# Schläfli symbol

The dodecahedron is a regular polyhedron with Schläfli symbol {5,3}, having 3 pentagons around each vertex.

In geometry, the Schläfli symbol is a notation of the form {p,q,r,...} that defines regular polytopes and tessellations.

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

## Description

The Schläfli symbol is a recursive description, starting with a p-sided regular polygon as {p}. For example, {3} is an equilateral triangle, {4} is a square and so on.

A regular polyhedron which has q regular p-sided polygon faces around each vertex is represented by {p,q}. For example, the cube has 3 squares around each vertex and is represented by {4,3}.

A regular 4-dimensional polytope, with r {p,q} regular polyhedral cells around each edge is represented by {p,q,r}, and so on.

Regular polytopes can have star polygon elements, like the pentagram, with symbol {5/2}, represented by the vertices of a pentagon but connected alternately.

A facet of a regular polytope {p,q,r,...,y,z} is {p,q,r,...,y}.

A regular polytope has a regular vertex figure. The vertex figure of a regular polytope {p,q,r,...y,z} is {q,r,...y,z}.

The Schläfli symbol can represent a finite convex polyhedron, an infinite tessellation of Euclidean space, or an infinite tessellation of hyperbolic space, depending on the angle defect of the construction. A positive angle defect allows the vertex figure to fold into a higher dimension and loops back into itself as a polytope. A zero angle defect will tessellate space of the same dimension as the facets. A negative angle defect can't exist in ordinary space, but can be constructed in hyperbolic space.

Usually, a facet or a vertex figure is assumed to be a finite polytope, but can sometimes be considered a tessellation itself.

A regular polytope also has a dual polytope, represented by the Schläfli symbol elements in reverse order. A self-dual regular polytope will have a symmetric Schläfli symbol.

## Symmetry groups

A Schläfli symbol is closely related to reflection symmetry groups, also called Coxeter groups, given with the same indices, but square brackets instead [p,q,r,...]. Such groups are often named by the regular polytopes they generate. For example [3,3] is the Coxeter group for reflective tetrahedral symmetry, and [3,4] is reflective octahedral symmetry, and [3,5] is reflective icosahedral symmetry.

## Regular polygons (plane)

Regular convex and star polygons with 3 to 12 vertices labelled with their Schläfli symbols

The Schläfli symbol of a regular polygon with p edges is {p}.

For example, a regular pentagon is represented by {5}.

See the convex regular polygon and nonconvex star polygon.

For example, {5/2} is the pentagram.

## Regular polyhedra (3-space)

The Schläfli symbol of a regular 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).

For example {5,3} is the regular dodecahedron. It has pentagonal (5 edges) faces, and 3 pentagons around each vertex.

See the 5 convex Platonic solids, the 4 nonconvex Kepler-Poinsot polyhedra.

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

For example, the hexagonal tiling is represented by {6,3}.

## Regular polychora (4-space)

The Schläfli symbol of a regular 4-polytope is of the form {p,q,r}. Its (two-dimensional) faces are regular p-gons ({p}), the cells are regular polyhedra of type {p,q}, the vertex figures are regular polyhedra of type {q,r}, and the edge figures are regular r-gons (type {r}).

See the six convex regular and 10 nonconvex polychora.

For example, the 120-cell is represented by {5,3,3}. It is made of dodecahedron cells {5,3}, and has 3 cells around each edge.

There is also one regular tessellation of Euclidean 3-space: the cubic honeycomb, with a Schläfli symbol of {4,3,4}, made of cubic cells, and 4 cubes around each edge.

There are also 4 regular compact hyperbolic tessellations including {5,3,4}, the Hyperbolic small dodecahedral honeycomb, which fills space with dodecahedron cells.

## Higher dimensions

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

Notice that a vertex figure of a facet of a polytope and a facet of a vertex figure of the same polytope are the same: {p2,p3, ..., pn − 2}.

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 hypercube, {4,3,3,...,3}. There are no non-convex regular polytopes above 4 dimensions.

## Dual polytopes

If a polytope of dimension ≥ 2 has Schläfli symbol {p1,p2, ..., pn − 1} then its dual has Schläfli symbol {pn − 1, ..., p2,p1}.

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

## Prismatic polytopes

Uniform prismatic polytopes can be defined and named as a Cartesian product of lower-dimensional regular polytopes:

• In 2D, a rectangle is represented as { } × { }. The symbol { } means a digon or line segment. Its Coxeter diagram is .
• In 3D, a p-gonal prism is represented as { } × {p}. Its Coxeter diagram is .
• In 4D, a uniform {p,q}-hedral prism as { } × {p,q}. Its Coxeter diagram is .
• In 4D, a uniform p-q duoprism as {p} × {q}. Its Coxeter diagram is .

The prismatic duals, or bipyramids can also be represented as composite symbols, but with the addition operator, +.

Pyramids containing vertices on two parallel hyperplanes. These can be represented in a join operator, ∨. A point is represented ( ). Every pair of vertices between joined figures are connected by edges.

