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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

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

<0-Entry

0-Entry

1-Entry

2-Entry