Regular polytopes

It is fairly well known that there are five regular solids in three dimensions: the tetrahedron, the octahedron, the icosahedron, the cube, and the dodecahedron. The two-dimensional case is straightforward; here, the regular shapes are simply the regular n-gons: equilateral triangles, squares, regular pentagons, regular hexagons, and so on. But what hap…