Uniform polyhedron

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.

Platonic solid: Tetrahedron
Uniform star polyhedron: Snub dodecadodecahedron

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:

  • Hidden faces. Some polyhedra have faces that are hidden, in the sense that no points of their interior can be seen from the outside. These are usually not counted as uniform polyhedra.
  • Degenerate compounds. Some polyhedra have multiple edges and their faces are the faces of two or more polyhedra, though these are not compounds in the previous sense since the polyhedra share edges.
  • Double covers. There are some non-orientable polyhedra that have double covers satisfying the definition of a uniform polyhedron. There double covers have doubled faces, edges and vertices. They are usually not counted as uniform polyhedra.
  • Double faces. There are several polyhedra with doubled faces produced by Wythoff's construction. Most authors do not allow doubled faces and remove them as part of the construction.
  • Double edges. Skilling's figure has the property that it has double edges (as in the degenerate uniform polyhedra) but its faces cannot be written as a union of two uniform polyhedra.

History

Regular convex polyhedra

Nonregular uniform convex polyhedra

Regular star polyhedra

Other 53 nonregular star polyhedra
  • Of the remaining 53, Edmund Hess (1878) discovered two, Albert Badoureau (1881) discovered 36 more, and Pitsch (1881) independently discovered 18, of which 3 had not previously been discovered. Together these gave 41 polyhedra.
  • 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. Longuet-Higgins and H.C. Longuet-Higgins independently discovered eleven of these. Lesavre and Mercier rediscovered five of them in 1947.
  • 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 drew 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..[3]
  • 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.[4]
  • Also in 1993, R. Mäder ported this Kaleido solution to Mathematica with a slightly different indexing system.[5]
  • 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.[6]

Uniform star polyhedra
Main article: Uniform star 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
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



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)		 p 2 p p q q q 2 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)
3.3.3
3.6.6
3.3.3.3
3.6.6
3.3.3
3.4.3.4
4.6.6
3.3.3.3.3
Octahedral
(4 3 2)
4.4.4
3.8.8
3.4.3.4
4.6.6
3.3.3.3
3.4.4.4
4.6.8
3.3.3.3.4
Icosahedral
(5 3 2)
5.5.5
3.10.10
3.5.3.5
5.6.6
3.3.3.3.3
3.4.5.4
4.6.10
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
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 p 2 p p 2 2 2 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)
{2,2}
2.4.4
2.2.2.2
4.4.2
2.2
2.4.2.4
4.4.4
3.3.3.2
Dihedral
(3 2 2)
3.3
2.6.6
2.3.2.3
4.4.3
2.2.2
2.4.3.4
4.4.6
3.3.3.3
Dihedral
(4 2 2)
4.4		 2.8.8
2.4.2.4
4.4.4
2.2.2.2
2.4.4.4
4.4.8
3.3.3.4
Dihedral
(5 2 2)
5.5		 2.10.10
2.5.2.5
4.4.5
2.2.2.2.2
2.4.5.4
4.4.10
3.3.3.5
Dihedral
(6 2 2)
6.6
2.12.12
2.6.2.6
4.4.6
2.2.2.2.2.2
2.4.6.4
4.4.12
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: .

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

# Name Graph
A3		 Graph
A2		 Picture Tiling Vertex
figure		 Coxeter
and Schläfli
symbols		 Face counts by position Element counts
Pos. 2

[3]
(4)		 Pos. 1

[2]
(6)		 Pos. 0

[3]
(4)		 Faces Edges Vertices
1 Tetrahedron
{3,3}
{3}		 4 6 4
[1] Birectified tetrahedron
(same as tetrahedron)
t2{3,3}={3,3}
{3}		 4 6 4
2 Rectified tetrahedron
Tetratetrahedron
(same as octahedron)
t1{3,3}=r{3,3}
{3}
{3}		 8 12 6
3 Truncated tetrahedron
t0,1{3,3}=t{3,3}
{6}
{3}		 8 18 12
[3] Bitruncated tetrahedron
(same as truncated tetrahedron)
t1,2{3,3}=t{3,3}
{3}
{6}		 8 18 12
4 Cantellated tetrahedron
Rhombitetratetrahedron
(same as cuboctahedron)
t0,2{3,3}=rr{3,3}
{3}
{4}
{3}		 14 24 12
5 Omnitruncated tetrahedron
Truncated tetratetrahedron
(same as truncated octahedron)
t0,1,2{3,3}=tr{3,3}
{6}
{4}
{6}		 14 36 24
6 Snub tetratetrahedron
(same as icosahedron)
sr{3,3}
{3}
2 {3}
{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: .

