View source for Uniform polyhedron

{{Short description|Isogonal polyhedron with regular faces}}
[[File:Tetrahedron.jpg|150px|thumb|[[Platonic solid|Platonic solid]]: [[Tetrahedron|Tetrahedron]]]]
[[File:Snub dodecadodecahedron.png|150px|thumb|[[Uniform star polyhedron|Uniform star polyhedron]]: [[Snub dodecadodecahedron|Snub dodecadodecahedron]]]]

In [[Geometry|geometry]], a '''uniform polyhedron''' has [[Regular polygon|regular polygon]]s as [[Face (geometry)|faces]] and is vertex-transitive (i.e., there is an [[Isometry|isometry]] mapping any vertex onto any other). It follows that all vertices are [[Congruence (geometry)|congruent]].

Uniform polyhedra may be [[Regular polyhedron|regular]] (if also [[Isohedral figure|face-]] and [[Isotoxal figure|edge-transitive]]), [[Quasiregular polyhedron|quasi-regular]] (if also edge-transitive but not face-transitive), or [[Semiregular polyhedron|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 polyhedron|star polyhedra]].

There are two infinite classes of uniform polyhedra, together with 75 other polyhedra:
*Infinite classes:
**[[Prism (geometry)|prisms]],
**[[Antiprism|antiprism]]s.
* Convex exceptional:
** 5 [[Platonic solid]]s: regular convex polyhedra,
** 13 [[Archimedean solid]]s: 2 [[Quasiregular polyhedron|quasiregular]] and 11 [[Semiregular polyhedron|semiregular]] convex polyhedra.
* Star (nonconvex) exceptional:
** 4 Kepler–Poinsot polyhedra: regular nonconvex polyhedra,
** 53 uniform star polyhedra: 14 [[Quasiregular polyhedron#Nonconvex examples|quasiregular]] and 39 semiregular.

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|great disnub dirhombidodecahedron]] (Skilling's figure).

Dual polyhedra to uniform polyhedra are face-transitive (isohedral) and have regular [[Vertex figure|vertex figure]]s, 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 ==
{{quotebox|thumb|width=33%|source={{harvs|first=Branko |last=Grünbaum|year=1994}}|quote=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.}}

{{harvtxt|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.  {{harvtxt|Grünbaum|1994}} gave a rather complicated definition of a polyhedron, while
{{harvtxt|McMullen|Schulte|2002}} gave a simpler and more general definition of a polyhedron: in their terminology, a polyhedron is a 2-dimensional [[Abstract polytope|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 ===
* The [[Platonic solid]]s date back to the classical Greeks and were studied by the [[Philosophy:Pythagoreanism|Pythagorean]]s, [[Biography:Plato|Plato]] (c. 424 – 348 BC), [[Biography:Theaetetus (mathematician)|Theaetetus]] (c. 417 BC – 369 BC), [[Biography:Timaeus of Locri|Timaeus of Locri]] (c. 420–380 BC), and [[Biography:Euclid|Euclid]] (fl. 300 BC). The Etruscans discovered the regular dodecahedron before 500 BC.<ref>Regular Polytopes, p.13</ref>

=== Nonregular uniform convex polyhedra ===
* The [[Cuboctahedron|cuboctahedron]] was known by [[Biography:Plato|Plato]].
* [[Biography:Archimedes|Archimedes]] (287 BC – 212 BC) discovered all of the 13 [[Archimedean solid]]s. His original book on the subject was lost, but [[Biography:Pappus of Alexandria|Pappus of Alexandria]] (c. 290 – c. 350 AD) mentioned Archimedes listed 13 polyhedra.
* [[Biography:Piero della Francesca|Piero della Francesca]] (1415 – 1492) rediscovered the five truncations of the Platonic solids—truncated tetrahedron, truncated octahedron, truncated cube, truncated dodecahedron, and truncated icosahedron—and included illustrations and calculations of their metric properties in his book ''[[De quinque corporibus regularibus]]''. He also discussed the cuboctahedron in a different book.<ref>[http://www.georgehart.com/virtual-polyhedra/piero.html Piero della Francesca's Polyhedra]</ref>
* [[Biography:Luca Pacioli|Luca Pacioli]] plagiarized Francesca's work in ''[[De divina proportione]]'' in 1509, adding the [[Rhombicuboctahedron|rhombicuboctahedron]], calling it an ''icosihexahedron'' for its 26 faces, which was drawn by [[Biography:Leonardo da Vinci|Leonardo da Vinci]].
*[[Biography:Johannes Kepler|Johannes Kepler]] (1571–1630) was the first to publish the complete list of [[Archimedean solid]]s, in 1619. He also identified the infinite families of uniform [[Prismatic uniform polyhedron|prisms and antiprisms]].

=== Regular star polyhedra ===
* [[Biography:Johannes Kepler|Kepler]] (1619) discovered two of the regular Kepler–Poinsot polyhedra, the [[Small stellated dodecahedron|small stellated dodecahedron]] and [[Great stellated dodecahedron|great stellated dodecahedron]].
*Louis Poinsot (1809) discovered the other two, the [[Great dodecahedron|great dodecahedron]] and [[Great icosahedron|great icosahedron]].
*The set of four was proven complete by [[Biography:Augustin-Louis Cauchy|Augustin-Louis Cauchy]] in 1813 and named by [[Biography:Arthur Cayley|Arthur Cayley]] in 1859.

=== Other 53 nonregular star polyhedra ===
* Of the remaining 53, Edmund Hess (1878) discovered 2, 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 [[Biography:Harold Scott MacDonald Coxeter|H.S.M. Coxeter]] discovered the remaining twelve in collaboration with [[Biography:J. C. P. Miller|J. C. P. Miller]] (1930–1932) but did not publish. [[Biography:Michael S. Longuet-Higgins|M.S. Longuet-Higgins]] and H.C. Longuet-Higgins independently discovered eleven of these. Lesavre and Mercier rediscovered five of them in 1947.
* {{harvtxt|Coxeter|Longuet-Higgins|Miller|1954}} published the list of uniform polyhedra.
* {{harvtxt|Sopov|1970}} proved their conjecture that the list was complete.
* In 1974, Magnus Wenninger published his book [[List of Wenninger polyhedron models|''Polyhedron models'']], which lists all 75 nonprismatic uniform polyhedra, with many previously unpublished names given to them by [[Biography:Norman Johnson (mathematician)|Norman Johnson]].
* {{harvtxt|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 (the [[Great disnub dirhombidodecahedron|great disnub dirhombidodecahedron]]).
* In 1987, [[Biography:Edmond Bonan|Edmond Bonan]] drew all the uniform polyhedra and their duals in 3D with a Turbo Pascal program called '''Polyca'''. Most of them were shown during the International Stereoscopic Union Congress held in 1993, at the Congress Theatre, Eastbourne, England; and again in 2005 at the Kursaal of Besançon, France.<ref>Edmond Bonan, "Polyèdres Eastbourne 1993", Stéréo-Club Français 1993, [https://www.image-en-relief.org/albums_3D/liste_images_repertoire.php?dirAlbum=Bonan_Edmond_1993-Polyedres_Eastbourne]</ref>
* In 1993, Zvi Har'El (1949–2008)<ref>[http://www.verniana.org/volumes/01/LetterSize/Tribute.pdf Dr. Zvi Har’El (December 14, 1949 – February 2, 2008) and International Jules Verne Studies - A Tribute]</ref> produced a complete kaleidoscopic construction of the uniform polyhedra and duals with a computer program called '''Kaleido''' and summarized it in a paper ''Uniform Solution for Uniform Polyhedra'', counting figures 1-80.<ref>{{cite journal
| last1=Har'el | first1=Zvi
| url=http://harel.org.il/zvi/docs/uniform.pdf
| title=Uniform Solution for Uniform Polyhedra
| journal=[[Geometriae Dedicata]]
| volume=47
| pages=57–110
| date=1993
| doi=10.1007/BF01263494 | doi-access=free}} [http://harel.org.il/zvi Zvi Har’El], [http://harel.org.il/zvi/kaleido/ Kaleido software], [http://harel.org.il/zvi/kaleido/poly.html Images], [http://harel.org.il/zvi/kaleido/dual.html dual images]</ref>
* Also in 1993, R. Mäder ported this Kaleido solution to [[Software:Mathematica|Mathematica]] with a slightly different indexing system.<ref>[http://www.mathconsult.ch/showroom/unipoly Mäder, R. E.] ''Uniform Polyhedra.'' Mathematica J. 3, 48-57, 1993. [http://library.wolfram.com/infocenter/Articles/2254]</ref>
* 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]].<ref>{{cite journal
| first1=Peter W. | last1=Messer
| title=Closed-Form Expressions for Uniform Polyhedra and Their Duals
| journal=Discrete & Computational Geometry
| volume=27
| pages=353–375
| year=2002
| issue=3
| doi=10.1007/s00454-001-0078-2
| doi-access=free}}</ref>

== Uniform star polyhedra ==
{{main|Uniform star polyhedron}}
[[File:Great dirhombicosidodecahedron.png|thumb|The great dirhombicosidodecahedron, the only non-Wythoffian uniform polyhedron]]
The 57 nonprismatic nonconvex forms, with exception of the [[Great dirhombicosidodecahedron|great dirhombicosidodecahedron]], are compiled by Wythoff constructions within [[Schwarz triangle]]s.

== Convex forms by Wythoff construction ==
[[File:Wythoffian construction diagram.svg|400px|right]]
[[File:Polyhedron truncation example3.png|400px|right|Example forms from the [[Cube|cube]] and [[Octahedron|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|octahedron]] is a regular polyhedron, and a triangular antiprism. The [[Octahedron|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 [[Spherical polyhedron|uniform tilings on the surface of a sphere]], so images of both are given. The spherical tilings including the set of [[Hosohedron|hosohedron]]s and [[Dihedron|dihedron]]s which are degenerate polyhedra.

