In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... A dodecahedron, one of the five Platonic solids. ...

It is defined as follows. The Schläfli symbol of a polygon with n edges is {n}. The Schläfli symbol of a polyhedron is {p,q} if its faces are p-gons, and each vertex is surrounded by q faces. Note that the Schläfli symbol is not well defined for polyhedra which are not (sufficiently) regular (such as the prism). Wiktionary has a definition of: Polygon A polygon (literally many angle, see Wiktionary for the etymology) is a closed planar path composed of a finite number of sequential line segments. ... In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ...

The Schläfli symbols of the Platonic solids are: A Platonic solid is a convex polyhedron whose faces all use the same regular polygon and such that the same number of faces meet at all its vertices. ...

• for the tetrahedron : {3,3}
• for the cube : {4,3}
• for the octahedron : {3,4}
• for the dodecahedron : {5,3}
• for the icosahedron : {3,5}

Schläfli symbols may also be defined for regular tessellations of euclidean or hyperbolic space in a similar way. A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... A dodecahedron is a Platonic solid composed of twelve pentagonal faces, with three meeting at each vertex. ... An icosahedron [ˌaıkəsəhiːdrən] noun (plural: -drons, -dra [-drə]) is a polyhedron having 20 faces. ... A tessellated plane A tessellation of the plane is a collection of plane figures that fill the plane with no overlaps and no gaps. ... In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ... A triangle immersed in a saddle-shape plane, as well as two diverging parallel lines. ...

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}. In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ... In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way. ...

The Schläfli symbol of a line segment is {}. If a polytope has Schläfli symbol {p₁,p₂,...,p_n-1} then its dual polytope has Schläfli symbol {p_n-1,...,p₂,p ₁}. In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the others. ...

Occasionally, you will see fractions in a Schläfli symbol. For example, there are several instances of ⁵/₂ in the list of regular polytopes. The symbol {^p/_q} means a planar figure with p vertexes where every q-th vertex is connected. Thus, ⁵/₂ is a five-pointed star shape. This page lists the regular polytopes in Euclidean space. ...

The Schläfli symbol is named after the 19th century mathematician Ludwig Schläfli who made important contributions in geometry and other areas. Geometry (from the Greek words Ge = earth and metro = measure) is the branch of mathematics first introduced by Theaetetus dealing with spatial relationships. ...