The first X-1 entries define a polytope, and the Xth entry defines how many are around each X-2-dimensional subfacet.
For example, defines a polytope where there are four triangles (schläfli symbol ) around each vertex; in this case, an octahedron. As another example, defines a polytope where there are four tetrahedra (schläfli symbol ) around each edge; in this case, a tetrarss.
Regular polygons of X sides have the schläfli symbol .
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 () is the tetrarss (). 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.
- - Point
- - Henagonal Henahedron
- - Henagonal Hosahedron
- - Henagonal Dihedron
- - Polygonal Dihedron
- - Tetrahedron
- - Cube
- - Great Stellated Dodecahedron
- - Dodecahedron
- - Hexagonal Tiling
- - Octahedron
- - Dodecadodecahedron
- - Great Icosahedron
- - Great Dodecahedron
- - Icosahedron
- - Medial Rhombic Triacontahedron
- - Small Stellated Dodecahedron
- - Medial Triambic Icosahedron
- - Ditrigonal Dodecadodecahedron
- - Excavated Dodecahedron
- - Polygonal Hosohedron