Schläfli symbol

Remarks

Example: the cube has square facets, with symbol $\set{4}$, and exactly $3$ facets contain each vertex, so its symbol is $\set{4, 3}$. The tesseract has cubes as facets and $3$ cubes around each edge, so its symbol is $\set{4, 3, 3}$. The regular $d$-simplex has symbol $\set{3, 3, \dots, 3}$ with $d-1$ entries.

Non-example: the string $\set{6, 3}$ is not the symbol of a convex polyhedron. Three regular hexagons at a vertex have angles summing to $3 \cdot 120^\circ = 360^\circ$, so they lie flat, and the pattern gives the hexagonal tiling of the plane instead of a polytope.

If $P$ has symbol $\set{p_1, \dots, p_{d-1}}$, its vertex figure has symbol $\set{p_2, \dots, p_{d-1}}$, and its dual polytope has symbol $\set{p_{d-1}, \dots, p_1}$. The same notation is used for regular tilings, such as $\set{4, 4}$ for the square tiling, and for star polytopes with fractional entries, such as $\set{5/2}$ for the pentagram.