The truncated icosahedron is iconically the archetypical design for the association football/soccerball. It is composed of 12 pentagons and 20 hexagons, with the former usually in black and the latter in white. However, if such a polyhedron makes for such a great ball in 3D, what might be the best choice in 4D?

The main features of the truncated icosahedron that are most strongly associated with its “soccerball vibe”, in my mind, are the fact that is composed of relatively round shapes (pentagons and hexagons) joined at simple vertices. In 4D, the “round” shapes would have to be themselves somewhat similar to soccerballs, composed mostly of pentagons and hexgons themselves, such as the Goldberg polyhedra (of which the truncated icosahedron is G(1,1)). However, the need to form facets of a polychoron means there is a limit to how shallow the angles between faces can be. Thus, it would seem the leading choice would be… the 120-cell? It is composed of dodecahedra (G(1,0) polyhedra), with the minimal vertex figure (tetrhedron), though it is somewhat surprising that the generalization of an archimedean solid is a regular.

There is, however, the question of how to color the cells of the 120-cell. For this, we note an interesting fact: it is possible to choose 24 cells from the 120-cell so that the cells are parallel to the cells of a regular 24-cell (equivalently, it is possible to vertex-inscribe a dual 24-cell into the dual 600-cell). Taking this choice of cells as the “black” panels, one gets a coloring that, as seen above, seems to pass the vibe check. Looking closely, the specific details that seem to have been carried over are:

  • The black “panels” are each completely surrounded by white “panels”
  • Each white “panel” touches at least one black “panel”

There are of course some details that don’t cerry over; some vertices are adjacent to no black cells, unlike with the truncated icosahedron, and more like the G(2,0) and G(2,1) goldberg polyhedra. However, since the former of those is often confusable with the standard soccerball, I’d say the vibe is well within needed margins. Thus, I hereby set forth the 24-colored 120-cell as the flagship soccerball of the 4th dimension.