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
Category:Shape Notation