Is there a general rule for how many faces/cells can fit around an edge in 4D polytopes?

Question

(For concreteness, when I say "face" I specifically mean the 2-dimensional kind. I will refer to "hyperfaces" as cells as this question only deals with polytopes in the fourth dimension.)

In 3-dimensional polyhedra, there are only ever 2 faces to an edge.

However in 4-dimensional regular polychora, there are 3 faces to an edge in the tesseract, octaplex (24-cell) and pentachoron(5-cell/4-simplex), but the orthoplex (16-cell) and 120-cell have 4 faces meeting at an edge, and the 600-cell has 5 meeting at an edge.

My first thought was that the angles in the faces would offer a clue, but that's obviously not true (tesseract/squares, pentachoron/triangles) and my next thought was that the angles between the faces of the cells would offer a clue but for the same reason that's obviously not true (pentachoron and 600-cell are both constructed of tetrahedra) but this did lead me to realise that the question of how many faces can fit around an edge is equivalent to how many cells can fit around an edge.

I suppose what I'm asking is, in general (i.e. not just for regular polychora), how do we know how many cells can "fit" around an edge and is there some neat way of expressing that in terms of the properties of the faces/cells?

The part that makes this confusing for me is that rotation works very differently in 4D to 3D. I can think how I would mathematically show how many cubes could fit around an edge in 3D (for a tesselation) but it seems like a nonsense question in 4D?

Answer 1

The main idea here is the right visual correspondance to 3D. In fact for a polyhedron to become convex, you need the local property that the sum of corner angles of the adjacent polygons at any vertex of the polyhedron has to be less than 360°.

The same applies then to 4D: For a polychoron to become convex, you need the local property that the sum of dihedral angles of the adjacent cells at any edge of the polychoron has to be less than 360°.

Within this generality you cannot tell more. But if you know something about your constituting cells, esp. about their dihedral angles, then you can run through the thereby provided possibilities.

Answer 2

The following example illustrates that there is not an upper limit to the number of 2-faces (or 3-faces) that an edge of a 4-d polytope (polychoron) can border.

Start with a pyramid P over an n-gon ($n \ge 3$). Then take a prism Q over P; Q is a polychoron. Consider the edge E of Q that connects the apex of P to the other apex. E borders n parallelogram faces of Q, and n triangular prism faces of Q.