Uniform polyhedron

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• Infinite sets of uniform prisms and antiprisms
  Prismatic uniform polyhedron
  
  A prismatic uniform polyhedron is a uniform polyhedron with Dihedral symmetry in three dimensions. They exist in two infinite families, the uniform Prism and the uniform antiprisms....
   (including star forms)
• 5 Platonic solid
  Platonic solid
  
  In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
  s - regular convex polyhedra
• 4 Kepler-Poinsot polyhedra - regular nonconvex polyhedra
• 13 Archimedean solid
  Archimedean solid
  
  In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
  s - quasiregular
  Quasiregular polyhedron
  
  A polyhedron which has regular faces and is transitive on its edges but not transitive on its faces is said to be quasiregular.A quasiregular polyhedron can have faces of only two kinds and these must alternate around each vertex....
   and semiregular
  Semiregular polyhedron
  
  A semiregular polyhedron is a polyhedron with regular polygon faces and a symmetry group which is transitive on its vertices. Or at least, that is what follows from Thorold Gosset's 1900 definition of the more general semiregular polytope....
   convex polyhedra
• 14 nonconvex polyhedra with convex faces
  List of uniform polyhedra
  
  Uniform polyhedra and tilings form a well studied group. They are listed here for quick comparison of their properties and varied naming schemes and symbols....
  
• 39 nonconvex polyhedra with nonconvex faces
  List of uniform polyhedra
  
  Uniform polyhedra and tilings form a well studied group. They are listed here for quick comparison of their properties and varied naming schemes and symbols....
  
• 1 polyhedron found by John Skilling
  John Skilling
  
  John Skilling was a civil engineer and architect, best known for being the chief structural engineer of the World Trade Center....
   with pairs of edges that coincide, called Great disnub dirhombidodecahedron
  Great disnub dirhombidodecahedron
  
  In geometry, the Great disnub dirhombidodecahedron, also called Skilling's figure, is a nonconvex uniform polyhedron.John Skilling discovered this one further uniform polyhedron, by relaxing the condition that only two faces may meet at an edge....
   (Skilling's figure).

History

• The Platonic solid
  Platonic solid
  
  In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
  s date back to the classical Greeks and were studied by Plato
  Plato
  
  Plato , was a Classical Greece Greeks philosopher, mathematician, writer of philosophical dialogues, and founder of the Platonic Academy in Ancient Athens, the first institution of higher learning in the western world....
  , Theaetetus
  Theaetetus (mathematician)
  
  Theaetetus of Athens, son of Euphronius, of the Athenian deme Sunium, was a classical Greece mathematician. His principal contributions were on irrational number lengths, which was included in Book X of Euclid's Elements, and proving that there are precisely five Platonic solid....
   and Euclid
  Euclid
  
  Euclid , floruit 300 BC, also known as Euclid of Alexandria, was a Greek mathematics and is often referred to as the Father of Geometry. He was active in Alexandria during the reign of Ptolemy I ....
  .
• Johannes Kepler
  Johannes Kepler
  
  Johannes Kepler was a Germans mathematician, astronomer and astrologer, and key figure in the 17th century Scientific revolution. He is best known for his eponymous Kepler's laws of planetary motion, codified by later astronomers based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astrononomy....
   (1571-1630) was the first to publish the complete list of Archimedean solid
  Archimedean solid
  
  In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
  s after the original work of Archimedes
  Archimedes
  
  Archimedes of Syracuse was a Greek mathematics, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity....
   was lost.
• Kepler
  Johannes Kepler
  
  Johannes Kepler was a Germans mathematician, astronomer and astrologer, and key figure in the 17th century Scientific revolution. He is best known for his eponymous Kepler's laws of planetary motion, codified by later astronomers based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astrononomy....
   (1619) discovered two of the regular Kepler-Poinsot polyhedra and Louis Poinsot
  Louis Poinsot
  
  Louis Poinsot was a France mathematician and physicist. Poinsot was the inventor of geometrical mechanics, showing how a system of forces acting on a rigid body could be resolved into a single force and a couple ....
   (1809) discovered the other two.
• Of the remaining 66, Albert Badoureau (1881) discovered 37. Edmund Hess
  Edmund Hess
  
  Edmund Hess was a Germany mathematician who discovered several regular polytopes.See also* Schl?fli-Hess polychoron* Hess polytope...
   (1878) discovered 2 more and Pitsch (1881) independently discovered 18, of which 15 had not previously been discovered.
• The geometer Donald Coxeter
  Harold Scott MacDonald Coxeter
  
