In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope.

The Schläfli symbol is named after the 19th-century mathematician Ludwig Schläfli who made important contributions in geometry and other areas.

See also list of regular polytopes.

Regular polygons (plane)

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The Schläfli symbol of a polygon with n edges is {n}. For example {5} is a pentagon.

Star polygons

There are nonconvex polygons which are considered regular. They are called star polygons. A star polygon with symbol {^p/_q} has p vertices where every qth vertex is connected. Thus, {⁵/₂} is a pentagram.

Star figures

If p and q have a common factor n, then {^p/_q} can be called a star figure which is a compound of n copies of {^p/n/_q/n} rotated relative to each other. For example {⁶/₂} is called a hexagram as a compound of two triangles {3/1} and {¹⁰/₄} as a compound of two pentagrams {⁵/₂}.

Regular polyhedra (3-space)

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The Schläfli symbol of a polyhedron is {p,q} if its faces are p-gons, and each vertex is surrounded by q faces (the vertex figure is a q-gon).

The Schläfli symbols of the Platonic solids are:

1. for the tetrahedron : {3,3}
2. for the cube : {4,3}
3. for the octahedron : {3,4}
4. for the dodecahedron : {5,3}
5. for the icosahedron : {3,5}

Schläfli symbols may also be defined for regular tessellations of Euclidean or hyperbolic space in a similar way.

In addition to the 5 convex regular polyhedra, there are four other nonconvex ones formed with star polygon faces or vertex figures.

The Schläfli symbols of the four Kepler-Poinsot solids are: (nonconvex regular polyhedra)

1. Small stellated dodecahedron : {⁵/₂,5}
2. Great dodecahedron : {5,⁵/₂}
3. Great stellated dodecahedron : {⁵/₂,3}
4. Great icosahedron ; {3,⁵/₂}

Three regular tilings of the Euclidean plane:

1. Hexagonal tiling : {6,3}
2. Square tiling : {4,4}
3. Triangular tiling : {3,6}

Regular polychora (4-space)

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The Schläfli symbol of a regular polychoron is of the form {p,q,r}. It has {p} regular polygonal faces, {p,q} cells, {q,r} regular polyhedral vertex figures, and {r} regular polygonal edge figures.

There are six convex regular and 10 nonconvex polychora. There is also one regular tesselation of Euclidean 3-space: the cubic honeycomb.

The smallest convex polychoron is {3,3,3}, the pentachoron, and the largest is {5,3,3}, the 120-cell.

All 10 nonconvex regular polychora are combinations of {3}, {5/2}, {5} sides.

Higher dimensions

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For higher dimensional polytopes, the Schläfli symbol is defined recursively as {p₁, p₂, ..., p_n − 1} if the facets have Schläfli symbol {p₁,p₂, ..., p_n − 2} and the vertex figures have Schläfli symbol {p₂,p₃, ..., p_n − 1}.

There are only 3 regular polytopes in 5 dimensions and above: the simplex, {3,3,3,...,3}; the cross-polytope, {3,3, ... ,3,4}; and the measure polytope, {4,3,3,...,3}. There are no non-convex regular polytopes above 4 dimensions.

Dual polytopes

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For dimension 2 or higher, every polytope has a dual.

If a polytope has Schläfli symbol {p₁,p₂, ..., p_n − 1} then its dual has Schläfli symbol {p_n − 1, ..., p₂,p₁}.

If the sequence is the same forwards and backwards, the polytope is self-dual. Every regular polytope in 2 dimensions (polygon) is self-dual.

Extended Schläfli symbols for polyhedra and tilings

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These extended Schläfli symbols were used in the 1954 paper by Coxeter enumerating the paper tiled uniform polyhedra.

Every regular polyhedron or tiling {p,q} has these five operations that create semiregular polyhedra. The short-hand notation is equivalent to the longer name. For instance, t{3,3} simply means truncated tetrahedron.

The vertical notation is used for dual-symmetric operations - those that generate the same polyhedron from {p,q} as {q,p}.

The second t indexed notation is a more general notation in parallel to the extended polychoron notation below.

