Classification of regular polytopes

Remarks

The facets and the vertex figures of a regular polytope are regular, so the possible symbols are built from shorter ones that are already known. The new condition at each step is an angle condition: the facets around a face of dimension $d-3$ must have dihedral angles that add up to less than $2\pi$. In dimension $3$ this gives $(p-2)(q-2) < 4$, which leaves the five Platonic symbols, as in the classification of the Platonic solids. Each surviving symbol is then realised by an explicit construction, and the symbol determines the polytope up to similarity.

Example: the regular dodecahedron $\set{5,3}$ has dihedral angle $\arccos(-1/\sqrt5) \approx 116.57^\circ$. Three dodecahedra around an edge use about $349.7^\circ < 360^\circ$, so $\set{5,3,3}$ is allowed, and it is the $120$-cell. Four would use more than $360^\circ$, so there is no $\set{5,3,4}$. For the cube, four around an edge use exactly $360^\circ$, so $\set{4,3,4}$ gives the cubic honeycomb of space and not a polytope.

Convexity is needed for the list: allowing star polytopes adds the four Kepler–Poinsot polyhedra in dimension $3$ and ten regular star polytopes in dimension $4$.

Going deeper

For a student

The theorem is proved by combining the Schläfli symbol with an angle count, and dimension four is special because the dodecahedron and icosahedron still fit in. The six four-dimensional polytopes include three without analogues elsewhere: the $24$-cell, which is self-dual, and the dual pair of the $120$-cell and the $600$-cell. The symmetry groups of the regular polytopes are finite reflection groups: the dihedral groups and the types $A_d$, $B_d$, $F_4$, $H_3$ and $H_4$. So the classification is tied to the classification of finite Coxeter groups.