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, ..., 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}.

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, ..., p_n − 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, ..., p_n − 1} then its dual has Schläfli symbol {p_n − 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, +.

In 2D, a rhombus, is represented as { } + { }. Its Coxeter diagram is .
In 3D, a p-gonal bipyramid, is represented as { } + {p}. Its Coxeter diagram is .
In 4D, a {p,q}-hedral bipyramids as { } + {p,q}. Its Coxeter diagram is .
In 4D, a p-q duopyramid as {p} + {q}. Its Coxeter diagram is .

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:

A digonal disphenoid can be represented as { } ∨ { } = [( ) ∨ ( )] ∨ [( ) ∨ ( )].
A p-gonal pyramids are represented as ( ) ∨ {p}.

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
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
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}	t_0,1{p,q}			Truncated cube
Bitruncation (Truncated dual)	$t\begin{Bmatrix} q , p \end{Bmatrix}$	2t{p,q}	t_1,2{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}	t_0,2{p,q}			Rhombicuboctahedron
Cantitruncated (Truncated rectified)	$t\begin{Bmatrix} p \\ q \end{Bmatrix}$	tr{p,q}	t_0,1,2{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}	ht_0,1{p,2q}	[p⁺,2q]
Snub dual regular	$s \begin{Bmatrix} q , 2p \end{Bmatrix}$	s{q,2p}	ht_1,2{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}	ht_0,2{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}	ht_0,1,2{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}	at_0,1{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
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}	t_0,1{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}	t_1,2{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}	t_2,3{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}	t_0,2{p,q,r}	Cantellated tesseract	=
Cantitruncated	$t\left\{\begin{array}{l}p\\q,r\end{array}\right\}$	tr{p,q,r}	t_0,1,2{p,q,r}	Cantitruncated tesseract	=
Runcinated (Expanded)	$e_3\begin{Bmatrix} p, q , r \end{Bmatrix}$	e₃{p,q,r}	t_0,3{p,q,r}	Runcinated tesseract
Runcitruncated			t_0,1,3{p,q,r}	Runcitruncated tesseract
Omnitruncated			t_0,1,2,3{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}	ht_0,1{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}	ht_0,1,2{p,q,r}	Snub 24-cell	=
Alternated duoprism		s{p}s{q}	ht_0,1,2,3{p,2,q}	Great duoantiprism

Bifurcating families
Form	Extended Schläfli symbol
Quasiregular	$\left\{p,{q\atop q}\right\}$	{p,q^1,1}	t₀{p,q^1,1}	16-cell
Truncated	$t\left\{p,{q\atop q}\right\}$	t{p,q^1,1}	t_0,1{p,q^1,1}	Truncated 16-cell
Rectified	$\left\{\begin{array}{l}p\\q\\q\end{array}\right\}$	r{p,q^1,1}	t₁{p,q^1,1}	24-cell
Cantellated	$r\left\{\begin{array}{l}p\\q\\q\end{array}\right\}$	rr{p,q^1,1}	t_0,2,3{p,q^1,1}	Cantellated 16-cell
Cantitruncated	$t\left\{\begin{array}{l}p\\q\\q\end{array}\right\}$	tr{p,q^1,1}	t_0,1,2,3{p,q^1,1}	Cantitruncated 16-cell
Snub rectified	$s\left\{\begin{array}{l}p\\q\\q\end{array}\right\}$	sr{p,q^1,1}	ht_0,1,2,3{p,q^1,1}	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}	t_0,1{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}	t_0,2,3{r,/q\,p}
Cantitruncated	$t\left\{\begin{array}{l}r\\p\\q\end{array}\right\}$	tr{r,/q\,p}	t_0,1,2,3{r,/q\,p}
Snub rectified	$s\left\{\begin{array}{l}p\\q\\r\end{array}\right\}$	sr{p,/q,\r}	ht_0,1,2,3{p,/q\,r}

References

Coxeter, H.S.M.; Regular Polytopes, (Methuen and Co., 1948). (pp. 14, 69, 149)
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]