Schläfli symbol

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The Schläfli symbol is a notation of the form {p,q,...} that defines regular tilings and polyhedra, as well as their multidimensional generalisations.

The Schläfli symbol of a regular polygon with p sides is {p}; for example, {7} is a regular heptagon. p may be made arbitrarily high; the limit, {∞}, denotes a regular apeirogon.

Tilings and polyhedra

For a regular tiling or a regular polyhedron that has q regular p-sided faces around each vertex, the Schläfli symbol is {p,q}. For example, the cube has Schläfli symbol {4,3}, because it has three square faces around each vertex.

The Schläfli symbols of the three regular tilings of the plane[note 1] are:

The equality (p - 2)(q - 2) = 4 holds for all three tilings. When (p - 2)(q - 2) < 4, the Schläfli symbol {p,q) denotes one of five Platonic solids (provided integer p and q greater than 2; there are four additional valid non-integer solutions, and an infinite number of solutions if either p or q is permitted to be 2). Finally, when (p - 2)(q - 2) > 4, the Schläfli symbol {p,q} denotes a regular hyperbolic tiling.

The definition is extended recursively for higher dimensions:

These can also be used to notate higher-dimensional hyperbolic honeycombs. Of these, only compact honeycombs are considered notable enough for study, as the infinite cells and vertex figures of paracompact and noncompact honeycombs do not lend themselves to interesting rules. While there are an infinite number of solutions in two dimensions, there are nine solutions in three and higher dimensions:

There are also four star solutions, however it is unknown how these differ behaviourally from the convex solutions.

See also

Notes

  1. ↑ Regular tilings of the sphere exist that would otherwise be degenerate as polytopes. These are the hosohedra {2,n} and their dual dihedra {n,2}. Coxeter calls these cases "improper" tessellations (Coxeter, pp. 66-67). There are also two improper regular tilings on the plane: {∞,2}, an apeirogonal dihedron, made from two apeirogons, each filling half the plane; and its dual, {2,∞}, an apeirogonal hosohedron, seen as an infinite set of parallel lines.

Further reading

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