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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 Edit

Arranged by size order, with the final entry taking precedence.

### <0-Entry Edit

• $()$ - Point

### 0-Entry Edit

• $\{\}$ - Line Segment

### 1-Entry Edit

• $\{1\}$ - Henagon
• $\{2\}$ - Digon
• $\{3\}$ - Triangle
• $\{4\}$ - Square
• $\{5/2\}$ - Pentagram
• $\{5\}$ - Pentagon
• $\{6\}$ - Hexagon
• $\{a\}$ - Polygon
• $\{{ \aleph }_{ 0 }\}$ - Apeirogon