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 22 2 | p2 | p 22 p | 2p | 2 2p 2 | 2p 2 2 || p 2 2
Vertex figure p22.2p.2pp.2.p.2p.4.42pp.4.2.44.2p.43.3.3.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
#NameGraph
A3
Graph
A2
PictureTiling Vertex
figure
Coxeter
and Schläfli
symbols
Face counts by positionElement 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)
FacesEdgesVertices
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}
464
[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}
464
2Rectified 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}
8126
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}
81812
[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}
81812
4Cantellated 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}
142412
5Omnitruncated 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}
143624
6Snub 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}
203012

(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
#NameGraph
B3
Graph
B2
PictureTiling Vertex
figure
Coxeter
and Schläfli
symbols
Face counts by positionElement 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)
FacesEdgesVertices
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}
6128
[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}
8126
[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}
142412
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}
143624
[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}
143624
9Cantellated 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}
264824
10Omnitruncated 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}
267248
[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}
203012
[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}
464
[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}
81812
[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}
142412
[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}
143624
[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}
264824
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}
386024

(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

#NameGraph
(A2)
[6]
Graph
(H3)
[10]
PictureTiling Vertex
figure
Coxeter
and Schläfli
symbols
Face counts by positionElement 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)
FacesEdgesVertices
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}
123020
[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}
203012
13Rectified 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}
326030
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}
329060
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}
329060
16Cantellated 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}
6212060
17Omnitruncated 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}
62180120
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}
9215060

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

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
#NamePictureTiling Vertex
figure
Coxeter
and Schläfli
symbols
Face counts by positionElement 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)
FacesEdgesVertices
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}
222
D4Truncated 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}
244
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}
6128
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}
 464

(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
#NamePictureTiling Vertex
figure
Coxeter
and Schläfli
symbols
Face counts by positionElement 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)
FacesEdgesVertices
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}
233
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}
332
D6Truncated 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}
266
P3Truncated 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}
596
P6Omnitruncated 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}
81812
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}
 8126
P3Cantic 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}
596

(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
#NamePictureTiling Vertex
figure
Coxeter
and Schläfli
symbols
Face counts by positionElement 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)
FacesEdgesVertices
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}
244
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}
442
D8Truncated 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}
288
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}
6128
D8Omnitruncated 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}
102416
A4Snub 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}
 10168
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}
6128
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}
464

(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
#NamePictureTiling Vertex
figure
Coxeter
and Schläfli
symbols
Face counts by positionElement 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)
FacesEdgesVertices
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}
255
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}
552
D10Truncated 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}
21010
P5Truncated 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}
71510
P10Omnitruncated 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}
123020
A5Snub 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}
 122010
P5Cantic 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}
71510

(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.

#NamePictureTiling Vertex
figure
Coxeter
and Schläfli
symbols
Face counts by positionElement 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)
FacesEdgesVertices
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}
266
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}
662
D12Truncated 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}
21212
H6Truncated 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}
81812
P12Omnitruncated 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}
143624
A6Snub 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}
 142412
P3Cantic 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}
596
P6Cantic 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}
81812
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}
8126

Wythoff construction operators

OperationSymbol Coxeter
diagram
Description
Parent{p,q}
t0{p,q}
CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngAny regular polyhedron or tiling
Rectified (r)r{p,q}
t1{p,q}
CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngThe 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.pngEach 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.pngA 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.pngIn 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.pngThe 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
OperationSymbol Coxeter
diagram
Description
Snub rectified (sr)sr{p,q}CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.pngThe 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.pngAlternated 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.pngOnly 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.pngSame 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.pngOnly 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.pngOnly 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, Zvi (1993). "Uniform Solution for Uniform Polyhedra" (PDF). Geometriae Dedicata . 47: 57–110. doi: 10.1007/BF01263494 . Zvi Har’El, Kaleido software, Images, dual images
  6. Mäder, R. E. Uniform Polyhedra. Mathematica J. 3, 48-57, 1993.
  7. Messer, Peter W. (2002). "Closed-Form Expressions for Uniform Polyhedra and Their Duals". Discrete & Computational Geometry . 27 (3): 353–375. doi: 10.1007/s00454-001-0078-2 .

Related Research Articles

<span class="mw-page-title-main">Archimedean solid</span> Polyhedra in which all vertices are the same

In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids, excluding the prisms and antiprisms, and excluding the pseudorhombicuboctahedron. They are a subset of the Johnson solids, whose regular polygonal faces do not need to meet in identical vertices.

<span class="mw-page-title-main">Cuboctahedron</span> Polyhedron with 8 triangular faces and 6 square faces

A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral.

<span class="mw-page-title-main">Octahedron</span> Polyhedron with 8 triangular faces

In geometry, an octahedron is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

<span class="mw-page-title-main">Polyhedron</span> 3D shape with flat faces, straight edges and sharp corners

In geometry, a polyhedron is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.

A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.

<span class="mw-page-title-main">Schläfli symbol</span> Notation that defines regular polytopes and tessellations

In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.

<span class="mw-page-title-main">Great icosahedron</span> Kepler-Poinsot polyhedron with 20 faces

In geometry, the great icosahedron is one of four Kepler–Poinsot polyhedra, with Schläfli symbol {3,52} and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence.

In geometry, a snub polyhedron is a polyhedron obtained by performing a snub operation: alternating a corresponding omnitruncated or truncated polyhedron, depending on the definition. Some, but not all, authors include antiprisms as snub polyhedra, as they are obtained by this construction from a degenerate "polyhedron" with only two faces.

<span class="mw-page-title-main">Alternation (geometry)</span> Removal of alternate vertices

In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.

In geometry, a polytope or a tiling is isotoxal or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation, and/or reflection that will move one edge to the other, while leaving the region occupied by the object unchanged.

<span class="mw-page-title-main">Uniform polytope</span> Isogonal polytope with uniform facets

In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons.

<span class="mw-page-title-main">Uniform star polyhedron</span> Self-intersecting uniform polyhedron

In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures, or both.

In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.

<span class="mw-page-title-main">Spherical polyhedron</span> Partition of a spheres surface into polygons

In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most conveniently derived in this way.

In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive.

<span class="mw-page-title-main">Uniform honeycombs in hyperbolic space</span> Tiling of hyperbolic 3-space by uniform polyhedra

In hyperbolic geometry, a uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as Wythoff constructions, and represented by permutations of rings of the Coxeter diagrams for each family.

<span class="mw-page-title-main">Icosahedron</span> Polyhedron with 20 faces

In geometry, an icosahedron is a polyhedron with 20 faces. The name comes from Ancient Greek εἴκοσι (eíkosi) 'twenty' and from Ancient Greek ἕδρα (hédra) ' seat'. The plural can be either "icosahedra" or "icosahedrons".

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