On the Number of Sides of a Petrie Polygon

Let {p, q, r} be the regular 4-dimensional poly tope for which each face is a {p, q} and each vertex figure is a {q, r}, where {p, q}, for example, is the regular polyhedron with p-gonal faces, q at each vertex. A Petrie polygon of {p, q} is a skew polygon made up of edges of {p, q} such that every two consecutive sides belong to the same face, but no three consecutive sides do. Then a Petrie polygon of {p, g, r} is defined by the property that every three consecutive sides belong to a Petrie polygon of a bounding {p, q}, but no four do. Let h Pqr be the number of sides of such a polygon, and g p,q,r the order of the group of symmetries of {p, g, r}.

The Pure Archimedean Polytopes in Six and Seven Dimensions

An Archimedean solid (in three dimensions) may be defined as a polyhedron whose faces are regular polygons of two or more kinds and whose vertices are all surrounded in the same way. For example, the “great rhombicosidodecahedron” is bounded by squares, hexagons and decagons, one of each occurring at each vertex. Thus any Archimedean solid is determined by the faces which meet at one vertex, and therefore by the shape and size of the “vertex figure,” which may be defined as follows. Suppose, for simplicity, that the length of each edge of the solid is unity. The further extremities of all the edges which meet at a particular vertex lie on a sphere of unit radius, and also on the circumscribing sphere of the solid, and therefore on a circle. These points form a polygon, called the “vertex figure,” whose sides correspond to the faces at a vertex and are of length 2 cos π/n for an n-gonal face. Thus the vertex figure of the great rhombicosi-dodecahedron is a scalene triangle of sides .