These symmetry groups are formed from the reflectional [[Point groups in three dimensions|point groups in three dimensions]], each represented by a fundamental triangle (''p''&nbsp;''q''&nbsp;''r''), where ''p''&nbsp;>&nbsp;1, ''q''&nbsp;>&nbsp;1, ''r''&nbsp;>&nbsp;1 and {{nowrap|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 ''n'' = 3,4,5,... – order 4''n''

The remaining nonreflective forms are constructed by [[Alternation (geometry)|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 ''[[Dihedron|dihedra]]'' and the ''[[Hosohedron|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:
# [[Hosohedron|Hosohedra]] ''H''<sub>2...</sub> (only as spherical tilings)
# [[Dihedron|Dihedra]] ''D''<sub>2...</sub> (only as spherical tilings)
# [[Prism (geometry)|Prisms]] ''P''<sub>3...</sub> (truncated hosohedra)
# [[Antiprism]]s ''A''<sub>3...</sub> (snub prisms)

=== Summary tables ===

{| class="wikitable" width=640
|- valign=top
![[Biography:Norman Johnson (mathematician)|Johnson]] name
!Parent
!Truncated
!Rectified
!Bitruncated<br />(tr. dual)
!Birectified<br />(dual)
!Cantellated
!Omnitruncated<br />(<small>cantitruncated</small>)
!Snub
|- valign=top
!Coxeter diagram
!{{CDD|node_1|p|node|q|node}}
!{{CDD|node_1|p|node_1|q|node}}
!{{CDD|node|p|node_1|q|node}}<br />{{CDD|node_1|split1-pq|nodes}}
!{{CDD|node|p|node_1|q|node_1}}
!{{CDD|node|p|node|q|node_1}}
!{{CDD|node_1|p|node|q|node_1}}<br />{{CDD|node|split1-pq|nodes_11}}
!{{CDD|node_1|p|node_1|q|node_1}}<br />{{CDD|node_1|split1-pq|nodes_11}}
!{{CDD|node_h|p|node_h|q|node_h}}<br />{{CDD|node_h|split1-pq|nodes_hh}}
|-
!rowspan=3|Extended<br />[[Schläfli symbol]]
!<math>\begin{Bmatrix} p , q \end{Bmatrix}</math>
!<math>t\begin{Bmatrix} p , q \end{Bmatrix}</math>
!<math>\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
!<math>t\begin{Bmatrix} q , p \end{Bmatrix}</math>
!<math>\begin{Bmatrix} q , p \end{Bmatrix}</math>
!<math>r\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
!<math>t\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
!<math>s\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
|-
!{p,q}
!t{p,q}
!r{p,q}
!2t{p,q}
!2r{p,q}
!rr{p,q}
!tr{p,q}
!sr{p,q}
|-
!t<sub>0</sub>{p,q}
!t<sub>0,1</sub>{p,q}
!t<sub>1</sub>{p,q}
!t<sub>1,2</sub>{p,q}
!t<sub>2</sub>{p,q}
!t<sub>0,2</sub>{p,q}
!t<sub>0,1,2</sub>{p,q}
!ht<sub>0,1,2</sub>{p,q}
|-
![[Wythoff symbol]]<br />(p q 2)
! q &#124; p 2
! 2 q &#124; p
! 2 &#124; p q
! 2 p &#124; q
! p &#124; q 2
! p q &#124; 2
! p q 2 &#124;
! &#124; p q 2
|-
![[Vertex configuration|Vertex figure]]
!p<sup>q</sup>
!q.2p.2p
!(p.q)<sup>2</sup>
!p.2q.2q
!q<sup>p</sup>
!p.4.q.4
!4.2p.2q
!3.3.p.3.q
|- align=center
|[[Tetrahedral symmetry|Tetrahedral]]<br />(3 3 2)
|[[File:Uniform polyhedron-33-t0.png|64px]]<br />[[Tetrahedron|3.3.3]]
|[[File:Uniform polyhedron-33-t01.png|64px]]<br />[[Truncated tetrahedron|3.6.6]]
|[[File:Uniform polyhedron-33-t1.png|64px]]<br />[[Octahedron|3.3.3.3]]
|[[File:Uniform polyhedron-33-t12.png|64px]]<br />[[Truncated tetrahedron|3.6.6]]
|[[File:Uniform polyhedron-33-t2.png|64px]]<br />[[Tetrahedron|3.3.3]]
|[[File:Uniform polyhedron-33-t02.png|64px]]<br />[[Cuboctahedron|3.4.3.4]]
|[[File:Uniform polyhedron-33-t012.png|64px]]<br />[[Truncated octahedron|4.6.6]]
|[[File:Uniform polyhedron-33-s012.svg|64px]]<br />[[Icosahedron|3.3.3.3.3]]
|- align=center
|[[Octahedral symmetry|Octahedral]]<br />(4 3 2)
|[[File:Uniform polyhedron-43-t0.svg|64px]]<br />[[Cube|4.4.4]]
|[[File:Uniform polyhedron-43-t01.svg|64px]]<br />[[Truncated cube|3.8.8]]
|[[File:Uniform polyhedron-43-t1.svg|64px]]<br />[[Cuboctahedron|3.4.3.4]]
|[[File:Uniform polyhedron-43-t12.svg|64px]]<br />[[Truncated octahedron|4.6.6]]
|[[File:Uniform polyhedron-43-t2.svg|64px]]<br />[[Octahedron|3.3.3.3]]
|[[File:Uniform polyhedron-43-t02.png|64px]]<br />3.4.4.4
|[[File:Uniform polyhedron-43-t012.png|64px]]<br />[[Truncated cuboctahedron|4.6.8]]
|[[File:Uniform polyhedron-43-s012.png|64px]]<br />[[Snub cube|3.3.3.3.4]]
|- align=center
|[[Icosahedral symmetry|Icosahedral]]<br />(5 3 2)
|[[File:Uniform polyhedron-53-t0.svg|64px]]<br />[[Dodecahedron|5.5.5]]
|[[File:Uniform polyhedron-53-t01.svg|64px]]<br />[[Truncated dodecahedron|3.10.10]]
|[[File:Uniform polyhedron-53-t1.svg|64px]]<br />[[Icosidodecahedron|3.5.3.5]]
|[[File:Uniform polyhedron-53-t12.svg|64px]]<br />[[Truncated icosahedron|5.6.6]]
|[[File:Uniform polyhedron-53-t2.svg|64px]]<br />[[Icosahedron|3.3.3.3.3]]
|[[File:Uniform polyhedron-53-t02.png|64px]]<br />[[Rhombicosidodecahedron|3.4.5.4]]
|[[File:Uniform polyhedron-53-t012.png|64px]]<br />[[Truncated icosidodecahedron|4.6.10]]
|[[File:Uniform polyhedron-53-s012.png|64px]]<br />[[Snub dodecahedron|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.)

{| class="wikitable" width=640
|-
!(p 2 2)
!Parent
!Truncated
!Rectified
!Bitruncated<br />(tr. dual)
!Birectified<br />(dual)
!Cantellated
!Omnitruncated<br />(<small>cantitruncated</small>)
!Snub
|-
!Coxeter diagram
!{{CDD|node_1|p|node|2|node}}
!{{CDD|node_1|p|node_1|2|node}}
!{{CDD|node|p|node_1|2|node}}
!{{CDD|node|p|node_1|2|node_1}}
!{{CDD|node|p|node|2|node_1}}
!{{CDD|node_1|p|node|2|node_1}}
!{{CDD|node_1|p|node_1|2|node_1}}
!{{CDD|node_h|p|node_h|2x|node_h}}
|-
!rowspan=3|Extended<br />[[Schläfli symbol]]
!<math>\begin{Bmatrix} p , 2 \end{Bmatrix}</math>
!<math>t\begin{Bmatrix} p , 2 \end{Bmatrix}</math>
!<math>\begin{Bmatrix} p \\ 2 \end{Bmatrix}</math>
!<math>t\begin{Bmatrix} 2 , p \end{Bmatrix}</math>
!<math>\begin{Bmatrix} 2 , p \end{Bmatrix}</math>
!<math>r\begin{Bmatrix} p \\ 2 \end{Bmatrix}</math>
!<math>t\begin{Bmatrix} p \\ 2 \end{Bmatrix}</math>
!<math>s\begin{Bmatrix} p \\ 2 \end{Bmatrix}</math>
|-
!{p,2}
!t{p,2}
!r{p,2}
!2t{p,2}
!2r{p,2}
!rr{p,2}
!tr{p,2}
!sr{p,2}
|-
!t<sub>0</sub>{p,2}
!t<sub>0,1</sub>{p,2}
!t<sub>1</sub>{p,2}
!t<sub>1,2</sub>{p,2}
!t<sub>2</sub>{p,2}
!t<sub>0,2</sub>{p,2}
!t<sub>0,1,2</sub>{p,2}
!ht<sub>0,1,2</sub>{p,2}
|-
![[Wythoff symbol]]
! 2 &#124; p 2
! 2 2 &#124; p
! 2 &#124; p 2
! 2 p &#124; 2
! p &#124; 2 2
! p 2 &#124; 2
! p 2 2 &#124;
! &#124; p 2 2
|-
![[Vertex configuration|Vertex figure]]
!p<sup>2</sup>
!2.2p.2p
!p.2.p.2
!p.4.4
!2<sup>p</sup>
!p.4.2.4
!4.2p.4
!3.3.3.p
|- align=center
|Dihedral<br />(2 2 2)
| [[File:Digonal dihedron.png|64px]]<br />[[Dihedron|{2,2}]]
| [[File:Tetragonal dihedron.png|64px]]<br />2.4.4
| [[File:Digonal dihedron.png|64px]]<br />2.2.2.2
| [[File:Tetragonal dihedron.png|64px]]<br />4.4.2
| [[File:Digonal dihedron.png|64px]]<br />[[Hosohedron|2.2]]
| [[File:Tetragonal dihedron.png|64px]]<br />2.4.2.4
| [[File:Spherical square prism2.png|64px]]<br />4.4.4
| [[File:Spherical digonal antiprism.svg|64px]]<br />3.3.3.2
|- align=center
|Dihedral<br />(3 2 2)
| [[File:Trigonal dihedron.png|64px]]<br />[[Dihedron|3.3]]
| [[File:Hexagonal dihedron.png|64px]]<br />2.6.6
| [[File:Trigonal dihedron.png|64px]]<br />2.3.2.3
| [[File:Spherical triangular prism.svg|64px]]<br />4.4.3
| [[File:Spherical trigonal hosohedron.svg|64px]]<br />[[Hosohedron|2.2.2]]
| [[File:Spherical triangular prism.svg|64px]]<br />2.4.3.4
| [[File:Spherical hexagonal prism2.png|64px]]<br />4.4.6
| [[File:Spherical trigonal antiprism.svg|64px]]<br />3.3.3.3
|- align=center
|Dihedral<br />(4 2 2)
| [[File:Tetragonal dihedron.png|64px]]<br />[[Dihedron|4.4]]
| 2.8.8
| [[File:Tetragonal dihedron.png|64px]]<br />2.4.2.4
| [[File:Spherical square prism.svg|64px]]<br />4.4.4
| [[File:Spherical square hosohedron.svg|64px]]<br />[[Hosohedron|2.2.2.2]]
| [[File:Spherical square prism.svg|64px]]<br />2.4.4.4
| [[File:Spherical octagonal prism2.png|64px]]<br />4.4.8
| [[File:Spherical square antiprism.svg|64px]]<br />3.3.3.4
|- align=center
|Dihedral<br />(5 2 2)
| [[File:Pentagonal dihedron.png|64px]]<br />[[Dihedron|5.5]]
| 2.10.10
| [[File:Pentagonal dihedron.png|64px]]<br />2.5.2.5
| [[File:Spherical pentagonal prism.svg|64px]]<br />4.4.5
| [[File:Spherical pentagonal hosohedron.svg|64px]]<br />[[Hosohedron|2.2.2.2.2]]
| [[File:Spherical pentagonal prism.svg|64px]]<br />2.4.5.4
| [[File:Spherical decagonal prism2.png|64px]]<br />4.4.10
| [[File:Spherical pentagonal antiprism.svg|64px]]<br />3.3.3.5
|- align=center
|Dihedral<br />(6 2 2)
| [[File:Hexagonal dihedron.png|64px]]<br />[[Dihedron|6.6]]
| [[File:Dodecagonal dihedron.png|64px]]<br />2.12.12
| [[File:Hexagonal dihedron.png|64px]]<br />2.6.2.6
| [[File:Spherical hexagonal prism.svg|64px]]<br />4.4.6
| [[File:Spherical hexagonal hosohedron.svg|64px]]<br />[[Hosohedron|2.2.2.2.2.2]]
| [[File:Spherical hexagonal prism.svg|64px]]<br />2.4.6.4
| [[File:Spherical dodecagonal prism2.png|64px]]<br />4.4.12
| [[File:Spherical hexagonal antiprism.svg|64px]]<br />3.3.3.6
|}