  Harold Scott MacDonald "Donald" Coxeter, Order of Canada is regarded as one of the great geometers of the 20th century. He was born in London but spent most of his life in Canada....
   discovered the remaining twelve in collaboration with J. C. P. Miller
  J. C. P. Miller
  
  Jeffrey Charles Percy Miller was an England mathematician and computing pioneer. He worked in number theory and on geometry, particularly polyhedra, where Miller's monster refers to the Great dirhombicosidodecahedron....
   (1930-1932) but did not publish. M.S. and H.C. Longuet-Higgins and independently discovered 11 of these.

• In 1974, Magnus Wenninger
  Magnus Wenninger
  
  Father Magnus J. Wenninger Order of Saint Benedict is a mathematician who works on uniform polyhedra, and wrote the first book on their construction....
   published his book Polyhedron models
  List of Wenninger polyhedron models
  
  This table contains an indexed list of the Uniform and stellated polyhedra from the book Polyhedron Models, by Magnus Wenninger.The book was written as a guide book to building polyhedra as physical models....
  , which lists all 75 nonprismatic uniform polyhedra, with many previously unpublished names given to them by Norman Johnson.

• 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
  Mathematica
  
  Mathematica is a computational software program used widely in scientific, engineering, and mathematical fields and other areas of technical computing....
   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
  Wythoff symbol
  
  In geometry, a Wythoff symbol is a short-hand notation, created by mathematician Willem Abraham Wythoff, for naming the regular and semiregular polyhedra using a Wythoff construction, by representing them as tilings on the surface of a sphere, Euclidean plane, or hyperbolic plane....
  .

Indexing

• [C] Coxeter et al., 1954, showed the convex forms as figures 15 through 32; three prismatic forms, figures 33–35; and the nonconvex forms, figures 36–92.
• [W] Wenninger, 1974, has 119 figures: 1-5 for the Platonic solids, 6-18 for the Archimedean solids, 19-66 for stellated forms including the 4 regular nonconvex polyhedra, and ended with 67-119 for the nonconvex uniform polyhedra.
• [K] Kaleido, 1993: The 80 figures were grouped by symmetry: 1-5 as representatives of the infinite families of prismatic forms with dihedral symmetry
  Dihedral symmetry in three dimensions
  
  This article deals with three infinite series of point groups in three dimensions which have a symmetry group which as abstract group is a dihedral group Dihn ....
  , 6-9 with tetrahedral symmetry
  Tetrahedral symmetry
  
  A regular tetrahedron has 12 rotational symmetries, and a total of 24 symmetries including transformations that combine a reflection and a rotation....
  , 10-26 with Octahedral symmetry
  Octahedral symmetry
  
  A regular octahedron has 24 rotational symmetries, and a total of 48 symmetries including transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the dual polyhedron of an octahedron....
  , 46-80 with icosahedral symmetry
  Icosahedral symmetry
  
  File:Soccer ball.svgA regular icosahedron has 60 rotational symmetries, and a total of 120 symmetries including transformations that combine a reflection and a rotation....
  .
• [U] Mathematica, 1993, follows the Kaleido series with the 5 prismatic forms moved to last, so that the nonprismatic forms become 1–75.

Nonconvex uniform polyhedra

Main article: Nonconvex uniform polyhedron

Nonconvex uniform polyhedron

In geometry, a 'nonconvex uniform polyhedron', or 'uniform star polyhedron', is a self-intersecting uniform polyhedron. Each can contain either star polygon faces, star polygon vertex figures or both....

Convex forms by Wythoff construction

• Tetrahedral symmetry
  Tetrahedral symmetry
  
  A regular tetrahedron has 12 rotational symmetries, and a total of 24 symmetries including transformations that combine a reflection and a rotation....
   (3 3 2) - order 24
• Octahedral symmetry
  Octahedral symmetry
  
  A regular octahedron has 24 rotational symmetries, and a total of 48 symmetries including transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the dual polyhedron of an octahedron....
   (4 3 2) - order 48
• Icosahedral symmetry
  Icosahedral symmetry
  
  File:Soccer ball.svgA regular icosahedron has 60 rotational symmetries, and a total of 120 symmetries including transformations that combine a reflection and a rotation....
   (5 3 2) - order 60
• Dihedral symmetry (n 2 2), for all n=3,4,5,... - order 4n