Operation	Extended Notation-1 Coxeter	Extended Notation-2 Johnson	Wythoff symbol	Face (2)	Face (1)	Face (0)
Parent	$\backslash begin\{Bmatrix\}\; p\; ,\; q\; \backslash end\{Bmatrix\}$	t0{p,q}	q | 2 p	{p}	--	--
Truncated	$t\backslash begin\{Bmatrix\}\; p\; ,\; q\; \backslash end\{Bmatrix\}$	t0,1{p,q}	2 q | p	t{p}	--	{q}
Rectified	$\backslash begin\{Bmatrix\}\; p\; \backslash \backslash \; q\; \backslash end\{Bmatrix\}$	t1{p,q}	2 | p q	{p}	--	{q}
Truncated dual	$t\backslash begin\{Bmatrix\}\; q\; ,\; p\; \backslash end\{Bmatrix\}$	t2,3{p,q}	2 p | q	{p}	--	t{q}
Dual	$\backslash begin\{Bmatrix\}\; q\; ,\; p\; \backslash end\{Bmatrix\}$	t2{p,q}	p | 2 q	--	--	{q}
Runcinated	$r\backslash begin\{Bmatrix\}\; p\; \backslash \backslash \; q\; \backslash end\{Bmatrix\}$	t0,2{p,q}	p q | 2	{p}	{4}	{q}
Omnitruncated	$t\backslash begin\{Bmatrix\}\; p\; \backslash \backslash \; q\; \backslash end\{Bmatrix\}$	t0,1,2{p,q}	2 p q |	t{p}	{4}	t{q}
Snub	$s\backslash begin\{Bmatrix\}\; p\; \backslash \backslash \; q\; \backslash end\{Bmatrix\}$	s{p,q}	| 2 p q	{p}	{3} {3}	{q}

Prisms

A p-gonal prism, with vertex figure p.4.4, can represented by either $t\backslash begin\{Bmatrix\}\; 2,p\; \backslash end\{Bmatrix\}$ or $\backslash begin\{Bmatrix\}\; p\; \backslash end\{Bmatrix\}\; \backslash times\; \backslash begin\{Bmatrix\}\backslash \; \backslash end\{Bmatrix\}$.

Extended for uniform polychora and 3-space honeycombs

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Every regular polytope can be seen as the images of a fundamental region in a small number of mirrors. In a 4-dimensional polytope (or 3-dimensional cubic honeycomb) the fundamental region is bounded by four mirrors. A mirror in 4-space is a three-dimensional hyperplane, but it is more convenient for our purposes to consider only its two-dimensional intersection with the three-dimensional surface of the hypersphere; thus the mirrors form an irregular tetrahedron.

Each of the sixteen regular polychora is generated by one of four symmetry groups, as follows:

• group ^*: the 5-cell {3,3,3}, which is self-dual;
• group ^*: 16-cell {3,3,4} and its dual tesseract {4,3,3};
• group ^*: the 24-cell {3,4,3}, self-dual;
• group ^*: 600-cell {3,3,5}, its dual 120-cell {5,3,3}, and their ten regular stellations.

(The groups are named in Coxeter notation.)

A set of up to 13 (nonregular) uniform polychora can be generated from each regular polychoron and its dual. Eight of the Andreini tessellations (uniform honeycombs in Euclidean 3-space) are analogously generated from the cubic honeycomb {4,3,4}.

For a given symmetry simplex, a generating point may be placed on any of the four vertices, 6 edges, 4 faces, or the interior volume. On each of these 15 elements there is a point whose images, reflected in the four mirrors, are the vertices of a uniform polychoron.

The extended Schläfli symbols are made by a t followed by inclusion of one to four subscripts 0,1,2,3. If there's one subscript, the generating point is on a corner of the fundamental region, i.e. a point where three mirrors meet. These corners are notated as

• 0: vertex of the parent polychoron (center of the dual's cell)
• 1: center of the parent's edge (center of the dual's face)
• 2: center of the parent's face (center of the dual's edge)
• 3: center of the parent's cell (vertex of the dual)

(For the two self-dual polychora, "dual" means a similar polychoron in dual position.) Two or more subscripts mean that the generating point is between the corners indicated.

The following table defines all 15 forms. Each trunction form can have from one to four cell types, located in positions 0,1,2,3 as defined above. The cells are labeled by polyhedral truncation notation.

• An n-gonal prism is represented as : $t\backslash begin\{Bmatrix\}\; 2,n\; \backslash end\{Bmatrix\}$
• The green background is shown on forms that are equivalent from either the parent or dual.
• The red background shows truncations of the parent, and blue as truncations of the dual.