=== (3 3 2) T<sub>d</sub> tetrahedral symmetry ===
The [[Tetrahedral symmetry|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<sub>2</sub> or [3,3], as well as a Coxeter diagram: {{CDD|node|3|node|3|node}}.

There are 24 triangles, visible in the faces of the [[Tetrakis hexahedron|tetrakis hexahedron]], and in the alternately colored triangles on a sphere:
:[[File:Tetrakishexahedron.jpg|100px]] 100px[[File:Sphere symmetry group td.png|100px]]

{| class="wikitable"
!rowspan=2|#
!rowspan=2|Name
!rowspan=2|Graph<br />A<sub>3</sub>
!rowspan=2|Graph<br />A<sub>2</sub>
!rowspan=2|Picture
!rowspan=2|Tiling
!rowspan=2|[[Vertex figure|Vertex<br />figure]]
!rowspan=2|Coxeter<br />and [[Schläfli symbol#Polychora and honeycombs|Schläfli]]<br />symbols
!colspan=3|Face counts by position
!colspan=3|Element counts
|-
! Pos. 2<br />{{CDD|node|3|node}}<br />[3]<br />(4)
! Pos. 1<br />{{CDD|node|2||node}}<br />[2]<br />(6)
! Pos. 0<br />{{CDD|2|node|3|node}}<br />[3]<br />(4)
! Faces
! Edges
! Vertices
|- bgcolor="#f0e0e0" align=center
!1
|align=center|[[Tetrahedron]]
|[[File:3-simplex t0.svg|50px]]
|[[File:3-simplex t0 A2.svg|50px]]
|[[File:Uniform polyhedron-33-t0.png|50px]]
|[[File:Uniform tiling 332-t0-1-.png|50px]]
|[[File:Tetrahedron vertfig.png|50px]]
|align=center|{{CDD|node_1|3|node|3|node}}<br />{3,3}
|[[File:Regular_polygon_3.svg|30px]]<br />[[Triangle|{3}]]
|
|
| 4
| 6
| 4
|- bgcolor="#e0e0f0" align=center
![1]
|align=center|Birectified tetrahedron<br />(same as [[Tetrahedron|tetrahedron]])
|[[File:3-simplex t0.svg|50px]]
|[[File:3-simplex t0 A2.svg|50px]]
|[[File:Uniform polyhedron-33-t2.png|50px]]
|[[File:Uniform tiling 332-t2.png|50px]]
|[[File:Tetrahedron vertfig.png|50px]]
|align=center|{{CDD|node|3|node|3|node_1}}<br />t<sub>2</sub>{3,3}={3,3}
|
|
|[[File:Regular_polygon_3.svg|30px]]<br />[[Triangle|{3}]]
| 4
| 6
| 4
|- bgcolor="#e0f0e0" align=center
!2
|align=center|Rectified tetrahedron<br />Tetratetrahedron<br />(same as [[Octahedron|octahedron]])
|[[File:3-simplex t1.svg|50px]]
|[[File:3-simplex t1 A2.svg|50px]]
|[[File:Uniform polyhedron-33-t1.png|50px]]
|[[File:Uniform tiling 332-t1-1-.png|50px]]
|[[File:Octahedron vertfig.svg|50px]]
|align=center|{{CDD|node|3|node_1|3|node}}<br />t<sub>1</sub>{3,3}=r{3,3}
| [[File:Regular_polygon_3.svg|30px]]<br />[[Triangle|{3}]]
|
| [[File:Regular_polygon_3.svg|30px]]<br />[[Triangle|{3}]]
| 8
| 12
| 6
|- bgcolor="#f0e0e0" align=center
!3
|align=center|[[Truncated tetrahedron]]
|[[File:3-simplex t01.svg|50px]]
|[[File:3-simplex t01 A2.svg|50px]]
|[[File:Uniform polyhedron-33-t01.png|50px]]
|[[File:Uniform tiling 332-t01-1-.png|50px]]
|[[File:Truncated tetrahedron vertfig.png|50px]]
|align=center|{{CDD|node_1|3|node_1|3|node}}<br />t<sub>0,1</sub>{3,3}=t{3,3}
| [[File:Regular_polygon_6.svg|30px]]<br />[[Hexagon|{6}]]
|
| [[File:Regular_polygon_3.svg|30px]]<br />[[Triangle|{3}]]
| 8
| 18
| 12
|- bgcolor="#e0e0f0" align=center
![3]
|align=center|Bitruncated tetrahedron<br />(same as [[Truncated tetrahedron|truncated tetrahedron]])
|[[File:3-simplex t01.svg|50px]]
|[[File:3-simplex t01 A2.svg|50px]]
|[[File:Uniform polyhedron-33-t12.png|50px]]
|[[File:Uniform tiling 332-t12.png|50px]]
|[[File:Truncated tetrahedron vertfig.png|50px]]
|align=center|{{CDD|node|3|node_1|3|node_1}}<br />t<sub>1,2</sub>{3,3}=t{3,3}
| [[File:Regular_polygon_3.svg|30px]]<br />[[Triangle|{3}]]
|
| [[File:Regular_polygon_6.svg|30px]]<br />[[Hexagon|{6}]]
| 8
| 18
| 12
|- bgcolor="#e0f0e0" align=center
!4
|align=center|Cantellated tetrahedron<br />Rhombitetratetrahedron<br />(same as [[Cuboctahedron|cuboctahedron]])
|[[File:3-simplex t02.svg|50px]]
|[[File:3-simplex t02 A2.svg|50px]]
|[[File:Uniform polyhedron-33-t02.png|50px]]
|[[File:Uniform tiling 332-t02.png|50px]]
|[[File:Cuboctahedron vertfig.png|50px]]
|align=center|{{CDD|node_1|3|node|3|node_1}}<br />t<sub>0,2</sub>{3,3}=rr{3,3}
| [[File:Regular_polygon_3.svg|30px]]<br />[[Triangle|{3}]]
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
| [[File:Regular_polygon_3.svg|30px]]<br />[[Triangle|{3}]]
| 14
| 24
| 12
|- bgcolor="#e0f0e0" align=center
!5
|align=center|Omnitruncated tetrahedron<br />Truncated tetratetrahedron<br />(same as [[Truncated octahedron|truncated octahedron]])
|[[File:3-simplex t012.svg|50px]]
|[[File:3-simplex t012 A2.svg|50px]]
|[[File:Uniform polyhedron-33-t012.png|50px]]
|[[File:Uniform tiling 332-t012.png|50px]]
|[[File:Truncated octahedron vertfig.png|50px]]
|align=center|{{CDD|node_1|3|node_1|3|node_1}}<br />t<sub>0,1,2</sub>{3,3}=tr{3,3}
| [[File:Regular_polygon_6.svg|30px]]<br />[[Hexagon|{6}]]
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
| [[File:Regular_polygon_6.svg|30px]]<br />[[Hexagon|{6}]]
| 14
| 36
| 24
|- bgcolor="#d0f0f0" align=center
!6
|align=center|Snub tetratetrahedron<br />(same as [[Icosahedron|icosahedron]])
|[[File:icosahedron graph A3.png|50px]]
|[[File:icosahedron graph A2.png|50px]]
|[[File:Uniform polyhedron-33-s012.svg|50px]]
|[[File:Spherical snub tetrahedron.png|50px]]
|[[File:Icosahedron vertfig.png|50px]]
|align=center|{{CDD|node_h|3|node_h|3|node_h}}<br />sr{3,3}
| [[File:Regular_polygon_3.svg|30px]]<br />[[Triangle|{3}]]
| 20px20px<br />2 [[Triangle|{3}]]
| [[File:Regular_polygon_3.svg|30px]]<br />[[Triangle|{3}]]
| 20
| 30
| 12
|}

=== (4 3 2) O<sub>h</sub> octahedral symmetry ===
The [[Octahedral symmetry|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]] B<sub>2</sub> or [4,3], as well as a Coxeter diagram: {{CDD|node|4|node|3|node}}.