1. Hosohedron
   Hosohedron
   
   An Polygon hosohedron is a tessellation of Lune s on a spherical surface, such that each lune shares the same two vertices. A regular n-gonal hosohedron has Schl?fli symbol ....
   s H_2... (Only as spherical tilings)
2. Dihedron
   Dihedron
   
   A dihedron is a type of polyhedron, made of two polygon faces which share the same set of edges. It is Mathematical degeneracy if its faces are flat....
   s D_2... (Only as spherical tilings)
3. Prisms
   Prism (geometry)
   
   In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....
    P_3... (Truncated hosohedrons)
4. Antiprism
   Antiprism
   
   An n-sided antiprism is a polyhedron composed of 2 parallel copies of some particular n-sided polygon, connected by an alternating band of triangles....
   s A_3... (Snub prisms)

Summary tables

	Parent	Truncated	Rectified	Bitruncated (tr. dual)	Birectified (dual)	Cantellated	Omnitruncated (Cantitruncated)	Snub
Extended Schläfli symbol Schläfli symbol In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
	t0	t0,1	t1	t1,2	t2	t0,2	t0,1,2	s
Wythoff symbol Wythoff symbol In geometry, a Wythoff symbol is a short-hand notation, created by mathematician Willem Abraham Wythoff, for naming the regular and semiregular polyhedra using a Wythoff construction, by representing them as tilings on the surface of a sphere, Euclidean plane, or hyperbolic plane.... 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-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction....
Vertex figure Vertex configuration In polyhedral geometry a vertex configuration is a short-hand notation for representing a polyhedron vertex figure as the sequence of faces around a vertex....	pq	(q.2p.2p)	(p.q.p.q)	(p.2q.2q)	qp	(p.4.q.4)	(4.2p.2q)	(3.3.p.3.q)
Tetrahedral Tetrahedral symmetry A regular tetrahedron has 12 rotational symmetries, and a total of 24 symmetries including transformations that combine a reflection and a rotation.... 3-3-2	Tetrahedron A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....	(3.6.6) Truncated tetrahedron The truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangle faces, 12 vertices and 18 edges....	(3.3.3.3) Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....	(3.6.6) Truncated tetrahedron The truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangle faces, 12 vertices and 18 edges....	Tetrahedron A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....	(3.4.3.4) Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....	(4.6.6) Truncated octahedron The truncated octahedron is an Archimedean solid. It has 8 regular hexagonal faces, 6 Square faces, 24 vertices and 36 edges. Since each of its faces has point symmetry the truncated octahedron is a zonohedron....	(3.3.3.3.3) Icosahedron In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
Octahedral Octahedral symmetry A regular octahedron has 24 rotational symmetries, and a total of 48 symmetries including transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the dual polyhedron of an octahedron.... 4-3-2	Cube A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....	(3.8.8) Truncated cube The truncated cube, or truncated hexahedron, is an Archimedean solid. It has 6 regular octagonal faces, 8 regular triangle faces, 24 vertices and 36 edges....	(3.4.3.4) Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....	(4.6.6) Truncated octahedron The truncated octahedron is an Archimedean solid. It has 8 regular hexagonal faces, 6 Square faces, 24 vertices and 36 edges. Since each of its faces has point symmetry the truncated octahedron is a zonohedron....	Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....	(3.4.4.4)	(4.6.8)	(3.3.3.3.4) Snub cube The snub cube, or snub cuboctahedron, is an Archimedean solid.The snub cube has 38 faces, 6 of which are square s and the other 32 are equilateral triangles....
Icosahedral Icosahedral symmetry File:Soccer ball.svgA regular icosahedron has 60 rotational symmetries, and a total of 120 symmetries including transformations that combine a reflection and a rotation.... 5-3-2	Dodecahedron A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....	(3.10.10) Truncated dodecahedron In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangle faces, 60 vertices and 90 edges....	(3.5.3.5) Icosidodecahedron An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....	(5.6.6) Truncated icosahedron The truncated icosahedron is an Archimedean solid. It comprises 12 regular pentagon faces, 20 regular hexagon faces, 60 vertices and 90 edges....	Icosahedron In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....	(3.4.5.4) Rhombicosidodecahedron The rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid. It has 20 regular triangle faces, 30 square faces, 12 regular pentagonal faces, 60 vertices and 120 edges....	(4.6.10) Truncated icosidodecahedron The truncated icosidodecahedron is an Archimedean solid. It has 30 square faces, 20 regular hexagonal faces, 12 regular decagonal faces, 120 vertices and 180 edges....	(3.3.3.3.5) Snub dodecahedron The snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid.The snub dodecahedron has 92 faces, of which 12 are pentagons and the other 80 are equilateral triangles....