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

# Name Graph
B3		 Graph
B2		 Picture Tiling Vertex
figure		 Coxeter
and Schläfli
symbols		 Face counts by position Element counts
Pos. 2

[4]
(6)		 Pos. 1

[2]
(12)		 Pos. 0

[3]
(8)		 Faces Edges Vertices
7 Cube
{4,3}
{4}		 6 12 8
[2] Octahedron
{3,4}
{3}		 8 12 6
[4] Rectified cube
Rectified octahedron
(Cuboctahedron)
{4,3}
{4}
{3}		 14 24 12
8 Truncated cube
t0,1{4,3}=t{4,3}
{8}
{3}		 14 36 24
[5] Truncated octahedron
t0,1{3,4}=t{3,4}
{4}
{6}		 14 36 24
9 Cantellated cube
Cantellated octahedron
Rhombicuboctahedron
t0,2{4,3}=rr{4,3}
{4}
{4}
{3}		 26 48 24
10 Omnitruncated cube
Omnitruncated octahedron
Truncated cuboctahedron
t0,1,2{4,3}=tr{4,3}
{8}
{4}
{6}		 26 72 48
[6] Snub octahedron
(same as Icosahedron)
=
s{3,4}=sr{3,3}
{3}
{3}		 20 30 12
[1] Half cube
(same as Tetrahedron)
=
h{4,3}={3,3}
1/2 {3}		 4 6 4
[2] Cantic cube
(same as Truncated tetrahedron)
=
h2{4,3}=t{3,3}
1/2 {6}
1/2 {3}		 8 18 12
[4] (same as Cuboctahedron)
=
rr{3,3}		 14 24 12
[5] (same as Truncated octahedron)
=
tr{3,3}		 14 36 24
[9] Cantic snub octahedron
(same as Rhombicuboctahedron)
s2{3,4}=rr{3,4}		 26 48 24
11 Snub cuboctahedron
sr{4,3}
{4}
2 {3}
{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: .

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

# Name Graph
(A2)
[6]		 Graph
(H3)
[10]		 Picture Tiling Vertex
figure		 Coxeter
and Schläfli
symbols		 Face counts by position Element counts
Pos. 2

[5]
(12)		 Pos. 1

[2]
(30)		 Pos. 0

[3]
(20)		 Faces Edges Vertices
12 Dodecahedron
{5,3}
{5}		 12 30 20
[6] Icosahedron
{3,5}
{3}		 20 30 12
13 Rectified dodecahedron
Rectified icosahedron
Icosidodecahedron
t1{5,3}=r{5,3}
{5}
{3}		 32 60 30
14 Truncated dodecahedron
t0,1{5,3}=t{5,3}
{10}
{3}		 32 90 60
15 Truncated icosahedron
t0,1{3,5}=t{3,5}
{5}
{6}		 32 90 60
16 Cantellated dodecahedron
Cantellated icosahedron
Rhombicosidodecahedron
t0,2{5,3}=rr{5,3}
{5}
{4}
{3}		 62 120 60
17 Omnitruncated dodecahedron
Omnitruncated icosahedron
Truncated icosidodecahedron
t0,1,2{5,3}=tr{5,3}
{10}
{4}
{6}		 62 180 120
18 Snub icosidodecahedron
sr{5,3}
{5}
2 {3}
{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: .

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:

# Name Picture Tiling Vertex
figure		 Coxeter
and Schläfli
symbols		 Face counts by position Element counts
Pos. 2

[2]
(2)		 Pos. 1

[2]
(2)		 Pos. 0

[2]
(2)		 Faces Edges Vertices
D2
H2		 Digonal dihedron,
digonal hosohedron
{2,2}
{2}		 2 2 2
D4 Truncated digonal dihedron
(same as square dihedron)
t{2,2}={4,2}
{4}		 2 4 4
P4
[7]		 Omnitruncated digonal dihedron
(same as cube)
t0,1,2{2,2}=tr{2,2}
{4}
{4}
{4}		 6 12 8
A2
[1]		 Snub digonal dihedron
(same as tetrahedron)
sr{2,2}
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:

# Name Picture Tiling Vertex
figure		 Coxeter
and Schläfli
symbols		 Face counts by position Element counts
Pos. 2

[3]
(2)		 Pos. 1

[2]
(3)		 Pos. 0

[2]
(3)		 Faces Edges Vertices
D3 Trigonal dihedron
{3,2}
{3}		 2 3 3
H3 Trigonal hosohedron
{2,3}
{2}		 3 3 2
D6 Truncated trigonal dihedron
(same as hexagonal dihedron)
t{3,2}
{6}		 2 6 6
P3 Truncated trigonal hosohedron
(Triangular prism)
t{2,3}
{3}
{4}		 5 9 6
P6 Omnitruncated trigonal dihedron
(Hexagonal prism)
t0,1,2{2,3}=tr{2,3}
{6}
{4}
{4}		 8 18 12
A3
[2]		 Snub trigonal dihedron
(same as Triangular antiprism)
(same as octahedron)
sr{2,3}
{3}
2 {3}		   8 12 6
P3 Cantic snub trigonal dihedron
(Triangular prism)
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:

# Name Picture Tiling Vertex
figure		 Coxeter
and Schläfli
symbols		 Face counts by position Element counts
Pos. 2

[4]
(2)		 Pos. 1

[2]
(4)		 Pos. 0

[2]
(4)		 Faces Edges Vertices
D4 square dihedron
{4,2}
{4}		 2 4 4
H4 square hosohedron
{2,4}
{2}		 4 4 2
D8 Truncated square dihedron
(same as octagonal dihedron)
t{4,2}
{8}		 2 8 8
P4
[7]		 Truncated square hosohedron
(Cube)
t{2,4}
{4}
{4}		 6 12 8
D8 Omnitruncated square dihedron
(Octagonal prism)
t0,1,2{2,4}=tr{2,4}
{8}
{4}
{4}		 10 24 16
A4 Snub square dihedron
(Square antiprism)
sr{2,4}
{4}
2 {3}		   10 16 8
P4
[7]		 Cantic snub square dihedron
(Cube)
s2{4,2}=t{2,4}		 6 12 8
A2
[1]		 Snub square hosohedron
(Digonal antiprism)
(Tetrahedron)
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:

# Name Picture Tiling Vertex
figure		 Coxeter
and Schläfli
symbols		 Face counts by position Element counts
Pos. 2

[5]
(2)		 Pos. 1

[2]
(5)		 Pos. 0

[2]
(5)		 Faces Edges Vertices
D5 Pentagonal dihedron
{5,2}
{5}		 2 5 5
H5 Pentagonal hosohedron
{2,5}
{2}		 5 5 2
D10 Truncated pentagonal dihedron
(same as decagonal dihedron)
t{5,2}
{10}		 2 10 10
P5 Truncated pentagonal hosohedron
(same as pentagonal prism)
t{2,5}
{5}
{4}		 7 15 10
P10 Omnitruncated pentagonal dihedron
(Decagonal prism)
t0,1,2{2,5}=tr{2,5}
{10}
{4}
{4}		 12 30 20
A5 Snub pentagonal dihedron
(Pentagonal antiprism)
sr{2,5}
{5}
2 {3}		   12 20 10
P5 Cantic snub pentagonal dihedron
(Pentagonal prism)
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

[6]
(2)		 Pos. 1

[2]
(6)		 Pos. 0

[2]
(6)		 Faces Edges Vertices
D6 Hexagonal dihedron
{6,2}
{6}		 2 6 6
H6 Hexagonal hosohedron
{2,6}
{2}		 6 6 2
D12 Truncated hexagonal dihedron
(same as dodecagonal dihedron)
t{6,2}
{12}		 2 12 12
H6 Truncated hexagonal hosohedron
(same as hexagonal prism)
t{2,6}
{6}
{4}		 8 18 12
P12 Omnitruncated hexagonal dihedron
(Dodecagonal prism)
t0,1,2{2,6}=tr{2,6}
{12}
{4}
{4}		 14 36 24
A6 Snub hexagonal dihedron
(Hexagonal antiprism)
sr{2,6}
{6}
2 {3}		   14 24 12
P3 Cantic hexagonal dihedron
(Triangular prism)		 =
h2{6,2}=t{2,3}		 5 9 6
P6 Cantic snub hexagonal dihedron
(Hexagonal prism)
s2{6,2}=t{2,6}		 8 18 12
A3
[2]		 Snub hexagonal hosohedron
(same as Triangular antiprism)
(same as octahedron)
s{2,6}=sr{2,3}		 8 12 6

Wythoff construction operators
Operation Symbol Coxeter
diagram		 Description
Parent {p,q}
t0{p,q}		 Any regular polyhedron or tiling
Rectified (r) r{p,q}
t1{p,q}		 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}
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}		 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.
Bitruncated (2t)
(also truncated dual)		 2t{p,q}
t1,2{p,q}		 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} 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}.
Cantitruncated (tr)
(Also omnitruncated)		 tr{p,q}
t0,1,2{p,q}		 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} 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.
Snub (s) s{p,2q} Alternated truncation
Cantic snub (s2) s2{p,2q}
Alternated cantellation (hrr) hrr{2p,2q} Only possible in uniform tilings (infinite polyhedra), alternation of
For example,
Half (h) h{2p,q} Alternation of , same as
Cantic (h2) h2{2p,q} Same as
Half rectified (hr) hr{2p,2q} Only possible in uniform tilings (infinite polyhedra), alternation of , same as or
For example, = or
Quarter (q) q{2p,2q} Only possible in uniform tilings (infinite polyhedra), same as
For example, = or

See also

Notes
  1. Regular Polytopes, p.13
  2. Piero della Francesca's Polyhedra
  3. "Stéréo-Club Français - Galerie : Polyedres".
  4. Har'El, Z. Uniform Solution for Uniform Polyhedra., Geometriae Dedicata 47, 57-110, 1993. Zvi Har’El, Kaleido software, Images, dual images
  5. Mäder, R. E. Uniform Polyhedra. Mathematica J. 3, 48-57, 1993.
  6. Closed-Form Expressions for Uniform Polyhedra and Their Duals, Peter W. Messer, Discrete Comput Geom 27:353–375 (2002)

References
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform 4-polytope 5-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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