There are 48 triangles, visible in the faces of the [[Disdyakis dodecahedron|disdyakis dodecahedron]], and in the alternately colored triangles on a sphere:
:[[File:Disdyakisdodecahedron.jpg|100px]] 100px[[File:Sphere symmetry group oh.png|100px]]

{| class="wikitable"
!rowspan=2|#
!rowspan=2|Name
!rowspan=2|Graph<br />B<sub>3</sub>
!rowspan=2|Graph<br />B<sub>2</sub>
!rowspan=2|Picture
!rowspan=2|Tiling
!rowspan=2|[[Vertex figure|Vertex<br />figure]]
!rowspan=2|Coxeter<br />and [[Schläfli symbol#Polychora and honeycombs|Schläfli]]<br />symbols
!colspan=3|Face counts by position
!colspan=3|Element counts
|-
! Pos. 2<br />{{CDD|node|4|node|2}}<br />[4]<br />(6)
! Pos. 1<br />{{CDD|node|2|2|node}}<br />[2]<br />(12)
! Pos. 0<br />{{CDD|2|node|3|node}}<br />[3]<br />(8)
! Faces
! Edges
! Vertices
|- bgcolor="#f0e0e0" align=center
!7
|align=center|[[Cube]]
|[[File:3-cube t0.svg|50px]]
|[[File:3-cube t0 B2.svg|50px]]
|[[File:Uniform polyhedron-43-t0.svg|50px]]
|[[File:Uniform tiling 432-t0.png|50px]]
|[[File:Cube vertfig.png|50px]]
|align=center|{{CDD|node_1|4|node|3|node}}<br />{4,3}
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
|
|
| 6
| 12
| 8
|- bgcolor="#e0e0f0" align=center
![2]
|align=center|[[Octahedron]]
|[[File:3-cube t2.svg|50px]]
|[[File:3-cube t2 B2.svg|50px]]
|[[File:Uniform polyhedron-43-t2.svg|50px]]
|[[File:Uniform tiling 432-t2.png|50px]]
|[[File:Octahedron vertfig.svg|50px]]
|align=center|{{CDD|node|4|node|3|node_1}}<br />{3,4}
|
|
| [[File:Regular_polygon_3.svg|30px]]<br />[[Triangle|{3}]]
| 8
| 12
| 6
|- bgcolor="#e0f0e0" align=center
![4]
|align=center|Rectified cube<br />Rectified octahedron<br />([[Cuboctahedron]])
|[[File:3-cube t1.svg|50px]]
|[[File:3-cube t1 B2.svg|50px]]
|[[File:Uniform polyhedron-43-t1.svg|50px]]
|[[File:Uniform tiling 432-t1.png|50px]]
|[[File:Cuboctahedron vertfig.png|50px]]
|align=center|{{CDD|node|4|node_1|3|node}}<br />{4,3}
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
|
| [[File:Regular_polygon_3.svg|30px]]<br />[[Triangle|{3}]]
| 14
| 24
| 12
|- bgcolor="#f0e0e0" align=center
!8
|align=center|[[Truncated cube]]
|[[File:3-cube t01.svg|50px]]
|[[File:3-cube t01 B2.svg|50px]]
|[[File:Uniform polyhedron-43-t01.svg|50px]]
|[[File:Uniform tiling 432-t01.png|50px]]
|[[File:Truncated cube vertfig.png|50px]]
|align=center|{{CDD|node_1|4|node_1|3|node}}<br />t<sub>0,1</sub>{4,3}=t{4,3}
| [[File:Regular_polygon_8.svg|30px]]<br />[[Octagon|{8}]]
|
| [[File:Regular_polygon_3.svg|30px]]<br />[[Triangle|{3}]]
| 14
| 36
| 24
|- bgcolor="#e0e0f0" align=center
![5]
|align=center|[[Truncated octahedron]]
|[[File:3-cube t12.svg|50px]]
|[[File:3-cube t12 B2.svg|50px]]
|[[File:Uniform polyhedron-43-t12.svg|50px]]
|[[File:Uniform tiling 432-t12.png|50px]]
|[[File:Truncated octahedron vertfig.png|50px]]
|align=center|{{CDD|node|4|node_1|3|node_1}}<br />t<sub>0,1</sub>{3,4}=t{3,4}
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
|
| [[File:Regular_polygon_6.svg|30px]]<br />[[Hexagon|{6}]]
| 14
| 36
| 24
|- bgcolor="#e0f0e0" align=center
!9
|align=center|Cantellated cube<br />Cantellated octahedron<br />[[Rhombicuboctahedron]]
|[[File:3-cube t02.svg|50px]]
|[[File:3-cube t02 B2.svg|50px]]
|[[File:Uniform polyhedron-43-t02.png|50px]]
|[[File:Uniform tiling 432-t02.png|50px]]
|[[File:Small rhombicuboctahedron vertfig.png|50px]]
|align=center|{{CDD|node_1|4|node|3|node_1}}<br />t<sub>0,2</sub>{4,3}=rr{4,3}
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
| [[File:Regular_polygon_3.svg|30px]]<br />[[Triangle|{3}]]
| 26
| 48
| 24
|- bgcolor="#e0f0e0" align=center
!10
|align=center|Omnitruncated cube<br />Omnitruncated octahedron<br />[[Truncated cuboctahedron]]
|[[File:3-cube t012.svg|50px]]
|[[File:3-cube t012 B2.svg|50px]]
|[[File:Uniform polyhedron-43-t012.png|50px]]
|[[File:Uniform tiling 432-t012.png|50px]]
|[[File:Great_rhombicuboctahedron_vertfig.png|50px]]
|align=center|{{CDD|node_1|4|node_1|3|node_1}}<br />t<sub>0,1,2</sub>{4,3}=tr{4,3}
| [[File:Regular_polygon_8.svg|30px]]<br />[[Octagon|{8}]]
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
| [[File:Regular_polygon_6.svg|30px]]<br />[[Hexagon|{6}]]
| 26
| 72
| 48
|- bgcolor="#d0f0f0" align=center
![6]
|align=center|Snub octahedron<br />(same as [[Icosahedron]])
|[[File:3-cube h01.svg|50px]]
|[[File:3-cube h01 B2.svg|50px]]
|[[File:Uniform polyhedron-43-h01.svg|50px]]
|[[File:Spherical alternated truncated octahedron.png|50px]]
|[[File:Icosahedron vertfig.png|50px]]
|align=center|{{CDD|node|4|node_h|3|node_h}}<br />= {{CDD|nodes_hh|split2|node_h}}<br />s{3,4}=sr{3,3}
|
| [[File:Regular_polygon_3.svg|30px]]<br />[[Triangle|{3}]]
| [[File:Regular_polygon_3.svg|30px]]<br />[[Triangle|{3}]]
| 20
| 30
| 12

|- bgcolor="#d0f0f0" align=center
![1]
|align=center|Half cube<br />(same as [[Tetrahedron]])
|[[File:3-simplex t0 A2.svg|50px]]
|[[File:3-simplex t0.svg|50px]]
|[[File:Uniform polyhedron-33-t2.png|50px]]
|[[File:Uniform tiling 332-t2.png|50px]]
|[[File:Tetrahedron vertfig.png|50px]]
|align=center|{{CDD|node_h1|4|node|3|node}}<br />= {{CDD|nodes_10ru|split2|node}}<br />h{4,3}={3,3}
| [[File:Regular_polygon_3.svg|30px]]<br /><sup>1</sup>/<sub>2</sub> [[Triangle|{3}]]
|
|
| 4
| 6
| 4

|- bgcolor="#d0f0f0" align=center
![2]
|align=center|Cantic cube<br />(same as [[Truncated tetrahedron]])
|[[File:3-simplex t01 A2.svg|50px]]
|[[File:3-simplex t01.svg|50px]]
|[[File:Uniform polyhedron-33-t12.png|50px]]
|[[File:Uniform tiling 332-t12.png|50px]]
|[[File:Truncated tetrahedron vertfig.png|50px]]
|align=center|{{CDD|node_h1|4|node|3|node_1}}<br />= {{CDD|nodes_10ru|split2|node_1}}<br />h<sub>2</sub>{4,3}=t{3,3}
| [[File:Regular_polygon_6.svg|30px]]<br /> <sup>1</sup>/<sub>2</sub> [[Hexagon|{6}]]
|
| [[File:Regular_polygon_3.svg|30px]]<br /><sup>1</sup>/<sub>2</sub> [[Triangle|{3}]]
| 8
| 18
| 12
|- bgcolor="#d0f0f0" align=center
![4]
| (same as [[Cuboctahedron]])
|[[File:3-simplex t02 A2.svg|50px]]
|[[File:3-simplex t02.svg|50px]]
|[[File:Uniform polyhedron-33-t02.png|50px]]
|[[File:Uniform tiling 332-t02.png|50px]]
|[[File:Cuboctahedron vertfig.png|50px]]
|align=center|{{CDD|node_h0|4|node_1|3|node}}<br />= {{CDD|nodes_11|split2|node}}<br />rr{3,3}
|
|
|
|14
|24
|12
|- bgcolor="#d0f0f0" align=center
![5]
| (same as [[Truncated octahedron]])
|[[File:3-simplex t012 A2.svg|50px]]
|[[File:3-simplex t012.svg|50px]]
|[[File:Uniform polyhedron-33-t012.png|50px]]
|[[File:Uniform tiling 332-t012.png|50px]]
|[[File:Truncated octahedron vertfig.png|50px]]
|align=center|{{CDD|node_h0|4|node_1|3|node_1}}<br />= {{CDD|nodes_11|split2|node_1}}<br />tr{3,3}
|
|
|
|14
|36
|24
|- bgcolor="#d0f0f0" align=center
![9]
| Cantic snub octahedron<br />(same as [[Rhombicuboctahedron]])
|[[File:3-cube t02.svg|50px]]
|[[File:3-cube t02 B2.svg|50px]]
|[[File:Rhombicuboctahedron uniform edge coloring.png|50px]]
|[[File:Uniform tiling 432-t02.png|50px]]
|[[File:Small rhombicuboctahedron vertfig.png|50px]]
|align=center|{{CDD|node_1|4|node_h|3|node_h}}<br />s<sub>2</sub>{3,4}=rr{3,4}
|
|
|
| 26
| 48
| 24
|- bgcolor="#d0f0f0" align=center
!11
|align=center|[[Snub cube|Snub cuboctahedron]]
|[[File:Snub cube_A2.png|50px]]
|[[File:Snub cube_B2.png|50px]]
|[[File:Uniform polyhedron-43-s012.png|50px]]
|[[File:Spherical snub cube.png|50px]]
|[[File:Snub cube vertfig.png|50px]]
|align=center|{{CDD|node_h|4|node_h|3|node_h}}<br />sr{4,3}
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
| 20px20px<br />2 [[Triangle|{3}]]
| [[File:Regular_polygon_3.svg|30px]]<br />[[Triangle|{3}]]
| 38
| 60
| 24
|}