(p 2 2)	Parent	Truncated	Rectified	Bitruncated (tr. dual)	Birectified (dual)	Cantellated	Omnitruncated (Cantitruncated)	Snub
Extended Schläfli symbol Schläfli symbol In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
	t0	t0,1	t1	t1,2	t2	t0,2	t0,1,2	s
Wythoff symbol Wythoff symbol In geometry, a Wythoff symbol is a short-hand notation, created by mathematician Willem Abraham Wythoff, for naming the regular and semiregular polyhedra using a Wythoff construction, by representing them as tilings on the surface of a sphere, Euclidean plane, or hyperbolic plane....	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 Coxeter-Dynkin diagram In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction....
Vertex figure Vertex configuration In polyhedral geometry a vertex configuration is a short-hand notation for representing a polyhedron vertex figure as the sequence of faces around a vertex....	p2	(2.2p.2p)	(p.2.p.2)	(p.4.4)	2p	(p.4.2.4)	(4.2p.4)	(3.3.p.3.2)
Dihedral (2 2 2)	Dihedron A dihedron is a type of polyhedron, made of two polygon faces which share the same set of edges. It is Mathematical degeneracy if its faces are flat....	2.4.4	2.2.2.2	4.4.2	Hosohedron An Polygon hosohedron is a tessellation of Lune s on a spherical surface, such that each lune shares the same two vertices. A regular n-gonal hosohedron has Schl?fli symbol ....	2.4.2.4	4.4.4	3.3.3.2
Dihedral (3 2 2)	Dihedron A dihedron is a type of polyhedron, made of two polygon faces which share the same set of edges. It is Mathematical degeneracy if its faces are flat....	2.6.6	2.3.2.3	4.4.3	Hosohedron An Polygon hosohedron is a tessellation of Lune s on a spherical surface, such that each lune shares the same two vertices. A regular n-gonal hosohedron has Schl?fli symbol ....	2.4.3.4	4.4.6	3.3.3.3
Dihedral (4 2 2)	Dihedron A dihedron is a type of polyhedron, made of two polygon faces which share the same set of edges. It is Mathematical degeneracy if its faces are flat....	2.8.8	2.4.2.4	4.4.4	Hosohedron An Polygon hosohedron is a tessellation of Lune s on a spherical surface, such that each lune shares the same two vertices. A regular n-gonal hosohedron has Schl?fli symbol ....	2.4.4.4	4.4.8	3.3.3.4
Dihedral (5 2 2)	Dihedron A dihedron is a type of polyhedron, made of two polygon faces which share the same set of edges. It is Mathematical degeneracy if its faces are flat....	2.10.10	2.5.2.5	4.4.5	Hosohedron An Polygon hosohedron is a tessellation of Lune s on a spherical surface, such that each lune shares the same two vertices. A regular n-gonal hosohedron has Schl?fli symbol ....	2.4.5.4	4.4.10	3.3.3.5
Dihedral (6 2 2)	Dihedron A dihedron is a type of polyhedron, made of two polygon faces which share the same set of edges. It is Mathematical degeneracy if its faces are flat....	2.12.12	2.6.2.6	4.4.6	Hosohedron An Polygon hosohedron is a tessellation of Lune s on a spherical surface, such that each lune shares the same two vertices. A regular n-gonal hosohedron has Schl?fli symbol ....	2.4.6.4	4.4.12	3.3.3.6

Wythoff construction operators

Example forms from the cube Cube A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....  and octahedron Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....

