Schläfli symbol

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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.

[edit] Regular polygons (plane)

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

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.

[edit] 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 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}.

[edit] Regular polychora (4-space)

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.

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 tesselation 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 hyperbolic tessellations including {5,3,4}, the Hyperbolic small dodecahedral honeycomb, which fills space with dodecahedron cells.

[edit] 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 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.

[edit] Dual polytopes

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.

[edit] Prismatic forms

Prismatic polytopes can be defined and named as a Cartesian product of lower dimensional polytopes:

A p-gonal prism, with vertex figure p.4.4 as $\begin{Bmatrix}\ \end{Bmatrix} \times \begin{Bmatrix} p \end{Bmatrix}$ .
A uniform {p,q}-hedral prism as $\begin{Bmatrix}\ \end{Bmatrix} \times \begin{Bmatrix} p,q \end{Bmatrix}$ .
A uniform p-q duoprism as $\begin{Bmatrix} p \end{Bmatrix} \times \begin{Bmatrix} q \end{Bmatrix}$ .

A prism can also be represented as the truncation of a hosohedron as $t\begin{Bmatrix} 2,p \end{Bmatrix}$ , and an antiprism (snub prism) as $s\begin{Bmatrix} 2 \\ p \end{Bmatrix}$ .

[edit] Extended Schläfli symbols for uniform polytopes

Uniform polytopes, made from a Wythoff construction, are represented by an extended truncation notation from a regular form {p,q,...}. There are a number of parallel symbolic forms that reference the elements of the Schläfli symbol, discussed by dimension below.

[edit] Uniform polyhedra and tilings

For polyhedra, one extended Schläfli symbol is used in the 1954 paper by Coxeter enumerating the paper tiled uniform polyhedra.

Every regular polyhedron or tiling {p,q} has 7 forms, including the regular form and its dual, corresponding to positions within the fundamental right triangle. An 8th special form, the snubs, correspond to an alternation of the omnitruncated form.

For instance, t{3,3} simply means truncated tetrahedron.

A second, more general notation, also used by Coxeter applies to all dimensions, and are specified by a t followed by a list of indices corresponding to Wythoff construction mirrors. (They also correspond to ringed nodes in a Coxeter-Dynkin diagram.)

For example, the truncated hexahedron can be represented by t_0,1{4,3} and it can be seen as midway between the cube, t₀{4,3}, and the cuboctahedron, t₁{4,3}.

In each a Wythoff construction operational name is given first. Second some have alternate terminology (given in parentheses) apply only for a given dimension. Specifically omnitruncation and expansion, as well as dual relations apply differently in each dimension.

Operation	Extended Schläfli Symbols		Wythoff symbol
Parent	$\begin{Bmatrix} p , q \end{Bmatrix}$	t₀{p,q}	q \| 2 p
Rectified (Quasiregular)	$\begin{Bmatrix} p \\ q \end{Bmatrix}$	t₁{p,q}	2 \| p q
Birectified (or dual)	$\begin{Bmatrix} q , p \end{Bmatrix}$	t₂{p,q}	p \| 2 q
Truncated	$t\begin{Bmatrix} p , q \end{Bmatrix}$	t_0,1{p,q}	2 q \| p
Bitruncated (or truncated dual)	$t\begin{Bmatrix} q , p \end{Bmatrix}$	t_2,3{p,q}	2 p \| q
Cantellated (or expanded)	$r\begin{Bmatrix} p \\ q \end{Bmatrix}$	t_0,2{p,q}	p q \| 2
Cantitruncated (or omnitruncated)	$t\begin{Bmatrix} p \\ q \end{Bmatrix}$	t_0,1,2{p,q}	2 p q \|
Snub	$s\begin{Bmatrix} p \\ q \end{Bmatrix}$	s{p,q}	\| 2 p q

[edit] Uniform polychora and honeycombs

There are up to 15 different truncation forms for polychora and honeycombs based on each {p,q,r} regular form.

See uniform polychoron and convex uniform honeycomb.

The subscripted-t notation is parallel to the graphical Coxeter-Dynkin diagram, with each graph node representing the 4 hyperplanes of the reflection mirrors in the fundamental domain.

Operation	Extended Schläfli symbols
Parent	$\begin{Bmatrix} p , q , r \end{Bmatrix}$	t₀{p,q,r}
Rectified	$\begin{Bmatrix} p \\ q , r \end{Bmatrix}$	t₁{p,q,r}
Birectified (or rectified dual)	$\begin{Bmatrix} q , p \\ r \end{Bmatrix}$	t₂{p,q,r}
Trirectifed (or dual)	$\begin{Bmatrix} r, q , p \end{Bmatrix}$	t₃{p,q,r}

Truncated		t_0,1{p,q,r}
Bitruncated		t_1,2{p,q,r}
Tritruncated (or truncated dual)		t_2,3{p,q,r}
Cantellated		t_0,2{p,q,r}
Bicantellated (or cantellated dual)		t_1,3{p,q,r}
Runcinated (or expanded)		t_0,3{p,q,r}
Cantitruncated		t_0,1,2{p,q,r}
Bicantitruncated (or cantitruncated dual)		t_1,2,3{p,q,r}
Runcitruncated		t_0,1,3{p,q,r}
Runcicantellated (or runcitruncated dual)		t_0,2,3{p,q,r}
Runcicantitruncated (or omnitruncated)		t_0,1,2,3{p,q,r}

[edit] References

The Beauty of Geometry: Twelve Essays (1999), Dover Publications ISBN 99-35678 (Chapter 3: Wythoff's construction for uniform polytopes, p41-53)
Johnson, N.W. Uniform Polytopes, Manuscript (1991)
Johnson, N.W. The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
Coxeter, H.S.M.; Regular Polytopes, (Methuen and Co., 1948). (pp. 14, 69, 149)
Coxeter, Longuet-Higgins, Miller, Uniform polyhedra, Phil. Trans. 1954, 246 A, 401-50. (Extended Schläfli notation defined: Table 1: p 403)