=== (5 3 2) I<sub>h</sub> icosahedral symmetry ===
The [[Icosahedral symmetry|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<sub>2</sub> or [5,3], as well as a Coxeter diagram: {{CDD|node|5|node|3|node}}.

There are 120 triangles, visible in the faces of the [[Disdyakis triacontahedron|disdyakis triacontahedron]], and in the alternately colored triangles on a sphere:<br>
[[File:Disdyakistriacontahedron.jpg|100px]] 100px[[File:Sphere symmetry group ih.png|100px]]

{| class="wikitable"
!rowspan=2|#
!rowspan=2|Name
!rowspan=2|Graph<br />(A<sub>2</sub>)<br />[6]
!rowspan=2|Graph<br />(H<sub>3</sub>)<br />[10]
!rowspan=2|Picture
!rowspan=2|Tiling
!rowspan=2|[[Vertex figure|Vertex<br />figure]]
!rowspan=2|Coxeter<br />and [[Schläfli symbol#Polychora and honeycombs|Schläfli]]<br />symbols
!colspan=3|Face counts by position
!colspan=3|Element counts
|-
! Pos. 2<br />{{CDD|node|5|node|2}}<br />[5]<br />(12)
! Pos. 1<br />{{CDD|node|2|node}}<br />[2]<br />(30)
! Pos. 0<br />{{CDD|2|node|3|node}}<br />[3]<br />(20)
! Faces
! Edges
! Vertices
|- bgcolor="#f0e0e0" align=center
!12
|align=center|[[Dodecahedron]]
|[[Image:Dodecahedron A2 projection.svg|50px]]
|[[File:Dodecahedron H3 projection.svg|50px]]
|[[File:Uniform polyhedron-53-t0.svg|50px]]
|[[File:Uniform tiling 532-t0.png|50px]]
|[[File:Dodecahedron vertfig.png|50px]]
|align=center|{{CDD|node_1|5|node|3|node}}<br />{5,3}
| [[File:Regular_polygon_5.svg|30px]]<br />[[Pentagon|{5}]]
|
|
| 12
| 30
| 20
|- bgcolor="#e0e0f0" align=center
![6]
|align=center|[[Icosahedron]]
|[[File:Icosahedron A2 projection.svg|50px]]
|[[File:Icosahedron H3 projection.svg|50px]]
|[[File:Uniform polyhedron-53-t2.svg|50px]]
|[[File:Uniform tiling 532-t2.png|50px]]
|[[File:Icosahedron vertfig.png|50px]]
|align=center|{{CDD|node|5|node|3|node_1}}<br />{3,5}
|
|
| [[File:Regular_polygon_3.svg|30px]]<br />[[Triangle|{3}]]
| 20
| 30
| 12
|- bgcolor="#e0f0e0" align=center
!13
|align=center|Rectified dodecahedron<br />Rectified icosahedron<br />[[Icosidodecahedron]]
|[[Image:Dodecahedron t1 A2.png|50px]]
|[[Image:Dodecahedron t1 H3.png|50px]]
|[[File:Uniform polyhedron-53-t1.svg|50px]]
|[[File:Uniform tiling 532-t1.png|50px]]
|[[File:Icosidodecahedron vertfig.png|50px]]
|align=center|{{CDD|node|5|node_1|3|node}}<br />t<sub>1</sub>{5,3}=r{5,3}
| [[File:Regular_polygon_5.svg|30px]]<br />[[Pentagon|{5}]]
|
| [[File:Regular_polygon_3.svg|30px]]<br />[[Triangle|{3}]]
| 32
| 60
| 30
|- bgcolor="#f0e0e0" align=center
!14
|align=center|[[Truncated dodecahedron]]
|[[Image:Dodecahedron t01 A2.png|50px]]
|[[Image:Dodecahedron t01 H3.png|50px]]
|[[File:Uniform polyhedron-53-t01.svg|50px]]
|[[File:Uniform tiling 532-t01.png|50px]]
|[[File:Truncated dodecahedron vertfig.png|50px]]
|align=center|{{CDD|node_1|5|node_1|3|node}}<br />t<sub>0,1</sub>{5,3}=t{5,3}
| [[File:Regular_polygon_10.svg|30px]]<br />[[Decagon|{10}]]
|
| [[File:Regular_polygon_3.svg|30px]]<br />[[Triangle|{3}]]
| 32
| 90
| 60
|- bgcolor="#e0e0f0" align=center
!15
|align=center|[[Truncated icosahedron]]
|[[Image:Icosahedron t01 A2.png|50px]]
|[[Image:Icosahedron t01 H3.png|50px]]
|[[File:Uniform polyhedron-53-t12.svg|50px]]
|[[File:Uniform tiling 532-t12.png|50px]]
|[[File:Truncated icosahedron vertfig.png|50px]]
|align=center|{{CDD|node|5|node_1|3|node_1}}<br />t<sub>0,1</sub>{3,5}=t{3,5}
| [[File:Regular_polygon_5.svg|30px]]<br />[[Pentagon|{5}]]
|
| [[File:Regular_polygon_6.svg|30px]]<br />[[Hexagon|{6}]]
| 32
| 90
| 60
|- bgcolor="#e0f0e0" align=center
!16
|align=center|Cantellated dodecahedron<br />Cantellated icosahedron<br />[[Rhombicosidodecahedron]]
|[[Image:Dodecahedron t02 A2.png|50px]]
|[[Image:Dodecahedron t02 H3.png|50px]]
|[[File:Uniform polyhedron-53-t02.png|50px]]
|[[File:Uniform tiling 532-t02.png|50px]]
|[[File:Small rhombicosidodecahedron vertfig.png|50px]]
|align=center|{{CDD|node_1|5|node|3|node_1}}<br />t<sub>0,2</sub>{5,3}=rr{5,3}
| [[File:Regular_polygon_5.svg|30px]]<br />[[Pentagon|{5}]]
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
| [[File:Regular_polygon_3.svg|30px]]<br />[[Triangle|{3}]]
| 62
| 120
| 60
|- bgcolor="#e0f0e0" align=center
!17
|align=center|Omnitruncated dodecahedron<br />Omnitruncated icosahedron<br />[[Truncated icosidodecahedron]]
|[[Image:Dodecahedron t012 A2.png|50px]]
|[[Image:Dodecahedron t012 H3.png|50px]]
|[[File:Uniform polyhedron-53-t012.png|50px]]
|[[File:Uniform tiling 532-t012.png|50px]]
|[[File:Great rhombicosidodecahedron vertfig.png|50px]]
|align=center|{{CDD|node_1|5|node_1|3|node_1}}<br />t<sub>0,1,2</sub>{5,3}=tr{5,3}
| [[File:Regular_polygon_10.svg|30px]]<br />[[Decagon|{10}]]
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
| [[File:Regular_polygon_6.svg|30px]]<br />[[Hexagon|{6}]]
| 62
| 180
| 120
|- bgcolor="#d0f0f0" align=center
!18
|align=center|[[Snub dodecahedron|Snub icosidodecahedron]]
|[[Image:Snub dodecahedron A2.png|50px]]
|[[Image:Snub dodecahedron H2.png|50px]]
|[[File:Uniform polyhedron-53-s012.png|50px]]
|[[File:Spherical snub dodecahedron.png|50px]]
|[[File:Snub dodecahedron vertfig.png|50px]]
|align=center|{{CDD|node_h|5|node_h|3|node_h}}<br />sr{5,3}
| [[File:Regular_polygon_5.svg|30px]]<br />[[Pentagon|{5}]]
| 20px20px<br />2 [[Triangle|{3}]]
| [[File:Regular_polygon_3.svg|30px]]<br />[[Triangle|{3}]]
| 92
| 150
| 60
|}

=== (p 2 2) Prismatic [p,2], I<sub>2</sub>(p) family (D<sub>''p''h</sub> dihedral symmetry) ===

{{main|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]] I<sub>2</sub>(p) or [n,2], as well as a prismatic Coxeter diagram: {{CDD|node|p|node|2|node}}.

