Wythoff symbol

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Wythoff symbol

In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a Wythoff symbol is a short-hand notation, created by mathematician Willem Abraham Wythoff

Willem Abraham Wythoff

Willem Abraham Wythoff was a Dutch mathematician and number theory. Wythoff is well-known for his study of a combinatorial game theory referred to as Wythoff's game and for the Wythoff construction and the Wythoff symbol utilised in this tesselation construction....
, for naming the regular and semiregular polyhedra using a kaleidoscopic construction

Wythoff construction

File:Wythoffian_construction_diagram.pngIn geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling....
, by representing them as tilings on the surface of a sphere

Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
, Euclidean plane, or hyperbolic plane

Hyperbolic plane

In mathematics, the term hyperbolic plane may refer to:* A two-dimensional quadratic space with a non-singular isotropic quadratic form* A plane in hyperbolic geometry...
.

The Wythoff symbol gives 3 numbers p,q,r and a positional vertical bar (|) which separate the numbers before or after it. Each number represents the order of mirrors at a vertex of the fundamental triangle.

Each symbol represents one uniform polyhedron

Uniform polyhedron

A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
or tiling, although the same tiling/polyhedron can have different Wythoff symbols from different symmetry generators.

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Encyclopedia

In geometry

Geometry

Willem Abraham Wythoff

Wythoff construction

Sphere

Hyperbolic plane

Uniform polyhedron

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....
can be represented by 3 | 4 2 with O_h symmetry, and 2 4 | 2 as a square prism

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....
with 2 colors and D_4h symmetry, as well as 2 2 2 | with 3 colors and D_2h symmetry.

Summary table

There are 7 generator points with each set of p,q,r: (And a few special forms)

General			Right triangle (r=2)
Description	Wythoff symbol	Vertex configuration 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....	Wythoff symbol	Vertex configuration
regular 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.... and 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....	q \| p r	(p.r)^q	q \| p 2	p^q
	p \| q r	(q.r)^p	p \| q 2	q^p
	r \| p q	(q.p)^r	2 \| p q	(q.p)²
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.... and 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 ....	q r \| p	q.2p.r.2p	q 2 \| p	q.2p.2p
	p r \| q	p.2q.r.2q	p 2 \| q	p.2q.2q
	p q \| r	2r.q.2r.p	p q \| 2	4.q.4.p
even-faced Zonohedron A zonohedron is a convex set polyhedron where every face is a polygon with point symmetry or, equivalently, symmetry under rotations through 180?....	p q r \|	2r.2q.2p	p q 2 \|	4.2q.2p
	p q (r s) \|	2p.2q.-2p.-2q	p 2 (r s) \|	2p.4.-2p.4/3
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....	\| p q r	3.r.3.q.3.p	\| p q 2	3.3.q.3.p
	\| p q r s	(4.p.4.q.4.r.4.s)/2	-	-

There are three special cases:

p q (r s) | - This is a mixture of p q r | and p q s |.
| p q r - Snub forms (alternated) are give this otherwise unused symbol.
| p q r s - A unique snub form for U75
Great dirhombicosidodecahedron

In geometry, the great dirhombicosidodecahedron is a nonconvex uniform polyhedron, indexed last as U75.This is the only uniform polyhedron with more than six faces meeting at a vertex....
that isn't Wythoff constructable.

Description

The numbers p,q,r describe the fundamental triangle of the symmetry group: at its vertices, the generating mirrors meet in angles of p/p, p/q, p/r. On the sphere there are 3 main symmetry types: (3 3 2), (4 3 2), (5 3 2), and one infinite family (p 2 2), for any p. (All simple families have one right angle and so r=2.)

The position of the vertical bar in the symbol specifies a categorical position of the generator point within the fundamental triangle. The generator point can either be on or off each mirror, activated or not. This distinction creates 8 (2³) possible forms, neglecting one where the generator point is on all the mirrors.

In this notation the mirrors are labeled by the reflection-order of the opposite vertex. The p,q,r values are listed before the bar if the corresponding mirror is active.

