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, 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.
defines a polytope where there are fourRegular 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
1-Entry
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 -