Uniform polyhedron
A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the polyhedron has a high degree of reflectional and rotational symmetry.
Uniform polyhedra may be regular (if also face and edge transitive), quasi-regular (if edge transitive but not face transitive) or semi-regular (if neither edge nor face transitive). The faces and vertices need not be convex, so many of the uniform polyhedra are also star polyhedra.
Excluding the infinite sets, there are 75 uniform polyhedra (or 76 if edges are allowed to coincide).
- Convex
- 5 Platonic solids – regular convex polyhedra
- 13 Archimedean solids – 2 quasiregular and 11 semiregular convex polyhedra
- Star
- 4 Kepler–Poinsot polyhedra – regular nonconvex polyhedra
- 53 uniform star polyhedra – 5 quasiregular and 48 semiregular
- 1 star polyhedron found by John Skilling with pairs of edges that coincide, called the great disnub dirhombidodecahedron (Skilling's figure).
There are also two infinite sets of uniform prisms and antiprisms, including convex and star forms.
Dual polyhedra to uniform polyhedra are face-transitive (isohedral) and have regular vertex figures, and are generally classified in parallel with their dual (uniform) polyhedron. The dual of a regular polyhedron is regular, while the dual of an Archimedean solid is a Catalan solid.
The concept of uniform polyhedron is a special case of the concept of uniform polytope, which also applies to shapes in higher-dimensional (or lower-dimensional) space.
History
- The Platonic solids date back to the classical Greeks and were studied by Plato, Theaetetus and Euclid.
- Johannes Kepler (1571–1630) was the first to publish the complete list of Archimedean solids after the original work of Archimedes was lost.
Regular star polyhedra:
- Kepler (1619) discovered two of the regular Kepler–Poinsot polyhedra and Louis Poinsot (1809) discovered the other two.
Other 53 nonregular star polyhedra:
- Of the remaining 53, Albert Badoureau (1881) discovered 36. Edmund Hess (1878) discovered 2 more and Pitsch (1881) independently discovered 18, of which 15 had not previously been discovered.
- The geometer H.S.M. Coxeter discovered the remaining twelve in collaboration with J. C. P. Miller (1930–1932) but did not publish. M.S. Longuet-Higgins and H.C. Longuet-Higgins and independently discovered 11 of these.
- Template:Harvtxt published the list of uniform polyhedra.
- Template:Harvtxt 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.
- Template:Harvtxt independently proved the completeness, and showed that if the definition of uniform polyhedron is relaxed to allow edges to coincide then there is just one extra possibility.
- In 1987, Edmond Bonan draw all the uniform polyhedra and their duals in 3D, with a Turbo Pascal program called Polyca : almost of them were shown during the International Stereoscopic Union Congress held at the Congress Theatre, Eastbourne, United Kingdom.
- In 1993, Zvi Har'El produced a complete kaleidoscopic construction of the uniform polyhedra and duals with a computer program called Kaleido, and summarized in a paper Uniform Solution for Uniform Polyhedra, counting figures 1-80.
- Also in 1993, R. Mäder ported this Kaleido solution to Mathematica with a slightly different indexing system.
- In 2002 Peter W. Messer discovered a minimal set of closed-form expressions for determining the main combinatorial and metrical quantities of any uniform polyhedron (and its dual) given only its Wythoff symbol.^{[1]}
Uniform star polyhedra
The 57 nonprismatic nonconvex forms are compiled by Wythoff constructions within Schwarz triangles.
{{#invoke:main|main}}
Convex forms by Wythoff construction
The convex uniform polyhedra can be named by Wythoff construction operations and can be named in relation to the regular form.
In more detail the convex uniform polyhedron are given below by their Wythoff construction within each symmetry group.
Within the Wythoff construction, there are repetitions created by lower symmetry forms. The cube is a regular polyhedron, and a square prism. The octahedron is a regular polyhedron, and a triangular antiprism. The octahedron is also a rectified tetrahedron. Many polyhedra are repeated from different construction sources and are colored differently.