Below are the first five dihedral symmetries: D<sub>2</sub> ... D<sub>6</sub>. The dihedral symmetry D<sub>p</sub> has order ''4n'', represented the faces of a [[Bipyramid|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:
:[[File:Octahedron.jpg|100px]] [[File:Sphere symmetry group d2h.png|100px]]

{| class="wikitable"
!rowspan=2|#
!rowspan=2|Name
!rowspan=2|Picture
!rowspan=2|Tiling
!rowspan=2|[[Vertex figure|Vertex<br />figure]]
!rowspan=2|Coxeter<br />and [[Schläfli symbol#Polychora and honeycombs|Schläfli]]<br />symbols
!colspan=3|Face counts by position
!colspan=3|Element counts
|-
! Pos. 2<br />{{CDD|node|2|node|2|}}<br />[2]<br />(2)
! Pos. 1<br />{{CDD|node|2|node}}<br />[2]<br />(2)
! Pos. 0<br />{{CDD|2|node|2|node}}<br />[2]<br />(2)
! Faces
! Edges
! Vertices
|- bgcolor="#f0e0e0"
!D<sub>2</sub><br />H<sub>2</sub>
|align=center|Digonal dihedron,<br />digonal hosohedron
|
|[[File:digonal dihedron.png|60px]]
|
|align=center|{{CDD|node_1|2|node|2|node}}<br />{2,2}
| [[File:Regular digon in spherical geometry-2.svg|30px]]<br />[[Digon|{2}]]
|
|
| 2
| 2
| 2
|- bgcolor="#e0f0e0"
!D<sub>4</sub>
|align=center|Truncated digonal dihedron<br />(same as square dihedron)
|
|[[File:tetragonal dihedron.png|60px]]
|
|align=center|{{CDD|node_1|2|node_1|2|node}}<br />t{2,2}={4,2}
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
|
|
| 2
| 4
| 4
|- bgcolor="#f0e0e0"
!P<sub>4</sub><br />[7]
|align=center|Omnitruncated digonal dihedron<br />(same as [[Cube|cube]])
|[[File:Uniform polyhedron 222-t012.png|60px]]
|[[File:Spherical square prism2.png|60px]]
|[[File:Cube vertfig.png|60px]]
|align=center|{{CDD|node_1|2|node_1|2|node_1}}<br />t<sub>0,1,2</sub>{2,2}=tr{2,2}
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
| 6
| 12
| 8
|- bgcolor="#d0f0f0"
!A<sub>2</sub><br />[1]
|align=center|Snub digonal dihedron<br />(same as [[Tetrahedron|tetrahedron]])
|[[File:Uniform polyhedron-33-t2.png|60px]]
|[[File:Spherical digonal antiprism.svg|60px]]
|[[File:Tetrahedron vertfig.png|60px]]
|align=center|{{CDD|node_h|2x|node_h|2x|node_h}}<br />sr{2,2}
|
| 20px20px<br />2 [[Triangle|{3}]]
| &nbsp;
| 4
| 6
| 4
|}

==== (3 2 2) D<sub>3h</sub> dihedral symmetry ====
There are 12 fundamental triangles, visible in the faces of the [[Hexagonal bipyramid|hexagonal bipyramid]] and alternately colored triangles on a sphere:
:[[File:Hexagonale bipiramide.png|100px]] [[File:Sphere symmetry group d3h.png|100px]]

{| class="wikitable"
!rowspan=2|#
!rowspan=2|Name
!rowspan=2|Picture
!rowspan=2|Tiling
!rowspan=2|[[Vertex figure|Vertex<br />figure]]
!rowspan=2|Coxeter<br />and [[Schläfli symbol#Polychora and honeycombs|Schläfli]]<br />symbols
!colspan=3|Face counts by position
!colspan=3|Element counts
|-
! Pos. 2<br />{{CDD|node|3|node|2}}<br />[3]<br />(2)
! Pos. 1<br />{{CDD|node|2|node}}<br />[2]<br />(3)
! Pos. 0<br />{{CDD|2|node|2|node}}<br />[2]<br />(3)
! Faces
! Edges
! Vertices
|- bgcolor="#f0e0e0"
!D<sub>3</sub>
|align=center|Trigonal dihedron
|
|[[File:Trigonal dihedron.png|60px]]
|
|align=center|{{CDD|node_1|3|node|2|node}}<br />{3,2}
| [[File:Regular_polygon_3.svg|30px]]<br />[[Triangle|{3}]]
|
|
| 2
| 3
| 3
|- bgcolor="#e0e0f0"
!H<sub>3</sub>
|align=center|Trigonal hosohedron
|
|[[File:Trigonal hosohedron.png|60px]]
|
|align=center|{{CDD|node|3|node|2|node_1}}<br />{2,3}
|
|
| [[File:Regular digon in spherical geometry-2.svg|30px]]<br />[[Digon|{2}]]
| 3
| 3
| 2
|- bgcolor="#e0f0e0"
!D<sub>6</sub>
|align=center|Truncated trigonal dihedron<br />(same as hexagonal dihedron)
|
|[[File:Hexagonal dihedron.png|60px]]
|
|align=center|{{CDD|node_1|3|node_1|2|node}}<br />t{3,2}
| [[File:Regular_polygon_6.svg|30px]]<br />[[Hexagon|{6}]]
|
|
| 2
| 6
| 6
|- bgcolor="#e0e0f0"
!P<sub>3</sub>
|align=center|Truncated trigonal hosohedron<br />([[Triangular prism]])
|[[File:Triangular prism.png|60px]]
|[[File:Spherical triangular prism.svg|60px]]
|[[File:Triangular prism vertfig.png|60px]]
|align=center|{{CDD|node|3|node_1|2|node_1}}<br />t{2,3}
| [[File:Regular_polygon_3.svg|30px]]<br />[[Triangle|{3}]]
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
|
| 5
| 9
| 6
|- bgcolor="#e0f0e0"
!P<sub>6</sub>
|align=center|Omnitruncated trigonal dihedron<br />([[Hexagonal prism]])
|[[File:Hexagonal prism.png|60px]]
|[[File:Spherical hexagonal prism2.png|60px]]
|[[File:Hexagonal prism vertfig.png|60px]]
|align=center|{{CDD|node_1|3|node_1|2|node_1}}<br />t<sub>0,1,2</sub>{2,3}=tr{2,3}
| [[File:Regular_polygon_6.svg|30px]]<br />[[Hexagon|{6}]]
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
| 8
| 18
| 12
|- bgcolor="#d0f0f0"
!A<sub>3</sub><br />[2]
|align=center|Snub trigonal dihedron<br />(same as Triangular antiprism)<br />(same as [[Octahedron|octahedron]])
|[[File:Trigonal antiprism.png|60px]]
|[[File:Spherical trigonal antiprism.svg|60px]]
|[[File:Octahedron vertfig.svg|60px]]
|align=center|{{CDD|node_h|3|node_h|2x|node_h}}<br />sr{2,3}
| [[File:Regular_polygon_3.svg|30px]]<br />[[Triangle|{3}]]
| 20px20px<br />2 [[Triangle|{3}]]
| &nbsp;
| 8
| 12
| 6
|- bgcolor="#d0f0f0"
!P<sub>3</sub>
|align=center|Cantic snub trigonal dihedron<br />([[Triangular prism]])
|[[File:Triangular prism.png|60px]]
|[[File:Spherical triangular prism.svg|60px]]
|[[File:Triangular prism vertfig.png|60px]]
|align=center|{{CDD|node_h|3|node_h|2x|node_1}}<br />s<sub>2</sub>{2,3}=t{2,3}
|
|
|
| 5
| 9
| 6
|}

==== (4 2 2) D<sub>4h</sub> dihedral symmetry ====
There are 16 fundamental triangles, visible in the faces of the [[Octagonal bipyramid|octagonal bipyramid]] and alternately colored triangles on a sphere:
:[[File:Octagonal bipyramid.png|80px]]

{| class="wikitable"
!rowspan=2|#
!rowspan=2|Name
!rowspan=2|Picture
!rowspan=2|Tiling
!rowspan=2|[[Vertex figure|Vertex<br />figure]]
!rowspan=2|Coxeter<br />and [[Schläfli symbol#Polychora and honeycombs|Schläfli]]<br />symbols
!colspan=3|Face counts by position
!colspan=3|Element counts
|-
! Pos. 2<br />{{CDD|node|4|node|2}}<br />[4]<br />(2)
! Pos. 1<br />{{CDD|node|2|node}}<br />[2]<br />(4)
! Pos. 0<br />{{CDD|2|node|2|node}}<br />[2]<br />(4)
! Faces
! Edges
! Vertices
|- bgcolor="#f0e0e0"
!D<sub>4</sub>
|align=center|square dihedron
|
|[[File:Tetragonal dihedron.png|60px]]
|
|align=center|{{CDD|node_1|4|node|2|node}}<br />{4,2}
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
|
|
| 2
| 4
| 4
|- bgcolor="#e0e0f0"
!H<sub>4</sub>
|align=center|square hosohedron
|
|[[File:Spherical square hosohedron.svg|60px]]
|
|align=center|{{CDD|node|4|node|2|node_1}}<br />{2,4}
|
|
| [[File:Regular digon in spherical geometry-2.svg|30px]]<br />[[Digon|{2}]]
| 4
| 4
| 2
|- bgcolor="#e0f0e0"
!D<sub>8</sub>
|align=center|Truncated square dihedron<br />(same as octagonal dihedron)
|
|
|
|align=center|{{CDD|node_1|4|node_1|2|node}}<br />t{4,2}
| [[File:Regular_polygon_8.svg|30px]]<br />[[Octagon|{8}]]
|
|
| 2
| 8
| 8
|- bgcolor="#e0e0f0"
!P<sub>4</sub><br />[7]
|align=center|Truncated square hosohedron<br />([[Cube]])
|[[File:Tetragonal prism.png|60px]]
|[[File:Spherical square prism.svg|60px]]
|[[File:Cube vertfig.png|60px]]
|align=center|{{CDD|node|4|node_1|2|node_1}}<br />t{2,4}
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
|
| 6
| 12
| 8
|- bgcolor="#e0f0e0"
!D<sub>8</sub>
|align=center|Omnitruncated square dihedron<br />([[Octagonal prism]])
|[[File:Octagonal prism.png|60px]]
|[[File:Spherical octagonal prism2.png|60px]]
|[[File:Octagonal prism vertfig.png|60px]]
|align=center|{{CDD|node_1|4|node_1|2|node_1}}<br />t<sub>0,1,2</sub>{2,4}=tr{2,4}
| [[File:Regular_polygon_8.svg|30px]]<br />[[Octagon|{8}]]
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
| 10
| 24
| 16
|- bgcolor="#d0f0f0"
!A<sub>4</sub>
|align=center|Snub square dihedron<br />([[Square antiprism]])
|[[File:Square antiprism.png|60px]]
|[[File:Spherical square antiprism.svg|60px]]
|[[File:Square antiprism vertfig.png|60px]]
|align=center|{{CDD|node_h|4|node_h|2x|node_h}}<br />sr{2,4}
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
| 20px20px<br />2 [[Triangle|{3}]]
| &nbsp;
| 10
| 16
| 8
|- bgcolor="#d0f0f0"
!P<sub>4</sub><br />[7]
|align=center|Cantic snub square dihedron<br />([[Cube]])
|[[File:Tetragonal prism.png|60px]]
|[[File:Spherical square prism.svg|60px]]
|[[File:Cube vertfig.png|60px]]
|align=center|{{CDD|node_h|4|node_h|2x|node_1}}<br />s<sub>2</sub>{4,2}=t{2,4}
|
|
|
| 6
| 12
| 8
|- bgcolor="#d0f0f0"
!A<sub>2</sub><br />[1]
|align=center|Snub square hosohedron<br />(Digonal antiprism)<br />([[Tetrahedron]])
|[[File:Uniform polyhedron-33-t2.png|60px]]
|[[File:Spherical digonal antiprism.svg|60px]]
|[[File:Tetrahedron vertfig.png|60px]]
|align=center|{{CDD|node|4|node_h|2x|node_h}}<br />s{2,4}=sr{2,2}
|
|
|
| 4
| 6
| 4
|}

