Symmetry groups 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.
Regular polygons (plane) 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 is the number of vertices skipped when drawing each edge of the star. For example, represents the
pentagram.
Regular polyhedra (3 dimensions) 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}.
Regular 4-polytopes (4 dimensions) 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.
Regular n-polytopes (higher dimensions) 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.
Dual polytopes 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.
Prismatic polytopes Uniform prismatic polytopes can be defined and named as a
Cartesian product (with operator "×") of lower-dimensional regular polytopes. • In 0D, a
point is represented by ( ). Its
Coxeter diagram is empty. Its
Coxeter notation symmetry is ][. • In 1D, a
line segment is represented by { }. Its
Coxeter diagram is . Its symmetry is [ ]. • In 2D, a
rectangle is represented as { } × { }. Its
Coxeter diagram is . Its symmetry is [2]. • In 3D, a
p-gonal
prism is represented as { } × {
p}. Its Coxeter diagram is . Its symmetry is [2,
p]. • In 4D, a uniform {
p,
q}-hedral prism is represented as { } × {
p,
q}. Its Coxeter diagram is . Its symmetry is [2,
p,
q]. • In 4D, a uniform
p-
q duoprism is represented as {
p} × {
q}. Its Coxeter diagram is . Its symmetry is [
p,2,
q]. The prismatic duals, or
bipyramids can be represented as composite symbols, but with the
addition operator, "+". • In 2D, a
rhombus is represented as { } + { }. Its Coxeter diagram is . Its symmetry is [2]. • In 3D, a
p-gonal bipyramid, is represented as { } + {
p}. Its Coxeter diagram is . Its symmetry is [2,
p]. • In 4D, a {
p,
q}-hedral bipyramid is represented as { } + {
p,
q}. Its Coxeter diagram is . Its symmetry is [
p,
q]. • In 4D, a
p-
q duopyramid is represented as {
p} + {
q}. Its Coxeter diagram is . Its symmetry is [
p,2,
q]. Pyramidal polytopes containing vertices orthogonally offset can be represented using a join operator, "∨". Every pair of vertices between joined figures are connected by edges. In 2D, an
isosceles triangle can be represented as ( ) ∨ { } = ( ) ∨ [( ) ∨ ( )]. In 3D: • A
digonal disphenoid can be represented as { } ∨ { } = [( ) ∨ ( )] ∨ [( ) ∨ ( )]. • A
p-gonal pyramid is represented as ( ) ∨ {
p}. In 4D: • A
p-q-hedral pyramid is represented as ( ) ∨ {
p,
q}. • A
5-cell is represented as ( ) ∨ [( ) ∨ {3}] or [( ) ∨ ( )] ∨ {3} = { } ∨ {3}. • A square pyramidal pyramid is represented as ( ) ∨ [( ) ∨ {4}] or [( ) ∨ ( )] ∨ {4} = { } ∨ {4}. When mixing operators, the
order of operations from highest to lowest is ×, +, ∨. Axial polytopes containing vertices on parallel offset hyperplanes can be represented by the ‖ operator. A uniform prism is {
n}‖{
n} and antiprism {
n}‖
r{
n}. == Extension of Schläfli symbols ==