The Wythoff construction applies equally to uniform polyhedra and uniform tilings on the surface of a sphere, so images of both are given. The spherical tilings including the set of hosohedrons and dihedrons which are degenerate polyhedra.
These symmetry groups are formed from the reflectional point groups in three dimensions, each represented by a fundamental triangle (p q r), where p>1, q>1, r>1 and 1/p+1/q+1/r<1.
- Tetrahedral symmetry (3 3 2) - order 24
- Octahedral symmetry (4 3 2) - order 48
- Icosahedral symmetry (5 3 2) - order 120
- Dihedral symmetry (n 2 2), for all n=3,4,5,... - order 4n
The remaining nonreflective forms are constructed by alternation operations applied to the polyhedra with an even number of sides.
Along with the prisms and their dihedral symmetry, the spherical Wythoff construction process adds two regular classes which become degenerate as polyhedra - the dihedra and hosohedra, the first having only two faces, and the second only two vertices. The truncation of the regular hosohedra creates the prisms.
Below the convex uniform polyhedra are indexed 1-18 for the nonprismatic forms as they are presented in the tables by symmetry form. Repeated forms are in brackets.
For the infinite set of prismatic forms, they are indexed in four families:
- Hosohedra H_{2...} (Only as spherical tilings)
- Dihedra D_{2...} (Only as spherical tilings)
- Prisms P_{3...} (Truncated hosohedra)
- Antiprisms A_{3...} (Snub prisms)
Summary tables
Johnson name | Parent | Truncated | Rectified | Bitruncated (tr. dual) |
Birectified (dual) |
Cantellated | Omnitruncated (Cantitruncated) |
Snub |
---|---|---|---|---|---|---|---|---|
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} | |
t_{0}{p,q} | t_{0,1}{p,q} | t_{1}{p,q} | t_{1,2}{p,q} | t_{2}{p,q} | t_{0,2}{p,q} | t_{0,1,2}{p,q} | ht_{0,1,2}{p,q} | |
Wythoff symbol (p q 2) |
q | p 2 | 2 q | p | 2 | p q | 2 p | q | p | q 2 | p q | 2 | p q 2 | | | p q 2 |
Coxeter diagram | Template:CDD | Template:CDD | Template:CDD Template:CDD |
Template:CDD | Template:CDD | Template:CDD Template:CDD |
Template:CDD Template:CDD |
Template:CDD Template:CDD |
Vertex figure | p^{q} | (q.2p.2p) | (p.q)^{2} | (p.2q.2q) | q^{p} | (p.4.q.4) | (4.2p.2q) | (3.3.p.3.q) |
Tetrahedral (3 3 2) |
{3,3} |
(3.6.6) |
(3.3.3.3) |
(3.6.6) |
{3,3} |
(3.4.3.4) |
(4.6.6) |
(3.3.3.3.3) |
Octahedral (4 3 2) |
{4,3} |
(3.8.8) |
(3.4.3.4) |
(4.6.6) |
{3,4} |
(3.4.4.4) |
(4.6.8) |
(3.3.3.3.4) |
Icosahedral (5 3 2) |
{5,3} |
(3.10.10) |
(3.5.3.5) |
(5.6.6) |
{3,5} |
(3.4.5.4) |
(4.6.10) |
(3.3.3.3.5) |
And a sampling of Dihedral symmetries:
(p 2 2) | Parent | Truncated | Rectified | Bitruncated (tr. dual) |
Birectified (dual) |
Cantellated | Omnitruncated (Cantitruncated) |
Snub |
---|---|---|---|---|---|---|---|---|
Extended Schläfli symbol |
||||||||
{p,2} | t{p,2} | r{p,2} | 2t{p,2} | 2r{p,2} | rr{p,2} | tr{p,2} | sr{p,2} | |
t_{0}{p,2} | t_{0,1}{p,2} | t_{1}{p,2} | t_{1,2}{p,2} | t_{2}{p,2} | t_{0,2}{p,2} | t_{0,1,2}{p,2} | ht_{0,1,2}{p,2} | |
