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Concept
Schläfli symbol
Summary
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.
Official source
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