{| class="wikitable" Operation Johnson Notation Cell
(3) Cell
(2) Cell
(1) Cell
(0) Parent t0{p,q,r} $\backslash begin\{Bmatrix\}\; p\; ,\; q\; \backslash end\{Bmatrix\}$ -- -- -- Truncated t0,1{p,q,r} $t\backslash begin\{Bmatrix\}\; p\; ,\; q\; \backslash end\{Bmatrix\}$ -- -- $\backslash begin\{Bmatrix\}\; q\; ,\; r\; \backslash end\{Bmatrix\}$ Rectified t1{p,q,r} $\backslash begin\{Bmatrix\}\; p\; \backslash \backslash \; q\; \backslash end\{Bmatrix\}$ -- -- $\backslash begin\{Bmatrix\}\; q\; ,\; r\; \backslash end\{Bmatrix\}$ Cantellated t0,2{p,q,r} $r\backslash begin\{Bmatrix\}\; p\; \backslash \backslash \; q\; \backslash end\{Bmatrix\}$ -- $t\backslash begin\{Bmatrix\}\; 2,r\; \backslash end\{Bmatrix\}$ $\backslash begin\{Bmatrix\}\; q\; \backslash \backslash \; r\; \backslash end\{Bmatrix\}$ Cantitruncated t0,1,2{p,q,r} $t\backslash begin\{Bmatrix\}\; p\; \backslash \backslash \; q\; \backslash end\{Bmatrix\}$ -- $t\backslash begin\{Bmatrix\}\; 2,r\; \backslash end\{Bmatrix\}$ $t\backslash begin\{Bmatrix\}\; q\; ,\; r\; \backslash end\{Bmatrix\}$ Bitruncated t1,2{p,q,r} $t\backslash begin\{Bmatrix\}\; q\; ,\; p\; \backslash end\{Bmatrix\}$ -- -- $t\backslash begin\{Bmatrix\}\; q\; ,\; r\; \backslash end\{Bmatrix\}$ Cantitruncated dual t1,2,3{p,q,r} $t\backslash begin\{Bmatrix\}\; q\; ,\; p\; \backslash end\{Bmatrix\}$ $t\backslash begin\{Bmatrix\}\; 2,p\; \backslash end\{Bmatrix\}$ -- $t\backslash begin\{Bmatrix\}\; q\; \backslash \backslash \; r\; \backslash end\{Bmatrix\}$ Cantellated dual t1,3{p,q,r} $\backslash begin\{Bmatrix\}\; p\; \backslash \backslash \; q\; \backslash end\{Bmatrix\}$ $t\backslash begin\{Bmatrix\}\; 2,p\; \backslash end\{Bmatrix\}$ -- $r\backslash begin\{Bmatrix\}\; q\; \backslash \backslash \; r\; \backslash end\{Bmatrix\}$ Rectified dual t2{p,q,r} $\backslash begin\{Bmatrix\}\; q\; ,\; p\; \backslash end\{Bmatrix\}$ -- -- $\backslash begin\{Bmatrix\}\; q\; \backslash \backslash \; r\; \backslash end\{Bmatrix\}$ Truncated dual t2,3{p,q,r} $\backslash begin\{Bmatrix\}\; q\; ,\; p\; \backslash end\{Bmatrix\}$ -- -- $t\backslash begin\{Bmatrix\}\; r\; ,\; q\; \backslash end\{Bmatrix\}$ Dual t3{p,q,r} -- -- -- $\backslash begin\{Bmatrix\}\; r\; ,\; q\; \backslash end\{Bmatrix\}$ Runcinated t0,3{p,q,r} $\backslash begin\{Bmatrix\}\; p\; ,\; q\; \backslash end\{Bmatrix\}$ $t\backslash begin\{Bmatrix\}\; 2,p\; \backslash end\{Bmatrix\}$ $t\backslash begin\{Bmatrix\}\; 2,r\; \backslash end\{Bmatrix\}$ $\backslash begin\{Bmatrix\}\; r\; ,\; q\; \backslash end\{Bmatrix\}$ Runcitruncated t0,1,3{p,q,r} $t\backslash begin\{Bmatrix\}\; p\; ,\; q\; \backslash end\{Bmatrix\}$ $t\backslash begin\{Bmatrix\}\; 2,p\; \backslash end\{Bmatrix\}$ $t\backslash begin\{Bmatrix\}\; 2,r\; \backslash end\{Bmatrix\}$ $r\backslash begin\{Bmatrix\}\; q\; \backslash \backslash \; r\; \backslash end\{Bmatrix\}$ Runcitruncated dual t0,2,3{p,q,r} $r\backslash begin\{Bmatrix\}\; p\; \backslash \backslash \; q\; \backslash end\{Bmatrix\}$ $t\backslash begin\{Bmatrix\}\; 2,p\; \backslash end\{Bmatrix\}$ $t\backslash begin\{Bmatrix\}\; 2,r\; \backslash end\{Bmatrix\}$ $t\backslash begin\{Bmatrix\}\; r\; ,\; q\; \backslash end\{Bmatrix\}$ Omnitruncated t0,1,2,3{p,q,r} $t\backslash begin\{Bmatrix\}\; p\; \backslash \backslash \; q\; \backslash end\{Bmatrix\}$ $t\backslash begin\{Bmatrix\}\; 2,p\; \backslash end\{Bmatrix\}$ $t\backslash begin\{Bmatrix\}\; 2,r\; \backslash end\{Bmatrix\}$ $t\backslash begin\{Bmatrix\}\; q\; \backslash \backslash \; r\; \backslash end\{Bmatrix\}$

External links

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• Wythoff Symbol and generalized Schläfli Symbols
• polyhedral names et notations

Polytopes | Mathematical notation

Símbolo de Schläfli