The one impossible symbol | p q r implies the generator point is on all mirrors, which is only possible if the triangle is degenerate, reduced to a point. This unused symbol is therefore arbitrarily reassigned to represent the case where all mirrors are active, but odd-numbered reflected images are ignored. The resulting figure has rotational symmetry only.

This symbol is functionally similar to the more general 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....
, in which each node represents a mirror and the arcs between them – marked with numbers – the angles between the mirrors. (An arc representing a right angle is omitted.) A node is circled if the generator point is not on the mirror.

Symmetry triangles

There are 4 symmetry classes of reflection on the sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
, and two for the Euclidean plane. A few of the infinitely many
List of regular polytopes

This page lists the regular polytopes in Euclidean geometry, spherical geometry and hyperbolic geometry spaces.The Schl?fli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each....
for the hyperbolic plane
Hyperbolic plane

In mathematics, the term hyperbolic plane may refer to:* A two-dimensional quadratic space with a non-singular isotropic quadratic form* A plane in hyperbolic geometry...
are also listed.

(p 2 2) 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 ....
p=2,3,4... (Order 4p)
(3 3 2) 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....
(Order 24)
(3 3 3) *333 symmetry (Euclidean plane)
(4 3 3) *433 symmetry (Hyperbolic plane)
(4 4 3) *443 symmetry (Hyperbolic plane)
(4 4 4) *444 symmetry (Hyperbolic plane)
(4 3 2) 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....
(Order 48)
(4 4 2) - *442 symmetry
Square tiling

In geometry, the Square tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of .John Horton Conway calls it a quadrille....
- 45-45-90 triangle (Includes square domain (2 2 2 2))
(5 3 2) 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....
(Order 120)
(5 4 2) - *542 symmetry
Order-4 pentagonal tiling

In geometry, the order-4 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schl?fli symbol of .The image shows a Poincar? disk model projection of the hyperbolic plane....
(Hyperbolic plane)
(5 5 2) - *552 symmetry (Hyperbolic plane)
(3 3 3) - *333 symmetry
Triangular tiling

In geometry, the triangular tiling is one of the three regular tessellations of the Euclidean plane. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees....
- 60-60-60 triangle
(6 3 2) - *632 symmetry
Hexagonal tiling

In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of or t .John Horton Conway calls it a hextille....
- 30-60-90 triangle
(7 3 2) - *732 symmetry
Order-3 heptagonal tiling

In geometry, the order-3 heptagonal tiling is a regular tiling of the hyperbolic plane. It has Schl?fli symbol of .The image shows a Poincar? disk model projection of the hyperbolic plane....
(Hyperbolic plane)
(8 3 2) - *832 symmetry (Hyperbolic plane)

Dihedral spherical Spherical
D_2h D_3h T_d O_h I_h
*222 *322 *332 *432 *532

(2 2 2)

(3 2 2)
( 3 3 2)
(4 3 2)
(5 3 2)

The above symmetry groups only includes the integer solutions on the sphere. The list of Schwarz triangle
Schwarz triangle

In mathematics, a Schwarz triangle is a spherical triangle that can be used to tessellation a sphere. Each Schwarz triangle defines a finite group — its triangle group....
s includes rational numbers, and determine the full set of solutions of uniform polyhedron
Uniform polyhedron

A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
s.

Euclidean plane
p4m p3m p6m
*442 *333 *632

(4 4 2)

(3 3 3)

(6 3 2)

Hyperbolic plane
*732 *542 *433

(7 3 2)
(5 4 2)
(4 3 3)

In the tilings above, each triangle is a fundamental domain, colored by even and odd reflections.

Summary spherical and plane tilings

Selected tilings created by the Wythoff construction are given below.

Spherical tilings (r=2)

(p q 2) Fund.
triangles Parent Truncated Rectified Bitruncated Birectified
(dual) Cantellated Omnitruncated
(Cantitruncated) Snub
Wythoff symbol
Wythoff construction

File:Wythoffian_construction_diagram.pngIn geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling....
q | p 2 2 q | p 2 | p q 2 p | q p | q 2 p q | 2 p q 2 | | p q 2
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 t_0,1 t₁ t_1,2 t₂ t_0,2 t_0,1,2 s
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....
p^q (q.2p.2p) (p.q.p.q) (p.2q.2q) q^p (p.4.q.4) (4.2p.2q) (3.3.p.3.q)
Tetrahedral
(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....