Wythoff symbol | 2 | p 2 | 2 2 | p | 2 | p 2 | 2 p | 2 | p | 2 2 | p 2 | 2 | p 2 2 | | | p 2 2 |
Coxeter-Dynkin diagram | Template:CDD | Template:CDD | Template:CDD | Template:CDD | Template:CDD | Template:CDD | Template:CDD | Template:CDD |
Vertex figure | p^{2} | (2.2p.2p) | (p. 2.p. 2) | (p. 4.4) | 2^{p} | (p. 4.2.4) | (4.2p.4) | (3.3.p. 3.2) |
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) |
{3,2} |
2.6.6 |
2.3.2.3 | 4.4.3 |
{2,3} |
2.4.3.4 |
4.4.6 |
3.3.3.3 |
Dihedral (4 2 2) |
{4,2} | 2.8.8 | 2.4.2.4 | 4.4.4 |
{2,4} |
2.4.4.4 |
4.4.8 |
3.3.3.4 |
Dihedral (5 2 2) |
{5,2} | 2.10.10 | 2.5.2.5 | 4.4.5 |
{2,5} |
2.4.5.4 |
4.4.10 |
3.3.3.5 |
Dihedral (6 2 2) |
{6,2} |
2.12.12 |
2.6.2.6 |
4.4.6 |
{2,6} |
2.4.6.4 |
4.4.12 |
3.3.3.6 |
Wythoff construction operators
Operation | Symbol | Coxeter diagram |
Description |
---|---|---|---|
Parent | {p,q} t_{0}{p,q} |
Template:CDD | Any regular polyhedron or tiling |
Rectified (r) | r{p,q} t_{1}{p,q} |
Template:CDD | The edges are fully truncated into single points. The polyhedron now has the combined faces of the parent and dual. |
Birectified (2r) (also dual) |
2r{p,q} t_{2}{p,q} |
Template:CDD | The birectified (dual) is a further truncation so that the original faces are reduced to points. New faces are formed under each parent vertex. The number of edges is unchanged and are rotated 90 degrees. The dual of the regular polyhedron {p, q} is also a regular polyhedron {q, p}. |
Truncated (t) | t{p,q} t_{0,1}{p,q} |
Template:CDD | 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} t_{1,2}{p,q} |
Template:CDD | Same as truncated dual. |
Cantellated (rr) (Also expanded) |
rr{p,q} | Template:CDD | 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. |
Cantitruncated (tr) (Also omnitruncated) |
tr{p,q} t_{0,1,2}{p,q} |
Template:CDD | 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. |
Operation | Symbol | Coxeter diagram |
Description |
---|---|---|---|
Snub rectified (sr) | sr{p,q} | Template:CDD | 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} | Template:CDD | Alternated truncation |
Cantic snub (s_{2}) | s_{2}{p,2q} | Template:CDD | |
Alternated cantellation (hrr) | hrr{2p,2q} | Template:CDD | Only possible in uniform tilings (infinite polyhedra), alternation of Template:CDD For example, Template:CDD |
Half (h) | h{2p,q} | Template:CDD | Alternation of Template:CDD, same as Template:CDD |
Cantic (h_{2}) | h_{2}{2p,q} | Template:CDD | Same as Template:CDD |
Half rectified (hr) | hr{2p,2q} | Template:CDD | Only possible in uniform tilings (infinite polyhedra), alternation of Template:CDD, same as Template:CDD or Template:CDD For example, Template:CDD = Template:CDD or Template:CDD |
Quarter (q) | q{2p,2q} | Template:CDD | Only possible in uniform tilings (infinite polyhedra), same as Template:CDD For example, Template:CDD = Template:CDD or Template:CDD |
(3 3 2) T_{d} 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 A_{2} or [3,3], as well as a Coxeter-Dynkin diagram: Template:CDD.
