Schläfli symbol

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