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Notation that defines regular polytopes and tessellations

In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more than three dimensions and discovered all their convex regular polytopes, including the six that occur in four dimensions.

The Schläfli symbol is a recursive description, starting with {p} for a p sided regular polygon that is convex. For example, {3} is an equilateral triangle, {4} is a square, {5} a convex regular pentagon, etc. Regular star polygons are not convex, and their Schläfli symbols {p/q} contain irreducible fractions p/q, where p is the number of vertices, and q is their turning number. Equivalently, {p/q} is created from the vertices of {p}, connected every q. For example, is a pentagram; is a pentagon. A regular polyhedron that 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}. For example, a tesseract, {4,3,3}, has 3 cubes, {4,3}, around an edge. In general, a regular polytope {p,q,r, ,y,z} has z {p,q,r, ,y} facets around every peak, where a peak is a vertex in a polyhedron, an edge in a 4 polytope, a face in a 5 polytope, and an (n 3) face in an n polytope.

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}. Regular polytopes can have star polygon elements, like the pentagram, with symbol , represented by the vertices of a pentagon but connected alternately. 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 tessellates space of the same dimension as the facets. A negative angle defect cannot 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 itself be considered a tessellation. 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. In addition to describing Euclidean polytopes, Schläfli symbols can be used to describe spherical polytopes or spherical honeycombs.

Schläfli's work was almost unknown in his lifetime, and his notation for describing polytopes was rediscovered independently by several others. In particular, Thorold Gosset rediscovered the Schläfli symbol which he wrote as |p|q|r| |z| rather than with brackets and commas as Schläfli did. Gosset's form has greater symmetry, so the number of dimensions is the number of vertical bars, and the symbol exactly includes the sub symbols for facet and vertex figure. Gosset regarded |p as an operator, which can be applied to |q| |z| to produce a polytope with p gonal faces whose vertex figure is |q| |z|.

Schläfli symbols are closely related to (finite) reflection symmetry groups, which correspond precisely to the finite Coxeter groups and are specified 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, [3,4] is reflective octahedral symmetry, and [3,5] is reflective icosahedral symmetry.

The Schläfli symbol of a convex regular polygon with p edges is {p}. For example, a regular pentagon is represented by {5}. For nonconvex star polygons, the constructive notation is used, where p is the number of vertices and q−1 is the number of vertices skipped when drawing each edge of the star. For example, represents the pentagram.

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. Topologically, a regular 2 dimensional tessellation may be regarded as similar to a (3 dimensional) polyhedron, but such that the angular defect is zero. Thus, Schläfli symbols may also be defined for regular tessellations of Euclidean or hyperbolic space in a similar way as for polyhedra. The analogy holds for higher dimensions. For example, the hexagonal tiling is represented by {6,3}.

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 regular star 4 polytopes. 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 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. If a 4 polytope's symbol is palindromic (e. g. {3,3,3} or {3,4,3}), its bitruncation will only have truncated forms of the vertex figure as cells.

For higher dimensional regular polytopes, the Schläfli symbol is defined recursively as if the facets have Schläfli symbol and the vertex figures have Schläfli symbol
A vertex figure of a facet of a polytope and a facet of a vertex figure of the same polytope are the same:
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.

If a polytope of dimension n≥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.