There are 24 triangles, visible in the faces of the tetrakis hexahedron and alternately colored triangles on a sphere:
# | Name | Graph A_{3} |
Graph A_{2} |
Picture | Tiling | Vertex figure |
Coxeter-Dynkin and Schläfli symbols |
Face counts by position | Element counts | ||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Pos. 2 Template:CDD [3] (4) |
Pos. 1 Template:CDD [2] (6) |
Pos. 0 Template:CDD [3] (4) |
Faces | Edges | Vertices | ||||||||
1 | Tetrahedron | Template:CDD {3,3} |
{3} |
4 | 6 | 4 | |||||||
[1] | Birectified tetrahedron (Same as tetrahedron) |
Template:CDD t_{2}{3,3}={3,3} |
{3} |
4 | 6 | 4 | |||||||
2 | Rectified tetrahedron (Same as octahedron) |
Template:CDD t_{1}{3,3}=r{3,3} |
{3} |
{3} |
8 | 12 | 6 | ||||||
3 | Truncated tetrahedron | Template:CDD t_{0,1}{3,3}=t{3,3} |
{6} |
{3} |
8 | 18 | 12 | ||||||
[3] | Bitruncated tetrahedron (Same as truncated tetrahedron) |
Template:CDD t_{1,2}{3,3}=t{3,3} |
{3} |
{6} |
8 | 18 | 12 | ||||||
4 | Rhombitetratetrahedron (Same as cuboctahedron) |
Template:CDD t_{0,2}{3,3}=rr{3,3} |
{3} |
{4} |
{3} |
14 | 24 | 12 | |||||
5 | Truncated tetratetrahedron (Same as truncated octahedron) |
Template:CDD t_{0,1,2}{3,3}=tr{3,3} |
{6} |
{4} |
{6} |
14 | 36 | 24 | |||||
6 | Snub tetratetrahedron (Same as icosahedron) |
Template:CDD sr{3,3} |
{3} |
2 {3} |
{3} |
20 | 30 | 12 |
(4 3 2) O_{h} Octahedral symmetry
The octahedral symmetry of the sphere generates 7 uniform polyhedra, and a 3 more by alternation. Four of these forms are repeated from the tetrahedral symmetry table above.
The octahedral symmetry is represented by a fundamental triangle (4 3 2) counting the mirrors at each vertex. It can also be represented by the Coxeter group B_{2} or [4,3], as well as a Coxeter-Dynkin diagram: Template:CDD.
There are 48 triangles, visible in the faces of the disdyakis dodecahedron and alternately colored triangles on a sphere:
# | Name | Graph B_{3} |
Graph B_{2} |
Picture | Tiling | Vertex figure |
Coxeter-Dynkin and Schläfli symbols |
Face counts by position | Element counts | ||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Pos. 2 Template:CDD [4] (8) |
Pos. 1 Template:CDD [2] (12) |
Pos. 0 Template:CDD [3] (6) |
Faces | Edges | Vertices | ||||||||
7 | Cube | Template:CDD {4,3} |
{4} |
6 | 12 | 8 | |||||||
[2] | Octahedron | Template:CDD {3,4} |
{3} |
8 | 12 | 6 | |||||||
[4] | rectified cube rectified octahedron (Cuboctahedron) |
Template:CDD {4,3} |
{4} |
{3} |
14 | 24 | 12 | ||||||
8 | Truncated cube | Template:CDD t_{0,1}{4,3}=t{4,3} |
{8} |
{3} |
14 | 36 | 24 | ||||||
[5] | Truncated octahedron | Template:CDD t_{0,1}{3,4}=t{3,4} |
{4} |
{6} |
14 | 36 | 24 | ||||||
9 | Cantellated cube cantellated octahedron Rhombicuboctahedron |
Template:CDD t_{0,2}{4,3}=rr{4,3} |