In 2D, a isosceles triangle can be represented as ( ) ∨ { } = ( ) ∨ [( ) ∨ ( )].

In 3D:

In 4D:

• A p-q-hedral pyramids as ( ) ∨ {p,q}.
• A 5-cell as ( ) ∨ [( ) ∨ {3}] or [( ) ∨ ( )] ∨ {3} = { } ∨ {3}.
• A square pyramidal pyramid as ( ) ∨ [( ) ∨ {4}] or [( ) ∨ ( )] ∨ {4} = { } ∨ {4}.

When mixing operators, the order of operations from highest to lowest is: ×, +, and ∨.

## Extension of Schläfli symbols

### Polygons and circle tilings

A truncated regular polygon doubles in sides. A regular polygon with even sides can be halfed. An altered even-sided regular 2n-gon generates a star figure compound, 2{n}.

Form Schläfli symbol Symmetry Coxeter diagram Example, {6}
Regular {p} [p] Hexagon
Truncated t{p} = {2p} [[p]] = [2p] = Truncated hexagon
(Dodecagon)
=
Altered a{2p} [2p] Altered hexagon
(Hexagram)
Half h{2p} = {p} [1+,2p] = [p] = Half hexagon
(Triangle)
=

### Polyhedra and tilings

Coxeter expanded his usage of the Schläfli symbol to quasiregular polyhedra by adding a vertical dimension to the symbol. It was a starting point toward the more general Coxeter diagram. Norman Johnson simplified the notation for vertical symbols with an r prefix. The t-notation is the most general, and directly corresponds to the rings of the Coxeter diagram. Symbols have a corresponding alternation, replacing rings with holes in a Coxeter diagram and h prefix standing for half, construction limited by the requirement that neighboring branches must be even-ordered and cuts the symmetry order in half. A related operator, a for altered, is shown with two nested holes, represents a compound polyhedra with both alternated halves, retaining the original full symmetry. A snub is a half form of a truncation, and a holosnub is both halves of an alternated truncation.

Form Schläfli symbols Symmetry Coxeter diagram Example, {4,3}
Regular $\begin{Bmatrix} p , q \end{Bmatrix}$ {p,q} t0{p,q} [p,q]
or
[(p,q,2)]
Cube
Truncated $t\begin{Bmatrix} p , q \end{Bmatrix}$ t{p,q} t0,1{p,q} Truncated cube
Bitruncation
(Truncated dual)
$t\begin{Bmatrix} q , p \end{Bmatrix}$ 2t{p,q} t1,2{p,q} Truncated octahedron
Rectified
(Quasiregular)
$\begin{Bmatrix} p \\ q \end{Bmatrix}$ r{p,q} t1{p,q} Cuboctahedron
Birectification
(Regular dual)
$\begin{Bmatrix} q , p \end{Bmatrix}$ 2r{p,q} t2{p,q} Octahedron
Cantellated
(Rectified rectified)
$r\begin{Bmatrix} p \\ q \end{Bmatrix}$ rr{p,q} t0,2{p,q} Rhombicuboctahedron
Cantitruncated
(Truncated rectified)
$t\begin{Bmatrix} p \\ q \end{Bmatrix}$ tr{p,q} t0,1,2{p,q} Truncated cuboctahedron
Alternations
Alternated (half) regular $h \begin{Bmatrix} 2p , q \end{Bmatrix}$ h{2p,q} ht0{2p,q} [1+,2p,q] or Demicube
(Tetrahedron)
Snub regular $s\begin{Bmatrix} p , 2q \end{Bmatrix}$ s{p,2q} ht0,1{p,2q} [p+,2q]
Snub dual regular $s \begin{Bmatrix} q , 2p \end{Bmatrix}$ s{q,2p} ht1,2{2p,q} [2p,q+] Snub octahedron
(Icosahedron)
Alternated dual regular $h \begin{Bmatrix} 2q , p \end{Bmatrix}$ h{2q,p} ht2{p,2q} [p,2q,1+]
Alternated rectified
(p and q are even)
$h \begin{Bmatrix} p \\ q \end{Bmatrix}$ hr{p,q} ht1{p,q} [p,1+,q]
Alternated rectified rectified
(p and q are even)
$hr \begin{Bmatrix} p \\ q \end{Bmatrix}$ hrr{p,q} ht0,2{p,q} [(p,q,2+)]
Quartered
(p and q are even)
$q\begin{Bmatrix} p \\ q \end{Bmatrix}$ q{p,q} ht0ht2{p,q} [1+,p,q,1+]
Snub rectified
Snub quasiregular
$s\begin{Bmatrix} p \\ q \end{Bmatrix}$ sr{p,q} ht0,1,2{p,q} [p,q]+ Snub cuboctahedron
(Snub cube)
Altered and holosnubbed
Altered regular $a \begin{Bmatrix} p , q \end{Bmatrix}$ a{p,q} at0{p,q} [p,q] Stellated octahedron
Holosnub dual regular ß$\begin{Bmatrix} q , p \end{Bmatrix}$ ß{q,p} at0,1{q,p} [p,q] Compound of two icosahedra
ß, looking similar to the greek letter beta (β), is the German alphabet letter eszett.