Operation	Extended Schläfli symbols Schläfli symbol In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....		Coxeter- Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction....	Description
Parent	t0			Any regular polyhedron or tiling
Rectified Rectification (geometry) In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points....	t1			The edges are fully truncated into single points. The polyhedron now has the combined faces of the parent and dual.
Birectified Also Dual Dual polyhedron In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....	t2			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 is also a regular polyhedron .
Truncated Truncation (geometry) In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new Facet in place of each vertex....	t0,1			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	t1,2			Same as truncated dual.
Cantellated Cantellation (geometry) In geometry, a cantellation is an operation in any dimension that cuts a regular polytope at its edges and vertices, creating a new facet in place of each edge and vertex.... (or rhombated) (Also expanded Expansion (geometry) In geometry, expansion is a polytope operation where Facet are separated and moved radially apart, and new facets are formed at separated elements .... )	t0,2			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.
Omnitruncated Omnitruncation (geometry) In geometry, an omnitruncation is an operation applied to a regular polytope in a Wythoff construction that creates a maximum number of facets.... (or cantitruncated) (or rhombitruncated)	t0,1,2			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.
Snub Snub (geometry) In geometry, an alternation is an operation on a polyhedron or tessellation that fully truncates alternate vertices. Only even-sided polyhedra can be alternated, for example the zonohedron....	s			The snub takes the omnitruncated form and rectifies alternate vertices. (This operation is only possible for polyhedra with all even-sided faces.) 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.

(3 3 2) T_d Tetrahedral symmetry

#	Name	Picture	Tiling	Coxeter-Dynkin Coxeter-Dynkin diagram In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction.... and Schläfli Schläfli symbol In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas.... symbols	Face counts by position			Element counts
					Pos. 2 [3] (4)	Pos. 1 [ ]x[ ] (6)	Pos. 0 [3] (4)	Faces	Edges	Vertices
1	tetrahedron Tetrahedron A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....				Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....			4	6	4
[1]	Birectified tetrahedron (Same as tetrahedron Tetrahedron A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids.... )			t2			Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....	4	6	4
2	rectified tetrahedron (Same as octahedron Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex.... )			t1	Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....		Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....	8	12	6
3	truncated tetrahedron Truncated tetrahedron The truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangle faces, 12 vertices and 18 edges....			t0,1	Hexagon In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....		Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....	8	18	12
[3]	Bitruncated tetrahedron (Same as truncated tetrahedron Truncated tetrahedron The truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangle faces, 12 vertices and 18 edges.... )			t1,2	Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....		Hexagon In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....	8	18	12
4	cantellated tetrahedron (Same as cuboctahedron Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square.... )			t0,2	Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....	Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....	Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....	14	24	12
5	omnitruncated tetrahedron (Same as truncated octahedron Truncated octahedron The truncated octahedron is an Archimedean solid. It has 8 regular hexagonal faces, 6 Square faces, 24 vertices and 36 edges. Since each of its faces has point symmetry the truncated octahedron is a zonohedron.... )			t0,1,2	Hexagon In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....	Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....	Hexagon In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....	14	36	24
6	Snub tetrahedron (Same as icosahedron Icosahedron In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces.... )			s	Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....	2 Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....	Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....	20	30	12

(4 3 2) O_h Octahedral symmetry

#	Name	Picture	Tiling	Coxeter-Dynkin Coxeter-Dynkin diagram In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction.... and Schläfli Schläfli symbol In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas.... symbols	Face counts by position			Element counts
					Pos. 2 [4] (8)	Pos. 1 [ ]x[ ] (6)	Pos. 0 [3] (12)	Faces	Edges	Vertices
7	Cube Cube A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....				Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....			6	12	8
[2]	Octahedron Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....						Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....	8	12	6
[4]	rectified cube rectified octahedron (Cuboctahedron Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square.... )				Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....		Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....	14	24	12
8	Truncated cube Truncated cube The truncated cube, or truncated hexahedron, is an Archimedean solid. It has 6 regular octagonal faces, 8 regular triangle faces, 24 vertices and 36 edges....			t0,1	Octagon In geometry, an octagon is a polygon that has 8 sides. A regular octagon is represented by the Schl?fli symbol ....		Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....	14	36	24
[5]	Truncated octahedron Truncated octahedron The truncated octahedron is an Archimedean solid. It has 8 regular hexagonal faces, 6 Square faces, 24 vertices and 36 edges. Since each of its faces has point symmetry the truncated octahedron is a zonohedron....			t0,1	Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....		Hexagon In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....	14	36	24
9	Cantellated cube cantellated octahedron Rhombicuboctahedron Rhombicuboctahedron The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangle and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each....			t0,2	Octagon In geometry, an octagon is a polygon that has 8 sides. A regular octagon is represented by the Schl?fli symbol ....	Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....	Hexagon In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....	26	48	24
10	Omnitruncated cube omnitruncated octahedron Truncated cuboctahedron Truncated cuboctahedron The truncated cuboctahedron is an Archimedean solid. It has 12 Square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges....			t0,1,2	Octagon In geometry, an octagon is a polygon that has 8 sides. A regular octagon is represented by the Schl?fli symbol ....	Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....	Hexagon In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....	26	72	48
[6]	Alternated truncated octahedron (Same as Icosahedron Icosahedron In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces.... )			h0,1		Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....	Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....	20	30	12
[1]	Alternated cube (Same as tetrahedron Tetrahedron A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids.... )			h			1/2 Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....	6	12	8
11	Snub cube Snub cube The snub cube, or snub cuboctahedron, is an Archimedean solid.The snub cube has 38 faces, 6 of which are square s and the other 32 are equilateral triangles....			s	Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....	2 Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....	Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....	38	60	24

