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
- $ () $ - Point
0-Entry
- $ \{\} $ - Line Segment
1-Entry
- $ \{1\} $ - Henagon
- $ \{2\} $ - Digon
- $ \{3\} $ - Triangle
- $ \{4\} $ - Square
- $ \{5/2\} $ - Pentagram
- $ \{5\} $ - Pentagon
- $ \{6\} $ - Hexagon
- $ \{a\} $ - Polygon
- $ \{{ \aleph }_{ 0 }\} $ - Apeirogon
2-Entry
- $ \{1, 1\} $ - Henagonal Henahedron
- $ \{2, 1\} $ - Henagonal Hosahedron
- $ \{1, 2\} $ - Henagonal Dihedron
- $ \{a, 2\} $ - Polygonal Dihedron
- $ \{3, 3\} $ - Tetrahedron
- $ \{4, 3\} $ - Cube
- $ \{5/2, 3\} $ - Great Stellated Dodecahedron
- $ \{5, 3\} $ - Dodecahedron
- $ \{6, 3\} $ - Hexagonal Tiling
- $ \{3, 4\} $ - Octahedron
- $ \{5, 4\} $ - Dodecadodecahedron
- $ \{3, 5/2\} $ - Great Icosahedron
- $ \{5, 5/2\} $ - Great Dodecahedron
- $ \{3, 5\} $ - Icosahedron
- $ \{4, 5\} $ - Medial Rhombic Triacontahedron
- $ \{5/2, 5\} $ - Small Stellated Dodecahedron
- $ \{6, 5\} $ - Medial Triambic Icosahedron
- $ \{5, 6\} $ - Ditrigonal Dodecadodecahedron
- $ \{6, 6\} $ - Excavated Dodecahedron
- $ \{2, b\} $ - Polygonal Hosohedron