==== (5 2 2) D<sub>5h</sub> dihedral symmetry ====
There are 20 fundamental triangles, visible in the faces of the [[Decagonal bipyramid|decagonal bipyramid]] and alternately colored triangles on a sphere:
:[[File:Decagonal bipyramid.png|60px]]

{| class="wikitable"
!rowspan=2|#
!rowspan=2|Name
!rowspan=2|Picture
!rowspan=2|Tiling
!rowspan=2|[[Vertex figure|Vertex<br />figure]]
!rowspan=2|Coxeter<br />and [[Schläfli symbol#Polychora and honeycombs|Schläfli]]<br />symbols
!colspan=3|Face counts by position
!colspan=3|Element counts
|-
! Pos. 2<br />{{CDD|node|5|node|2}}<br />[5]<br />(2)
! Pos. 1<br />{{CDD|node|2|node}}<br />[2]<br />(5)
! Pos. 0<br />{{CDD|2|node|2|node}}<br />[2]<br />(5)
! Faces
! Edges
! Vertices
|- bgcolor="#f0e0e0"
!D<sub>5</sub>
|align=center|Pentagonal dihedron
|
|[[File:Pentagonal dihedron.png|60px]]
|
|align=center|{{CDD|node_1|5|node|2|node}}<br />{5,2}
| [[File:Regular_polygon_5.svg|30px]]<br />[[Pentagon|{5}]]
|
|
| 2
| 5
| 5
|- bgcolor="#e0e0f0"
!H<sub>5</sub>
|align=center|Pentagonal hosohedron
|
|[[File:Spherical pentagonal hosohedron.svg|60px]]
|
|align=center|{{CDD|node|5|node|2|node_1}}<br />{2,5}
|
|
| [[File:Regular digon in spherical geometry-2.svg|30px]]<br />[[Digon|{2}]]
| 5
| 5
| 2
|- bgcolor="#e0f0e0"
!D<sub>10</sub>
|align=center|Truncated pentagonal dihedron<br />(same as decagonal dihedron)
|
|
|
|align=center|{{CDD|node_1|5|node_1|2|node}}<br />t{5,2}
| [[File:Regular_polygon_10.svg|30px]]<br />[[Decagon|{10}]]
|
|
| 2
| 10
| 10
|- bgcolor="#e0e0f0"
!P<sub>5</sub>
|align=center|Truncated pentagonal hosohedron<br />(same as [[Pentagonal prism|pentagonal prism]])
|[[File:Pentagonal prism.png|60px]]
|[[File:Spherical pentagonal prism.svg|60px]]
|[[File:Pentagonal prism vertfig.png|60px]]
|align=center|{{CDD|node|5|node_1|2|node_1}}<br />t{2,5}
| [[File:Regular_polygon_5.svg|30px]]<br />[[Pentagon|{5}]]
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
|
| 7
| 15
| 10
|- bgcolor="#e0f0e0"
!P<sub>10</sub>
|align=center|Omnitruncated pentagonal dihedron<br />([[Decagonal prism]])
|[[File:Decagonal prism.png|60px]]
|[[File:Spherical decagonal prism2.png|60px]]
|[[File:Decagonal prism vf.png|60px]]
|align=center|{{CDD|node_1|5|node_1|2|node_1}}<br />t<sub>0,1,2</sub>{2,5}=tr{2,5}
| [[File:Regular_polygon_10.svg|30px]]<br />[[Decagon|{10}]]
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
| 12
| 30
| 20
|- bgcolor="#d0f0f0"
!A<sub>5</sub>
|align=center|Snub pentagonal dihedron<br />([[Pentagonal antiprism]])
|[[File:Pentagonal antiprism.png|60px]]
|[[File:Spherical pentagonal antiprism.svg|60px]]
|[[File:Pentagonal antiprism vertfig.png|60px]]
|align=center|{{CDD|node_h|5|node_h|2x|node_h}}<br />sr{2,5}
| [[File:Regular_polygon_5.svg|30px]]<br />[[Pentagon|{5}]]
| 20px20px<br />2 [[Triangle|{3}]]
| &nbsp;
| 12
| 20
| 10
|- bgcolor="#d0f0f0"
!P<sub>5</sub>
|align=center|Cantic snub pentagonal dihedron<br />([[Pentagonal prism]])
|[[File:Pentagonal prism.png|60px]]
|[[File:Spherical pentagonal prism.svg|60px]]
|[[File:Pentagonal prism vertfig.png|60px]]
|align=center|{{CDD|node_h|5|node_h|2x|node_1}}<br />s<sub>2</sub>{5,2}=t{2,5}
|
|
|
| 7
| 15
| 10
|}

==== (6 2 2) D<sub>6h</sub> dihedral symmetry ====
There are 24 fundamental triangles, visible in the faces of the dodecagonal bipyramid and alternately colored triangles on a sphere.

{| class="wikitable"
!rowspan=2|#
!rowspan=2|Name
!rowspan=2|Picture
!rowspan=2|Tiling
!rowspan=2|[[Vertex figure|Vertex<br />figure]]
!rowspan=2|Coxeter<br />and [[Schläfli symbol#Polychora and honeycombs|Schläfli]]<br />symbols
!colspan=3|Face counts by position
!colspan=3|Element counts
|-
! Pos. 2<br />{{CDD|node|6|node|2}}<br />[6]<br />(2)
! Pos. 1<br />{{CDD|node|2|node}}<br />[2]<br />(6)
! Pos. 0<br />{{CDD|2|node|2|node}}<br />[2]<br />(6)
! Faces
! Edges
! Vertices
|- bgcolor="#f0e0e0"
!D<sub>6</sub>
|align=center|Hexagonal dihedron
|
|[[File:Hexagonal dihedron.png|60px]]
|
|align=center|{{CDD|node_1|6|node|2|node}}<br />{6,2}
| [[File:Regular_polygon_6.svg|30px]]<br />[[Hexagon|{6}]]
|
|
| 2
| 6
| 6
|- bgcolor="#e0e0f0"
!H<sub>6</sub>
|align=center|Hexagonal hosohedron
|
|[[File:Hexagonal hosohedron.png|60px]]
|
|align=center|{{CDD|node|6|node|2|node_1}}<br />{2,6}
|
|
| [[File:Regular digon in spherical geometry-2.svg|30px]]<br />[[Digon|{2}]]
| 6
| 6
| 2
|- bgcolor="#e0f0e0"
!D<sub>12</sub>
|align=center|Truncated hexagonal dihedron<br />(same as dodecagonal dihedron)
|
|[[File:Dodecagonal dihedron.png|60px]]
|
|align=center|{{CDD|node_1|6|node_1|2|node}}<br />t{6,2}
| [[File:Regular_polygon_10.svg|30px]]<br />[[Dodecagon|{12}]]
|
|
| 2
| 12
| 12
|- bgcolor="#e0e0f0"
!H<sub>6</sub>
|align=center|Truncated hexagonal hosohedron<br />(same as [[Hexagonal prism|hexagonal prism]])
|[[File:Hexagonal prism.png|60px]]
|[[File:Spherical hexagonal prism.svg|60px]]
|[[File:Hexagonal prism vertfig.png|60px]]
|align=center|{{CDD|node|6|node_1|2|node_1}}<br />t{2,6}
| [[File:Regular_polygon_6.svg|30px]]<br />[[Hexagon|{6}]]
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
|
| 8
| 18
| 12
|- bgcolor="#e0f0e0"
!P<sub>12</sub>
|align=center|Omnitruncated hexagonal dihedron<br />([[Dodecagonal prism]])
|[[File:Dodecagonal prism.png|60px]]
|[[File:Spherical truncated hexagonal prism.png|60px]]
|[[File:Dodecagonal prism vf.png|60px]]
|align=center|{{CDD|node_1|6|node_1|2|node_1}}<br />t<sub>0,1,2</sub>{2,6}=tr{2,6}
| [[File:Regular_polygon_10.svg|30px]]<br />[[Dodecagon|{12}]]
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
| [[File:Regular_polygon_4.svg|30px]]<br />[[Square|{4}]]
| 14
| 36
| 24
|- bgcolor="#d0f0f0"
!A<sub>6</sub>
|align=center|Snub hexagonal dihedron<br />([[Hexagonal antiprism]])
|[[File:Hexagonal antiprism.png|60px]]
|[[File:Spherical hexagonal antiprism.svg|60px]]
|[[File:Hexagonal antiprism vertfig.png|60px]]
|align=center|{{CDD|node_h|6|node_h|2x|node_h}}<br />sr{2,6}
| [[File:Regular_polygon_6.svg|30px]]<br />[[Hexagon|{6}]]
| 20px20px<br />2 [[Triangle|{3}]]
|&nbsp;
| 14
| 24
| 12
|- bgcolor="#d0f0f0"
!P<sub>3</sub>
|align=center|Cantic hexagonal dihedron<br />([[Triangular prism]])
|[[File:Triangular prism.png|60px]]
|[[File:Spherical triangular prism.svg|60px]]
|[[File:Triangular prism vertfig.png|60px]]
|align=center|{{CDD|node_h1|6|node|2|node_1}} = {{CDD|node_1|3|node|2|node_1}}<br />h<sub>2</sub>{6,2}=t{2,3}
|
|
|
|5
|9
|6
|- bgcolor="#d0f0f0"
!P<sub>6</sub>
|align=center|Cantic snub hexagonal dihedron<br />([[Hexagonal prism]])
|[[File:Hexagonal prism.png|60px]]
|[[File:Spherical hexagonal prism.svg|60px]]
|[[File:Hexagonal prism vertfig.png|60px]]
|align=center|{{CDD|node_h|6|node_h|2x|node_1}}<br />s<sub>2</sub>{6,2}=t{2,6}
|
|
|
| 8
| 18
| 12
|- bgcolor="#d0f0f0"
!A<sub>3</sub><br />[2]
|align=center|Snub hexagonal hosohedron<br />(same as Triangular antiprism)<br />(same as [[Octahedron|octahedron]])
|[[File:Trigonal antiprism.png|60px]]
|[[File:Spherical trigonal antiprism.svg|60px]]
|[[File:Octahedron vertfig.svg|60px]]
|align=center|{{CDD|node|6|node_h|2x|node_h}}<br />s{2,6}=sr{2,3}
|
|
|
| 8
| 12
| 6
|}