(3.3a.3.3a)
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....

(3a.4.3b.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....

(4.6a.6b)
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....

(3.3.3a.3.3b)
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
(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....

(3.4.4a.4)
(4.6.8)
(3.3.3a.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
(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....

(3.3.3a.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....

Dihedral symmetry (q=r=2)

Spherical tilings with dihedral symmetry exist for all p=2,3,4,... many with digon
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....
faces which become degenerate polyhedra. Two of the eight forms (Rectified and cantellated) are replications and are skipped in the table.

(p 2 2) Fund.
triangles Parent Truncated Bitruncated
(truncated dual) Birectified
(dual) 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 t_0,1 t_1,2 t₂ t_0,1,2 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 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.p) 2^p (4.2p.4) (3.3.p.3)
(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
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....
4.4.2
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....

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 ....

4.4.4
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....

3.3.3.2
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 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
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....

4.4.3
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....

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 ....

4.4.6
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...

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....
(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
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....

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 ....

4.4.8

3.3.3.4
(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
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....

4.4.5
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....

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 ....

4.4.10
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....

3.3.3.5
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....
(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
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....

4.4.6
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...

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 ....

4.4.12
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....

3.3.3.6
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....
...

Planar tilings (r=2)

One representative hyperbolic tiling is given, and shown as a Poincaré disk

Poincaré disk model

In geometry, the Poincar? disk model, also called the conformal disk model, is a model of n-dimensional hyperbolic geometry in which the points of the geometry are in an n-dimensional disk, or ball , and the straight lines of the hyperbolic geometry are segments of circles contained in the disk orthogonal to the boundary of the disk...
projection.

(p q 2) Fund.
triangles Parent Truncated Rectified Bitruncated Birectified
(dual) Cantellated Omnitruncated
(Cantitruncated) Snub
Wythoff symbol
Wythoff construction

File:Wythoffian_construction_diagram.pngIn geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling....
q | p 2 2 q | p 2 | p q 2 p | q p | q 2 p q | 2 p q 2 | | p q 2
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 t_0,1 t₁ t_1,2 t₂ t_0,2 t_0,1,2 s
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....
p^q (q.2p.2p) (p.q.p.q) (p.2q.2q) q^p (p.4.q.4) (4.2p.2q) (3.3.p.3.q)
Square tiling
Square tiling

In geometry, the Square tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of .John Horton Conway calls it a quadrille....

(4 4 2)
V4.8.8
Square tiling

In geometry, the Square tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of .John Horton Conway calls it a quadrille....

4.8.8
Truncated square tiling

In geometry, the truncated square tiling is a semiregular tiling of the Euclidean plane. There is one square and two octagons on each vertex . This is the only edge-to-edge tiling by regular convex polygons which contains an octagon....

4.4a.4.4a
Square tiling

In geometry, the Square tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of .John Horton Conway calls it a quadrille....

4.8.8
Truncated square tiling

In geometry, the truncated square tiling is a semiregular tiling of the Euclidean plane. There is one square and two octagons on each vertex . This is the only edge-to-edge tiling by regular convex polygons which contains an octagon....

Square tiling

In geometry, the Square tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of .John Horton Conway calls it a quadrille....

4.4a.4b.4a
Square tiling

In geometry, the Square tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of .John Horton Conway calls it a quadrille....

4.8.8
Truncated square tiling

In geometry, the truncated square tiling is a semiregular tiling of the Euclidean plane. There is one square and two octagons on each vertex . This is the only edge-to-edge tiling by regular convex polygons which contains an octagon....

3.3.4a.3.4b
Snub square tiling

In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex . It has Schl?fli symbol of s....
(Hyperbolic plane)
(5 4 2)

4.10.10
4.5.4.5
5.8.8

4.4.5.4
4.8.10
3.3.4.3.5
(Hyperbolic plane)
(5 5 2)

5.10.10
5.5.5.5
5.10.10

4.4.5.4
4.10.10
3.3.5.3.5
Hexagonal tiling
Hexagonal tiling

In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of or t .John Horton Conway calls it a hextille....

