Schläfli symbol

A notation that encodes the regularity and incidence structure of regular polytopes. Its entries describe how faces meet around lower-dimensional parts.

Polytope: A geometric object with flat faces that generalizes polygons and polyhedra to arbitrary dimensions. Schläfli symbols classify regular polytopes across dimensions.
Regular polygon: A polygon with equal sides and equal angles. The one-entry symbol {p} identifies a regular polygon with p sides.
Tetrahedron: A polyhedron with four triangular faces, four vertices, and six edges. Its regular form is represented by {3,3}.
Coxeter notation: A notation for reflection groups and regular or uniform polytopes based on Coxeter diagrams and related symbols. It emphasizes reflection relations and can encode symmetry information beyond the face-incidence entries.
Regular polytope: A polytope whose symmetries act transitively on its flags. These are the objects the notation is designed to classify.
Vertex figure: A geometric section that records the arrangement of faces around a chosen vertex. The second entry in {p,q} counts faces meeting at each vertex.
Cube: A regular hexahedron with six square faces, twelve edges, and eight vertices. Its symbol {4,3} states that three square faces meet at each vertex.
Wythoff symbol: A notation that identifies a uniform polytope through its Coxeter diagram and a marked construction point. Unlike a Schläfli symbol, it distinguishes different uniform constructions with the same symmetry group.
Polygon: A plane figure bounded by a finite closed chain of straight line segments. The symbol {p} records the number of sides of a regular polygon.
Coxeter group: A group defined by generators that are involutions and relations between their pairwise products. Regular-polytope symmetries are encoded by Coxeter groups associated with the symbol.

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