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.

List of Shapes By Schläfi Symbol Edit

Arranged by size order, with the final entry taking precedence.

<0-Entry Edit

0-Entry Edit

1-Entry Edit

2-Entry Edit