{8} |
{4} |
{6} |
26 | 48 | 24 | |||||
10 | Omnitruncated cube omnitruncated octahedron Truncated cuboctahedron |
Template:CDD t_{0,1,2}{4,3}=tr{4,3} |
{8} |
{4} |
{6} |
26 | 72 | 48 | |||||
[6] | Alternated truncated octahedron (Same as Icosahedron) |
Template:CDD = Template:CDD s{3,4}=sr{3,3} |
{3} |
{3} |
20 | 30 | 12 | ||||||
[1] | Half cube (Same as tetrahedron) |
Template:CDD = Template:CDD h{4,3}={3,3} |
^{1}/_{2} {3} |
4 | 6 | 4 | |||||||
[2] | Cantic cube (Same as Truncated tetrahedron) |
Template:CDD = Template:CDD h_{2}{4,3}=t{3,3} |
^{1}/_{2} {6} |
^{1}/_{2} {3} |
8 | 18 | 12 | ||||||
11 | Snub cube | Template:CDD sr{4,3} |
{4} |
2 {3} |
{3} |
38 | 60 | 24 |
(5 3 2) I_{h} 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 G_{2} or [5,3], as well as a Coxeter-Dynkin diagram: Template:CDD.
There are 120 triangles, visible in the faces of the disdyakis triacontahedron and alternately colored triangles on a sphere:
# | Name | Graph (A_{2}) [6] |
Graph (H_{3}) [10] |
Picture | Tiling | Vertex figure |
Coxeter-Dynkin and Schläfli symbols |
Face counts by position | Element counts | ||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Pos. 2 Template:CDD [5] (12) |
Pos. 1 Template:CDD [2] (30) |
Pos. 0 Template:CDD [3] (20) |
Faces | Edges | Vertices | ||||||||
12 | Dodecahedron | Template:CDD {5,3} |
{5} |
12 | 30 | 20 | |||||||
[6] | Icosahedron | Template:CDD {3,5} |
{3} |
20 | 30 | 12 | |||||||
13 | Rectified dodecahedron Rectified icosahedron Icosidodecahedron |
Template:CDD t_{1}{5,3}=r{5,3} |
{5} |
{3} |
32 | 60 | 30 | ||||||
14 | Truncated dodecahedron | Template:CDD t_{0,1}{5,3}=t{5,3} |
{10} |
{3} |
32 | 90 | 60 | ||||||
15 | Truncated icosahedron | Template:CDD t_{0,1}{3,5}=t{3,5} |
{5} |
{6} |
32 | 90 | 60 | ||||||
16 | Cantellated dodecahedron Cantellated icosahedron Rhombicosidodecahedron |
Template:CDD t_{0,2}{5,3}=rr{5,3} |
{5} |
{4} |
{3} |
62 | 120 | 60 | |||||
17 | Omnitruncated dodecahedron Omnitruncated icosahedron Truncated icosidodecahedron |
Template:CDD t_{0,1,2}{5,3}=tr{5,3} |
{10} |
{4} |
{6} |
62 | 180 | 120 | |||||
18 | Snub dodecahedron Snub icosahedron |
Template:CDD sr{5,3} |
{5} |
2 {3} |
{3} |
92 | 150 | 60 |
(p 2 2) Prismatic [p,2], I_{2}(p) family (D_{ph} Dihedral symmetry)
{{#invoke:main|main}}
The dihedral symmetry of the sphere generates two infinite sets of uniform polyhedra, prisms and antiprisms, and two more infinite set of degenerate polygons, the hosohedrons and dihedrons which exists as tilings on the sphere.
The dihedral symmetry is represented by a fundamental triangle (p 2 2) counting the mirrors at each vertex. It can also be represented by the Coxeter group I_{2}(p) or [n,2], as well as a prismatic Coxeter-Dynkin diagram: Template:CDD.