(6 3 2)

V4.6.12
Hexagonal tiling

In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of or t .John Horton Conway calls it a hextille....

3.12.12
Truncated hexagonal tiling

In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons and one triangle on each vertex ....

3.6.3.6
Trihexagonal tiling

In geometry, the trihexagonal tiling is a semiregular tiling of the Euclidean plane. There are two triangles and two hexagons alternating on each vertex ....

6.6.6
Hexagonal tiling

In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of or t .John Horton Conway calls it a hextille....

Triangular tiling

In geometry, the triangular tiling is one of the three regular tessellations of the Euclidean plane. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees....

3.4.6.4
Small rhombitrihexagonal tiling

In geometry, the small rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two Square s, and one hexagon on each vertex ....

4.6.12
Great rhombitrihexagonal tiling

In geometry, the Great rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one square, one hexagon, and one dodecagon on each vertex ....

3.3.3.3.6
Snub hexagonal tiling

In geometry, the Snub hexagonal tiling is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex ....
(Hyperbolic plane)
(7 3 2)
Order-3 heptagonal tiling

In geometry, the order-3 heptagonal tiling is a regular tiling of the hyperbolic plane. It has Schl?fli symbol of .The image shows a Poincar? disk model projection of the hyperbolic plane....

3.14.14
3.7.3.7
Triheptagonal tiling

In geometry, the triheptagonal tiling is a semiregular tiling of the hyperbolic plane. There are two triangles and two heptagons alternating on each vertex ....

7.6.6
Order-7 triangular tiling

In geometry, the order-7 triangular tiling is a regular tiling of the hyperbolic plane with a Schl?fli symbol of .The image shows a Poincar? disk model projection of the hyperbolic plane....

3.4.7.4
4.6.14
Great rhombitriheptagonal tiling

In geometry, the great rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one square, one hexagon, and one tetrakaidecagon on each vertex ....

3.3.3.3.7
(Hyperbolic plane)
(8 3 2)

3.16.16
3.8.3.8
8.6.6

3.4.8.4
4.6.16
3.3.3.3.8

Planar tilings (r>2)

The 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....
is given in a linear form, although it is actually a triangle, with the trailing segment r connecting to the first node.

Wythoff symbol
Wythoff construction

File:Wythoffian_construction_diagram.pngIn geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling....

(p q r) Fund.
triangles q | p r r q | p r | p q r p | q p | q r p q | r p q r | | p q r
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....
(p.q)^r (r.2p.q.2p) (p.r)^q (q.2r.p.2r) (q.r)^p (q.2r.p.2r) (r.2q.p.2q) (3.r.3.q.3.p)
Triangular
(3 3 3)

(3.3)³
Triangular tiling

In geometry, the triangular tiling is one of the three regular tessellations of the Euclidean plane. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees....

3.6.3.6
Trihexagonal tiling

In geometry, the trihexagonal tiling is a semiregular tiling of the Euclidean plane. There are two triangles and two hexagons alternating on each vertex ....

(3.3)³
Triangular tiling

In geometry, the triangular tiling is one of the three regular tessellations of the Euclidean plane. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees....

3.6.3.6
Trihexagonal tiling

In geometry, the trihexagonal tiling is a semiregular tiling of the Euclidean plane. There are two triangles and two hexagons alternating on each vertex ....

(3.3)³
Triangular tiling

In geometry, the triangular tiling is one of the three regular tessellations of the Euclidean plane. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees....

3.6.3.6
Trihexagonal tiling

In geometry, the trihexagonal tiling is a semiregular tiling of the Euclidean plane. There are two triangles and two hexagons alternating on each vertex ....

6.6.6
Hexagonal tiling

In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of or t .John Horton Conway calls it a hextille....