(5 3 2) I_h Icosahedral symmetry

#	Name	Picture	Tiling	Coxeter-Dynkin Coxeter-Dynkin diagram In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction.... and Schläfli Schläfli symbol In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas.... symbols	Face counts by position			Element counts
					Pos. 2 [5] (12)	Pos. 1 [ ]x[ ] (30)	Pos. 0 [3] (20)	Faces	Edges	Vertices
12	Dodecahedron Dodecahedron A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....				Pentagon In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....			12	30	20
[6]	Icosahedron Icosahedron In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....						Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....	20	30	12
13	Rectified dodecahedron Rectified icosahedron Icosidodecahedron Icosidodecahedron An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....			t1	Pentagon In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....		Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....	32	60	30
14	Truncated dodecahedron Truncated dodecahedron In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangle faces, 60 vertices and 90 edges....			t0,1	Decagon In geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular polygon decagon, having all sides of equal length and all internal angles equal to 4π/5 ....		Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....	32	90	60
15	Truncated icosahedron Truncated icosahedron The truncated icosahedron is an Archimedean solid. It comprises 12 regular pentagon faces, 20 regular hexagon faces, 60 vertices and 90 edges....			t0,1	Pentagon In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....		Hexagon In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....	32	90	60
16	Cantellated dodecahedron Cantellated icosahedron Rhombicosidodecahedron Rhombicosidodecahedron The rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid. It has 20 regular triangle faces, 30 square faces, 12 regular pentagonal faces, 60 vertices and 120 edges....			t0,2	Pentagon In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....	Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....	Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....	62	120	60
17	Omnitruncated dodecahedron Omnitruncated icosahedron Truncated icosidodecahedron Truncated icosidodecahedron The truncated icosidodecahedron is an Archimedean solid. It has 30 square faces, 20 regular hexagonal faces, 12 regular decagonal faces, 120 vertices and 180 edges....			t0,1,2	Decagon In geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular polygon decagon, having all sides of equal length and all internal angles equal to 4π/5 ....	Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....	Hexagon In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....	62	180	120
18	Snub dodecahedron Snub dodecahedron The snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid.The snub dodecahedron has 92 faces, of which 12 are pentagons and the other 80 are equilateral triangles.... Snub icosahedron			s	Pentagon In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....	2 Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....	Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....	92	150	60

(p 2 2) Prismatic [p,2], I₂(p) family (D_ph Dihedral symmetry)

Main article: Prismatic uniform polyhedron

Prismatic uniform polyhedron

A prismatic uniform polyhedron is a uniform polyhedron with Dihedral symmetry in three dimensions. They exist in two infinite families, the uniform Prism and the uniform antiprisms....

(2 2 2) dihedral symmetry

#	Name	Picture	Tiling	Coxeter-Dynkin Coxeter-Dynkin diagram In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction.... and Schläfli Schläfli symbol In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas.... symbols	Face counts by position			Element counts
					Pos. 2 [2] (2)	Pos. 1 [ ]x[ ] (2)	Pos. 0 [ ]x[ ] (2)	Faces	Edges	Vertices
D2 H2	digonal dihedron digonal hosohedron				Digon In geometry a digon is a Degeneracy polygon with two sides and two Vertex .A digon must be Regular polygon because its two edges are the same length....			2	2	2
D4	truncated digonal dihedron (Same as square dihedron)			t=	Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....			2	4	4
P4 [7]	omnitruncated digonal dihedron (Same as cube Cube A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids.... )			t0,1,2	Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....	Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....	Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....	6	12	8
A2 [1]	snub digonal dihedron (Same as tetrahedron Tetrahedron A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids.... )			s		Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....		4	6	4