Below are the first five dihedral symmetries: D_{2} ... D_{6}. The dihedral symmetry D_{p} 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-Dynkin and Schläfli symbols |
Face counts by position | Element counts | ||||
---|---|---|---|---|---|---|---|---|---|---|---|
Pos. 2 Template:CDD [2] (2) |
Pos. 1 Template:CDD [2] (2) |
Pos. 0 Template:CDD [2] (2) |
Faces | Edges | Vertices | ||||||
D_{2} H_{2} |
digonal dihedron digonal hosohedron |
Template:CDD {2,2} |
{2} |
2 | 2 | 2 | |||||
D_{4} | truncated digonal dihedron (Same as square dihedron) |
Template:CDD t{2,2}={4,2} |
{4} |
2 | 4 | 4 | |||||
P_{4} [7] |
omnitruncated digonal dihedron (Same as cube) |
Template:CDD t_{0,1,2}{2,2}=tr{2,2} |
{4} |
{4} |
{4} |
6 | 12 | 8 | |||
A_{2} [1] |
snub digonal dihedron (Same as tetrahedron) |
Template:CDD sr{2,2} |
2 {3} |
4 | 6 | 4 |
(3 2 2) D_{3h}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-Dynkin and Schläfli symbols |
Face counts by position | Element counts | ||||
---|---|---|---|---|---|---|---|---|---|---|---|
Pos. 2 Template:CDD [3] (2) |
Pos. 1 Template:CDD [2] (3) |
Pos. 0 Template:CDD [2] (3) |
Faces | Edges | Vertices | ||||||
D_{3} | Trigonal dihedron | Template:CDD {3,2} |
{3} |
2 | 3 | 3 | |||||
H_{3} | Trigonal hosohedron | Template:CDD {2,3} |
{2} |
3 | 3 | 2 | |||||
D_{6} | Truncated trigonal dihedron (Same as hexagonal dihedron) |
Template:CDD t{3,2} |
{6} |
2 | 6 | 6 | |||||
P_{3} | Truncated trigonal hosohedron (Triangular prism) |
Template:CDD t{2,3} |
{3} |
{4} |
5 | 9 | 6 | ||||
P_{6} | Omnitruncated trigonal dihedron (Hexagonal prism) |
Template:CDD t_{0,1,2}{2,3}=tr{2,3} |
{6} |
{4} |
{4} |
8 | 18 | 12 | |||
A_{3} [2] |
Snub trigonal dihedron (Same as Triangular antiprism) (Same as octahedron) |
Template:CDD sr{2,3} |
{3} |
2 {3} |
8 | 12 | 6 |
(4 2 2) D_{4h}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-Dynkin and Schläfli symbols |
Face counts by position | Element counts | ||||
---|---|---|---|---|---|---|---|---|---|---|---|
Pos. 2 Template:CDD [4] (2) |
Pos. 1 Template:CDD [2] (4) |
Pos. 0 Template:CDD [2] (4) |
Faces | Edges | Vertices | ||||||
D_{4} | square dihedron | Template:CDD {4,2} |
{4} |
2 | 4 | 4 | |||||
H_{4} | square hosohedron | Template:CDD {2,4} |
{2} |
4 | 4 | 2 | |||||
D_{8} | Truncated square dihedron (Same as octagonal dihedron) |
Template:CDD t{4,2} |
{8} |
2 | 8 | 8 | |||||
P_{4} [7] |
Truncated square hosohedron (Cube) |
Template:CDD t{2,4} |
{4} |
{4} |
6 | 12 | 8 | ||||
D_{8} | Omnitruncated square dihedron (Octagonal prism) |
Template:CDD t_{0,1,2}{2,4}=tr{2,4} |
{8} |
{4} |
{4} |
10 | 24 | 16 | |||
A_{4} | Snub square dihedron (Square antiprism) |
Template:CDD sr{2,4} |
{4} |
2 {3} |
10 | 16 | 8 |
(5 2 2) D_{5h} 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-Dynkin and Schläfli symbols |
