A Schläfli Symbol is a notation to define regular polytopes and tilings. They appear in the form .

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.

### <0-Entry

• - Point

### 0-Entry

• - Line Segment

### 1-Entry

• - Henagon
• - Digon
• - Triangle
• - Square
• - Pentagram
• - Pentagon
• - Hexagon
• - Polygon
• - Apeirogon

### 2-Entry

• - 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