Schläfli symbol

A Schläfli Symbol is a notation to define regular polytopes and tilings. They appear in the form $\{ a, b, c, \cdots \}$ .

The first X-1 entries define a polytope, and the Xth entry defines how many are around each X-2-dimensional subfacet.

For example, $\{3, 4\}$ defines a polytope where there are four triangles (schläfli symbol $\{3\}$ ) around each vertex; in this case, an octahedron. As another example, $\{3, 3, 4\}$ defines a polytope where there are four tetrahedra (schläfli symbol $\{3, 3\}$ ) around each edge; in this case, a tetrarss.

Regular polygons of X sides have the schläfli symbol $\{X\}$ .

An X-dimensional polytope will always have a schläfli symbol with X-1 entries, though not all schläfli symbols with X-1 entries define an X-dimensional polytope; this is because tilings in X dimensions have X entries in their schläfli symbols.

To find the dual of a regular polytope, simply reverse its schläfli symbol. For example, the dual of the tesseract ( $\{4, 3, 3\}$ ) is the tetrarss ( $\{3, 3, 4\}$ ). This also means that polytopes with palindromic schläfli symbols, such as the tetrahedron, are self-dual.