Chapter 9: Fundamental Physics

Section 7: Space as a Network

Regular polytopes

In 3D, of the five regular polyhedra, only the tetrahedron, cube and dodecahedron have three edges meeting at each vertex, corresponding to a trivalent network. (Of the 13 additional Archimedean solids, 7 yield trivalent networks.) In 4D the six regular polytopes have 4, 4, 6, 8, 4 and 12 edges meeting at each vertex, and in higher dimensions the simplex (d+1 vertices) and hypercube (2^d vertices) have d edges meeting at each vertex, while the co-cube (2d vertices) has 2(d-1). (See also symmetric graphs on page 1032, and page 929.)

From Stephen Wolfram: A New Kind of Science [citation]