(3 2 2) D_3hdihedral symmetry

#	Name	Picture	Tiling	Coxeter-Dynkin Coxeter-Dynkin diagram In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction.... and Schläfli Schläfli symbol In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas.... symbols	Face counts by position			Element counts
					Pos. 2 [3] (2)	Pos. 1 [ ]x[ ] (3)	Pos. 0 [ ]x[ ] (3)	Faces	Edges	Vertices
D3	Trigonal dihedron				Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....			2	3	3
H3	Trigonal hosohedron						Digon In geometry a digon is a Degeneracy polygon with two sides and two Vertex .A digon must be Regular polygon because its two edges are the same length....	3	3	2
D6	Truncated trigonal dihedron (Same as hexagonal dihedron)			t	Hexagon In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....			2	6	6
P3	Truncated trigonal hosohedron (Triangular prism Triangular prism In geometry, a triangular prism or three-sided prism is a type of Prism ; it is a polyhedron made of a triangle base, a Translation copy, and 3 faces joining corresponding sides.... )			t	Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....	Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....		5	9	6
P6	Omnitruncated trigonal dihedron (Hexagonal prism Hexagonal prism In geometry, the hexagonal prism is a Prism with hexagonal base.It is an octahedron. However, the term octahedron is mainly used with "regular" in front or implied, hence not meaning a hexagonal prism; in the general meaning the term octahedron it is not much used because there are different types which have not much in common exce... )			t	Hexagon In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....	Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....	Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....	8	18	12
A3 [2]	Snub trigonal dihedron (Same as Triangular antiprism) (Same as octahedron Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex.... )			s	Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....	Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....		8	12	6

(4 2 2) D_4hdihedral symmetry

#	Name	Picture	Tiling	Coxeter-Dynkin Coxeter-Dynkin diagram In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction.... and Schläfli Schläfli symbol In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas.... symbols	Face counts by position			Element counts
					Pos. 2 [4] (2)	Pos. 1 [ ]x[ ] (4)	Pos. 0 [ ]x[ ] (4)	Faces	Edges	Vertices
D4	square dihedron				Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....			2	4	4
H4	square hosohedron						Digon In geometry a digon is a Degeneracy polygon with two sides and two Vertex .A digon must be Regular polygon because its two edges are the same length....	4	4	2
D8	Truncated square dihedron (Same as octagonal dihedron)			t	Octagon In geometry, an octagon is a polygon that has 8 sides. A regular octagon is represented by the Schl?fli symbol ....			2	8	8
P4 [7]	Truncated square hosohedron (Cube Cube A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids.... )			t	Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....	Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....		6	12	8
D8	Omnitruncated square dihedron (Octagonal prism Octagonal prism In geometry, the octagonal prism is the sixth in an infinite set of Prism formed by square sides and two regular polygon caps.If faces are all regular, it is a semiregular polyhedron.... )			t	Octagon In geometry, an octagon is a polygon that has 8 sides. A regular octagon is represented by the Schl?fli symbol ....	Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....	Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....	10	24	16
A4	Snub square dihedron (Square antiprism Square antiprism In geometry, the square antiprism is the second in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.... )			t	Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....	Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....		10	16	8

(5 2 2) D_5h dihedral symmetry

#	Name	Picture	Tiling	Coxeter-Dynkin Coxeter-Dynkin diagram In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction.... and Schläfli Schläfli symbol In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas.... symbols	Face counts by position			Element counts
					Pos. 2 [5] (2)	Pos. 1 [ ]x[ ] (5)	Pos. 0 [ ]x[ ] (5)	Faces	Edges	Vertices
D5	Pentagonal dihedron				Pentagon In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....			2	5	5
H5	Pentagonal hosohedron						Digon In geometry a digon is a Degeneracy polygon with two sides and two Vertex .A digon must be Regular polygon because its two edges are the same length....	5	5	2
D10	Truncated pentagonal dihedron (Same as decagonal dihedron)			t	Decagon In geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular polygon decagon, having all sides of equal length and all internal angles equal to 4π/5 ....			2	10	10
P5	Truncated pentagonal hosohedron (Same as pentagonal prism Pentagonal prism In geometry, the pentagonal prism is a Prism with a pentagonal base. It is a type of heptahedron.If faces are all regular, the pentagonal prism is a semiregular polyhedron, and the third in an infinite set of prisms formed by square sides and two regular polygon caps.... )			t	Pentagon In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....	Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....		7	15	10
P10	Omnitruncated pentagonal dihedron (Decagonal prism Decagonal prism In geometry, the decagonal prism is the eighth in an infinite set of Prism formed by square sides and two regular polygon caps.If faces are all regular, it is a semiregular polyhedron.... )			t	Decagon In geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular polygon decagon, having all sides of equal length and all internal angles equal to 4π/5 ....	Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....	Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....	12	30	20
A5	Snub pentagonal dihedron (Pentagonal antiprism Pentagonal antiprism In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.... )			t	Pentagon In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....	Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....		12	20	10

