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  "body": "\n## Definition\n\nThe *Schläfli symbol* of a convex [[regular polytope|F60PD912D5]] $P$ of dimension $d \\ge 2$ is defined by induction on $d$.\n\n1. If $d = 2$, then $P$ is a [[regular polygon|01KJJ9A400]] with $p$ sides for some $p \\ge 3$, and its symbol is $\\set{p}$.\n2. If $d \\ge 3$, then all [[facets|DPDXN2FJC4]] of $P$ are congruent regular polytopes of dimension $d-1$, with a common symbol $\\set{p_1, \\dots, p_{d-2}}$.\nLet $p_{d-1}$ be the number of facets of $P$ that contain a given [[face|KY7SB854VH]] of dimension $d-3$.\nBy regularity it does not depend on the face.\nThe symbol of $P$ is $\\set{p_1, \\dots, p_{d-2}, p_{d-1}}$.\n\n## Remarks\n\nExample: the cube has square facets, with symbol $\\set{4}$, and exactly $3$ facets contain each vertex, so its symbol is $\\set{4, 3}$.\nThe tesseract has cubes as facets and $3$ cubes around each edge, so its symbol is $\\set{4, 3, 3}$.\nThe regular $d$-simplex has symbol $\\set{3, 3, \\dots, 3}$ with $d-1$ entries.\n\nNon-example: the string $\\set{6, 3}$ is not the symbol of a convex polyhedron.\nThree 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.\n\nIf $P$ 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 $\\set{p_{d-1}, \\dots, p_1}$.\nThe 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.\n",
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