Schläfli symbol

From Schläfli (“a surname”) + symbol, after 19th-century Swiss mathematician Ludwig Schläfli (1814–1895).

Schläfli symbol (plural Schläfli symbols)

1. (geometry) A notation that recursively encodes certain properties of a specified regular polytope or tessellation.
   • 1995, Unnamed translator, E. B. Vinberg, Polynomial Group, article in M. Hazewinkel (editor), Encyclopaedia of Mathematics, Volume 4: Monge–Ampère Equation — Rings and Algebras, 2013, International Edition, page 481,

     The three-dimensional regular polytopes (Platonic solids) have the following Schläfli symbols: the tetrahedron — {3, 3}, the cube — {4, 3}, the octahedron — {3, 4} — the dodecahedron — {5, 3}, and the icosahedron — {3, 5}.

   • 2007, Michael E. Mortenson, chapter L, in Geometric Transformations for 3D Modeling‎, page 248:

     For the regular tiling of squares the Schläfli symbol is {4,4}, indicating that there are four squares surrounding each vertex. And for the regular tiling of hexagons the Schläfli symbol is {6,3}, indicating that there are three hexagons surrounding each vertex.

   • 2012, John Barnes, Gems of Geometry, 2nd edition, page 79:

     The 24-cell has has 24 octahedral cells, 24 vertices, 96 triangular faces and 96 edges. Its Schläfli symbol is {3, 4, 3} from which we see that the vertex figure is a cube.

• (notation encoding certain properties of a regular polytope or tessellation): Schläfli notation