### Polychora and honeycombs

Linear families
Form Schläfli symbol Coxeter diagram Example, {4,3,3}
Regular $\begin{Bmatrix} p, q , r \end{Bmatrix}$ {p,q,r} t0{p,q,r} Tesseract
Truncated $t\begin{Bmatrix} p, q , r \end{Bmatrix}$ t{p,q,r} t0,1{p,q,r} Truncated tesseract
Rectified $\left\{\begin{array}{l}p\\q,r\end{array}\right\}$ r{p,q,r} t1{p,q,r} Rectified tesseract =
Bitruncated 2t{p,q,r} t1,2{p,q,r} Bitruncated tesseract
Birectified
(Rectified dual)
$\left\{\begin{array}{l}q,p\\r\end{array}\right\}$ 2r{p,q,r} = r{r,q,p} t2{p,q,r} Rectified 16-cell =
Tritruncated
(Truncated dual)
$t\begin{Bmatrix} r, q , p \end{Bmatrix}$ 3t{p,q,r} = t{r,q,p} t2,3{p,q,r} Bitruncated tesseract
Trirectified
(Dual)
$\begin{Bmatrix} r, q , p \end{Bmatrix}$ 3r{p,q,r} = {r,q,p} t3{p,q,r} = {r,q,p} 16-cell
Cantellated $r\left\{\begin{array}{l}p\\q,r\end{array}\right\}$ rr{p,q,r} t0,2{p,q,r} Cantellated tesseract =
Cantitruncated $t\left\{\begin{array}{l}p\\q,r\end{array}\right\}$ tr{p,q,r} t0,1,2{p,q,r} Cantitruncated tesseract =
Runcinated
(Expanded)
$e_3\begin{Bmatrix} p, q , r \end{Bmatrix}$ e3{p,q,r} t0,3{p,q,r} Runcinated tesseract
Runcitruncated t0,1,3{p,q,r} Runcitruncated tesseract
Omnitruncated t0,1,2,3{p,q,r} Omnitruncated tesseract
Alternations
Half
p even
$h\begin{Bmatrix} p, q , r \end{Bmatrix}$ h{p,q,r} ht0{p,q,r} 16-cell
Quarter
p and r even
$q\begin{Bmatrix} p, q , r \end{Bmatrix}$ q{p,q,r} ht0ht3{p,q,r}
Snub
q even
$s\begin{Bmatrix} p, q , r \end{Bmatrix}$ s{p,q,r} ht0,1{p,q,r} Snub 24-cell
Snub rectified
r even
$s\left\{\begin{array}{l}p\\q,r\end{array}\right\}$ sr{p,q,r} ht0,1,2{p,q,r} Snub 24-cell =
Alternated duoprism s{p}s{q} ht0,1,2,3{p,2,q} Great duoantiprism
Bifurcating families
Form Extended Schläfli symbol Coxeter diagram Examples
Quasiregular $\left\{p,{q\atop q}\right\}$ {p,q1,1} t0{p,q1,1} 16-cell
Truncated $t\left\{p,{q\atop q}\right\}$ t{p,q1,1} t0,1{p,q1,1} Truncated 16-cell
Rectified $\left\{\begin{array}{l}p\\q\\q\end{array}\right\}$ r{p,q1,1} t1{p,q1,1} 24-cell
Cantellated $r\left\{\begin{array}{l}p\\q\\q\end{array}\right\}$ rr{p,q1,1} t0,2,3{p,q1,1} Cantellated 16-cell
Cantitruncated $t\left\{\begin{array}{l}p\\q\\q\end{array}\right\}$ tr{p,q1,1} t0,1,2,3{p,q1,1} Cantitruncated 16-cell
Snub rectified $s\left\{\begin{array}{l}p\\q\\q\end{array}\right\}$ sr{p,q1,1} ht0,1,2,3{p,q1,1} Snub 24-cell
Quasiregular $\left\{r,{p\atop q}\right\}$ {r,/q\,p} t0{r,/q\,p}
Truncated $t\left\{r,{p\atop q}\right\}$ t{r,/q\,p} t0,1{r,/q\,p}
Rectified $\left\{\begin{array}{l}r\\p\\q\end{array}\right\}$ r{r,/q\,p} t1{r,/q\,p}
Cantellated $r\left\{\begin{array}{l}r\\p\\q\end{array}\right\}$ rr{r,/q\,p} t0,2,3{r,/q\,p}
Cantitruncated $t\left\{\begin{array}{l}r\\p\\q\end{array}\right\}$ tr{r,/q\,p} t0,1,2,3{r,/q\,p}
Snub rectified $s\left\{\begin{array}{l}p\\q\\r\end{array}\right\}$ sr{p,/q,\r} ht0,1,2,3{p,/q\,r}

## References

• Coxeter, H.S.M.; Regular Polytopes, (Methuen and Co., 1948). (pp. 14, 69, 149) [1]
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]