(6 2 2) D_6hdihedral symmtry

#	Name	Picture	Tiling	Coxeter-Dynkin Coxeter-Dynkin diagram In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction.... and Schläfli Schläfli symbol In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas.... symbols	Face counts by position			Element counts
					Pos. 2 [6] (2)	Pos. 1 [ ]x[ ] (6)	Pos. 0 [ ]x[ ] (6)	Faces	Edges	Vertices
D6	Hexagonal dihedron				Hexagon In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....			2	6	6
H6	Hexagonal hosohedron						Digon In geometry a digon is a Degeneracy polygon with two sides and two Vertex .A digon must be Regular polygon because its two edges are the same length....	6	6	2
D12	Truncated hexagonal dihedron (Same as dodecagonal dihedron)			t	Dodecagon In geometry, a dodecagon is any polygon with 12 sides and twelve angles....			2	12	12
H6	Truncated hexagonal hosohedron (Same as hexagonal prism Hexagonal prism In geometry, the hexagonal prism is a Prism with hexagonal base.It is an octahedron. However, the term octahedron is mainly used with "regular" in front or implied, hence not meaning a hexagonal prism; in the general meaning the term octahedron it is not much used because there are different types which have not much in common exce... )			t	Hexagon In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....	Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....		8	18	12
P12	Omnitruncated hexagonal dihedron (Dodecagonal prism Dodecagonal prism In geometry, the dodecagonal prism is the tenth in an infinite set of Prism formed by square sides and two regular polygon caps.If faces are all regular, it is a semiregular polyhedron.... )			t	Dodecagon In geometry, a dodecagon is any polygon with 12 sides and twelve angles....	Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....	Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....	14	36	24
A6	Snub hexagonal dihedron (Hexagonal antiprism Hexagonal antiprism In geometry, the hexagonal antiprism is the 4th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by 2 polygon caps.... )			t	Hexagon In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....	Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....		14	24	12

See also

• Polyhedron
  Polyhedron
  
  |}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
  
• Regular polyhedron
  Regular polyhedron
  
  A regular polyhedron is a polyhedron whose faces are Congruence regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive - i.e....
  
• Quasiregular polyhedron
  Quasiregular polyhedron
  
  A polyhedron which has regular faces and is transitive on its edges but not transitive on its faces is said to be quasiregular.A quasiregular polyhedron can have faces of only two kinds and these must alternate around each vertex....
  
• Semiregular polyhedron
  Semiregular polyhedron
  
  A semiregular polyhedron is a polyhedron with regular polygon faces and a symmetry group which is transitive on its vertices. Or at least, that is what follows from Thorold Gosset's 1900 definition of the more general semiregular polytope....
  
• List of uniform polyhedra
  List of uniform polyhedra
  
  Uniform polyhedra and tilings form a well studied group. They are listed here for quick comparison of their properties and varied naming schemes and symbols....
  
• List of Wenninger polyhedron models
  List of Wenninger polyhedron models
  
  This table contains an indexed list of the Uniform and stellated polyhedra from the book Polyhedron Models, by Magnus Wenninger.The book was written as a guide book to building polyhedra as physical models....
  
• Polyhedron model
  Polyhedron model
  
  A polyhedron model is a physical construction of a polyhedron, constructed from cardboard, plastic board, wood board or other panel material, or, less commonly, solid material....
  
• List of uniform polyhedra by vertex figure
  List of uniform polyhedra by vertex figure
  
  There are many relations among the uniform polyhedron.Some are obtained by truncating the vertices of the regular or quasi-regular polyhedron.Others share the same vertices and edges as other polyhedron....
  
• List of uniform polyhedra by Wythoff symbol
  List of uniform polyhedra by Wythoff symbol
  
  There are many relations among the uniform polyhedron.Here they are grouped by the Wythoff symbol....
  

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