Face counts by position | Element counts | ||||
---|---|---|---|---|---|---|---|---|---|---|---|
Pos. 2 Template:CDD [5] (2) |
Pos. 1 Template:CDD [2] (5) |
Pos. 0 Template:CDD [2] (5) |
Faces | Edges | Vertices | ||||||
D_{5} | Pentagonal dihedron | Template:CDD {5,2} |
{5} |
2 | 5 | 5 | |||||
H_{5} | Pentagonal hosohedron | Template:CDD {2,5} |
{2} |
5 | 5 | 2 | |||||
D_{10} | Truncated pentagonal dihedron (Same as decagonal dihedron) |
Template:CDD t{5,2} |
{10} |
2 | 10 | 10 | |||||
P_{5} | Truncated pentagonal hosohedron (Same as pentagonal prism) |
Template:CDD t{2,5} |
{5} |
{4} |
7 | 15 | 10 | ||||
P_{10} | Omnitruncated pentagonal dihedron (Decagonal prism) |
Template:CDD t_{0,1,2}{2,5}=tr{2,5} |
{10} |
{4} |
{4} |
12 | 30 | 20 | |||
A_{5} | Snub pentagonal dihedron (Pentagonal antiprism) |
Template:CDD sr{2,5} |
{5} |
2 {3} |
12 | 20 | 10 |
(6 2 2) D_{6h}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-Dynkin and Schläfli symbols |
Face counts by position | Element counts | ||||
---|---|---|---|---|---|---|---|---|---|---|---|
Pos. 2 Template:CDD [6] (2) |
Pos. 1 Template:CDD [2] (6) |
Pos. 0 Template:CDD [2] (6) |
Faces | Edges | Vertices | ||||||
D_{6} | Hexagonal dihedron | Template:CDD {6,2} |
{6} |
2 | 6 | 6 | |||||
H_{6} | Hexagonal hosohedron | Template:CDD {2,6} |
{2} |
6 | 6 | 2 | |||||
D_{12} | Truncated hexagonal dihedron (Same as dodecagonal dihedron) |
Template:CDD t{6,2} |
{12} |
2 | 12 | 12 | |||||
H_{6} | Truncated hexagonal hosohedron (Same as hexagonal prism) |
Template:CDD t{2,6} |
{6} |
{4} |
8 | 18 | 12 | ||||
P_{12} | Omnitruncated hexagonal dihedron (Dodecagonal prism) |
Template:CDD t_{0,1,2}{2,6}=tr{2,6} |
{12} |
{4} |
{4} |
14 | 36 | 24 | |||
A_{6} | Snub hexagonal dihedron (Hexagonal antiprism) |
Template:CDD sr{2,6} |
{6} |
2 {3} |
14 | 24 | 12 |
See also
- Polyhedron
- List of uniform polyhedra
- List of Johnson solids
- List of Wenninger polyhedron models
- Polyhedron model
- List of uniform polyhedra by vertex figure
- List of uniform polyhedra by Wythoff symbol
- List of uniform polyhedra by Schwarz triangle
- Uniform tiling
- Uniform tilings in hyperbolic plane
Notes
References
- Brückner, M. Vielecke und vielflache. Theorie und geschichte.. Leipzig, Germany: Teubner, 1900. [1]
- {{#invoke:Citation/CS1|citation
|CitationClass=journal }}
- {{#invoke:Citation/CS1|citation
|CitationClass=journal }}
- {{#invoke:citation/CS1|citation
|CitationClass=book }}
- {{#invoke:Citation/CS1|citation
|CitationClass=journal }}
- Har'El, Z. Uniform Solution for Uniform Polyhedra., Geometriae Dedicata 47, 57-110, 1993. Zvi Har’El, Kaleido software, Images, dual images
- Mäder, R. E. Uniform Polyhedra. Mathematica J. 3, 48-57, 1993. [2]
- Messer, Peter W. Closed-Form Expressions for Uniform Polyhedra and Their Duals., Discrete & Computational Geometry 27:353-375 (2002).