== Wythoff construction operators ==
{| class="wikitable"
!Operation
!Symbol
!Coxeter<br />diagram
!Description
|-
!width=150|Parent
!width=60|{p,q}<br />t<sub>0</sub>{p,q}
|{{CDD|node_1|p|node|q|node}}
| Any regular polyhedron or tiling
|-
! [[Rectification (geometry)|Rectified]] (r)
!r{p,q}<br />t<sub>1</sub>{p,q}
|{{CDD|node|p|node_1|q|node}}
|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)<br />(also [[Dual polyhedron|dual]])
!2r{p,q}<br />t<sub>2</sub>{p,q}
|{{CDD|node|p|node|q|node_1}}
|[[File:Dual Cube-Octahedron.svg|100px|right]]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.
|-
![[Truncation (geometry)|Truncated]] (t)
!t{p,q}<br />t<sub>0,1</sub>{p,q}
|{{CDD|node_1|p|node_1|q|node}}
|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.<br />[[File:Cube truncation sequence.svg|400px]]
|-
!Bitruncated (2t)<br />(also truncated dual)
!2t{p,q}<br />t<sub>1,2</sub>{p,q}
|{{CDD|node|p|node_1|q|node_1}}
|A bitruncation can be seen as the truncation of the dual. A bitruncated cube is a truncated octahedron.
|-
! [[Cantellation (geometry)|Cantellated]] (rr)<br />(Also [[Expansion (geometry)|expanded]])
!rr{p,q}
|{{CDD|node_1|p|node|q|node_1}}
|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}.<br />[[File:Cube cantellation sequence.svg|400px]]
|-
!Cantitruncated (tr)<br />(Also [[Omnitruncated polyhedron|omnitruncated]])
!tr{p,q}<br />t<sub>0,1,2</sub>{p,q}
|{{CDD|node_1|p|node_1|q|node_1}}
|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.
|}

{| class="wikitable"
|+ Alternation operations
|-
!width=150|Operation
!width=70|Symbol
!Coxeter<br />diagram
!Description
|-

!Snub rectified (sr)
!sr{p,q}
|{{CDD|node_h|p|node_h|q|node_h}}
|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.<br />[[File:Snubcubes in grCO.svg|320px]]

|-
!Snub (s)
!s{p,2q}
|{{CDD|node_h|p|node_h|2x|q|node}}
|Alternated truncation
|-
!Cantic snub (s<sub>2</sub>)
!s<sub>2</sub>{p,2q}
|{{CDD|node_h|p|node_h|2x|q|node_1}}
|

|-
!Alternated cantellation (hrr)
!hrr{2p,2q}
|{{CDD|node_h|2x|p|node|2x|q|node_h}}
|Only possible in uniform tilings (infinite polyhedra), alternation of {{CDD|node_1|2x|p|node|2x|q|node_1}}<br />For example, {{CDD|node_h|4|node|4|node_h}}
|-
!Half (h)
!h{2p,q}
|{{CDD|node_h1|2x|p|node|q|node}}
|align=left|[[Alternation (geometry)|Alternation]] of {{CDD|node_1|2x|p|node|q|node}}, same as {{CDD|labelp|branch_10ru|split2-qq|node}}
|-
!Cantic (h<sub>2</sub>)
!h<sub>2</sub>{2p,q}
|{{CDD|node_h1|2x|p|node|q|node_1}}
|Same as {{CDD|labelp|branch_10ru|split2-qq|node_1}}
|-
!Half rectified (hr)
!hr{2p,2q}
|{{CDD|node|2x|p|node_h1|2x|q|node}}
|align=left|Only possible in uniform tilings (infinite polyhedra), alternation of {{CDD|node|2x|p|node_1|2x|q|node}}, same as {{CDD|labelp|branch_10ru|2a2b-cross|branch_10lu|labelq}} or {{CDD|labelp|branch_10r|iaib|branch_01l|labelq}}<br />For example, {{CDD|node|4|node_h1|4|node}} = {{CDD|nodes_10ru|2a2b-cross|nodes_10lu}} or {{CDD|nodes_11|iaib|nodes}}
|-
!Quarter (q)
!q{2p,2q}
|{{CDD|node_h1|2x|p|node|2x|q|node_h1}}
|Only possible in uniform tilings (infinite polyhedra), same as {{CDD|labelq|branch_11|papb-cross|branch_10l|labelq}}<br />For example, {{CDD|node_h1|4|node|4|node_h1}} = {{CDD|nodes_11|2a2b-cross|nodes_10lu}} or {{CDD|nodes_11|iaib|nodes_10l}}
|}

== See also ==
*[[Polyhedron]]
**[[Regular polyhedron]]
**[[Quasiregular polyhedron]]
**[[Semiregular polyhedron]]
*[[List of uniform polyhedra]]
**[[List of uniform polyhedra by vertex figure]]
**[[List of uniform polyhedra by Wythoff symbol]]
**[[List of uniform polyhedra by Schwarz triangle]]
*[[List of Johnson solids]]
*[[List of Wenninger polyhedron models]]
*[[Polyhedron model]]
*[[Uniform tiling]]
*[[Uniform tilings in hyperbolic plane]]
*Pseudo-uniform polyhedron
*List of shapes

== Notes ==
{{reflist}}

== References ==
*[[Biography:Max Brückner|Brückner, M.]] ''Vielecke und vielflache. Theorie und geschichte.''. Leipzig, Germany: Teubner, 1900. [http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=ABN8316.0001.001]
*{{Cite journal | last1=Coxeter | first1=Harold Scott MacDonald |  last2=Longuet-Higgins | first2=M. S. |  last3=Miller | first3=J. C. P. | author-link3=J. C. P. Miller| title=Uniform polyhedra | jstor=91532 | mr=0062446 | year=1954 | journal=Philosophical Transactions of the Royal Society A| issn=0080-4614 | volume=246 |issue=916 | pages=401–450 | doi=10.1098/rsta.1954.0003 | bibcode=1954RSPTA.246..401C | s2cid=202575183 |url=http://rsta.royalsocietypublishing.org/content/roypta/246/916/401.full.pdf}}
*{{citation|first= B.|last= Grünbaum |author-link1= Branko Grünbaum |chapter= Polyhedra with Hollow Faces
|editor1= Tibor Bisztriczky |editor2= Peter McMullen
|editor3= Rolf Schneider |display-editors = 3 |editor4= A Weiss
|title= Proceedings of the NATO Advanced Study Institute on Polytopes: Abstract, Convex and Computational
|year= 1994 |publisher= Springer |isbn= 978-94-010-4398-4|pages= 43–70
|doi= 10.1007/978-94-011-0924-6_3 }}
*{{citation|first1= Peter|last1= McMullen |author-link1=Peter McMullen
|title= Abstract Regular Polytopes
|year= 2002 |publisher= Cambridge University Press
|first2=Egon|last2=Schulte}}
*{{Cite journal | last1=Skilling | first1=J. | title=The complete set of uniform polyhedra | jstor=74475 | mr=0365333 | year=1975 | journal=Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences | issn=0080-4614 | volume=278 | pages=111–135 | doi=10.1098/rsta.1975.0022 | issue=1278| bibcode=1975RSPTA.278..111S | s2cid=122634260 }}
*{{Cite journal | last1=Sopov | first1=S. P. | title=A proof of the completeness on the list of elementary homogeneous polyhedra | mr=0326550 | year=1970 | journal=Ukrainskiui Geometricheskiui Sbornik | issue=8 | pages=139–156 }}
*{{cite book | first=Magnus | last=Wenninger |  title=Polyhedron Models | publisher=Cambridge University Press | year=1974 | isbn=978-0-521-09859-5 }}

== External links ==
* {{MathWorld | urlname=UniformPolyhedron | title=Uniform Polyhedron}}
*[http://harel.org.il/zvi/docs/uniform.pdf Uniform Solution for Uniform Polyhedra]
*[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
*[http://www.georgehart.com/virtual-polyhedra/uniform-info.html Virtual Polyhedra] Uniform Polyhedra
*[http://www.stereo-club.fr/GalerieSCF/main.php?cmd=album&var1=Edmond+Bonan%2FPolyedres/&var2=3 Uniform polyhedron gallery]
*[http://mathworld.wolfram.com/UniformPolyhedron.html Uniform Polyhedron -- from Wolfram MathWorld] ''Has a visual chart of all 75''

{{Polytopes}}

[[Category:Uniform polyhedra|*]]

{{Sourceattribution|Uniform polyhedron}}