M.E. Irizarry-Gelpí

Physics impostor. Mathematics interloper. Husband. Father.

Quartic Johnson Solids


I am searching for polyhedra that have four edges at every vertex. The simplest ones I have found so far are the \(n\)-antiprisms, with \(2n+2\) faces, \(2n\) vertices, and \(4n\) edges. Each convex polyhedron must satisfy the polyhedral formula:

$$V - E + F = 2$$

The triangular antiprism is the octahedron, which is a Platonic solid. There are four Archimedean solids that follow this rule too:

  • cuboctahedron (12 vertices)
  • rhombicuboctahedron (24 vertices)
  • icosidodecahedron (30 vertices)
  • rhombicosidodecahedron (60 vertices)

Besides the Platonic and Archimedean solids, you also have the Johnson solids. Here is a list of relevant Johnson solids:

The number of vertices is always a multiple of 4 or 5. According to WolframAlpha, the polyhedral graph for the polyhedra with odd number of vertices has edge chromatic number equal to 5. So \(J_{57}\), \(J_{32}\), \(J_{33}\), \(J_{40}\), and \(J_{41}\) do not work for the ABCD model.