3.3.3.3.3.3
Triangular tiling

In geometry, the triangular tiling is one of the three regular tessellations of the Euclidean plane. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees....
Hyperbolic
(4 3 3)
(3.4)³
3.8.3.8
(3.4)³
3.6.4.6
(3.3)⁴
3.6.4.6
6.6.8
3.3.3.3.3.4
Hyperbolic
(4 4 3)
(3.4)⁴
3.8.4.8
(3.4)⁴
3.6.4.6
(3.4)⁴
4.6.4.6
6.8.8
3.3.3.4.3.4
Hyperbolic
(4 4 4)
(4.4)⁴
4.8.4.8
(4.4)⁴
4.8.4.8
(4.4)⁴
4.8.4.8
8.8.8
3.4.3.4.3.4

Overlapping spherical tilings (r=2)

Tilings are shown as polyhedra

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....
. Some of the forms are degenerate, given with brackets for vertex figure

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off....
s, with overlapping edges or verices.

(p q 2)	Parent	Truncated	Rectified	Bitruncated	Birectified (dual)	Cantellated	Omnitruncated (Cantitruncated)	Snub
Wythoff symbol Wythoff construction File:Wythoffian_construction_diagram.pngIn geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling....	q \| p 2	2 q \| p	2 \| p q	2 p \| q	p \| q 2	p q \| 2	p q 2 \|	\| p q 2
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	t_0,1	t₁	t_1,2	t₂	t_0,2	t_0,1,2	s
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....	p^q	(q.2p.2p)	(p.q.p.q)	(p.2q.2q)	q^p	(p.4.q.4)	(4.2p.2q)	(3.3.p.3.q)
Icosahedral (5/2 3 2)	Great icosahedron In geometry, the great icosahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 20 intersecting triangular faces, with five triangles meeting at each vertex in a pentagrammic sequence....	(5/2.6.6) Truncated great icosahedron In geometry, the truncated great icosahedron is a nonconvex uniform polyhedron, indexed as U55.This polyhedron is the Truncation of the great icosahedron....	(3.5/2)² Great icosidodecahedron In geometry, the great icosidodecahedron is a nonconvex uniform polyhedron, indexed as U54.This polyhedron can be considered a Rectification great icosahedron....	[3.10/2.10/2]	Great stellated dodecahedron In geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra.It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at each vertex....	[3.4.5/2.4]	[4.10/2.6]	(3.3.3.3.5/2) Great snub icosidodecahedron In geometry, the great snub icosidodecahedron is a nonconvex uniform polyhedron, indexed as U57.This polyhedron can be considered a snub great icosahedron....
Icosahedral (5 5/2 2)	Great dodecahedron In geometry, the great dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces , with five pentagons meeting at each vertex, intersecting each other making a pentagrammic path....	(5/2.10.10) Truncated great dodecahedron In geometry, the truncated great dodecahedron is a nonconvex uniform polyhedron, indexed as U37.This polyhedron is the Truncation of the great dodecahedron....	(5/2.5)² Dodecadodecahedron In geometry, the dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U36.This polyhedron can be considered a Rectification great dodecahedron....	[5.10/2.10/2 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.... ]	Small stellated dodecahedron In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex....	(5/2.4.5.4) Rhombidodecadodecahedron In geometry, the rhombidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U38.By the Wythoff construction this polyhedron can also be named a Cantellation great dodecahedron....	[4.10/2.10]	(3.3.5/2.3.5) Snub dodecadodecahedron In geometry, the snub dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U40.This polyhedron can be considered a snub great dodecahedron....

External links

Free educational software for Windows by Jeffrey Weeks
Jeffrey Weeks (mathematician)

Jeffrey Renwick Weeks is an United States mathematician. He became a MacArthur Foundation in 1999. He received his A.B. from Dartmouth College in 1978, and his Ph.D....
that generated many of the images on the page.

Wythoff symbol

Summary table

Description

Symmetry triangles

Summary spherical and plane tilings

Spherical tilings (r=2)

Dihedral symmetry (q=r=2)

Planar tilings (r=2)

Planar tilings (r>2)

Overlapping spherical tilings (r=2)

See also

External links