Exceptional Object - Regular Polytopes

Regular Polytopes

The prototypical examples of exceptional objects arise when we classify the regular polytopes. In 2 dimensions we have a series of regular n-gons for n ≥ 3. In every dimension above 2 we find analogues of the cube, tetrahedron and octahedron. In 3 dimensions we find two more regular polyhedra – the dodecahedron (12-cell) and the icosahedron (20-cell) – making 5 Platonic solids. In 4 dimensions we have a total of 6 regular polytopes including the 120-cell, the 600-cell and the 24-cell. There are no other regular polytopes; in higher dimensions the only regular polytopes are of the hypercube, simplex, orthoplex series. So we have three series and 5 exceptional polytopes.

The pattern is similar if non-convex polytopes are included. In two dimensions there is a regular star polygon for every rational number p/q > 2. In three dimensions there are four Kepler–Poinsot polyhedra, and in four dimensions ten Schläfli–Hess polychora; in higher dimensions there are no non-convex regular figures.

These can be generalized to tessellations of other spaces, especially uniform tessellations, notably tilings of Euclidean space (honeycombs), which have exceptional objects, and tilings of hyperbolic space. There are various exceptional objects in dimension below 6, but in dimension 6 and above the only regular polyhedra/tilings/hyperbolic tilings are the simplex, hypercube, cross-polytope, and hypercube lattice.