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

(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?

• google.com/books/edition/Regular_Polytopes/iWvXsVInpgMC?hl=en Aug 24, 2021 at 22:04
• @WillJagy This is a well-known book. Do you have any specific chapter in mind for answering the question? OP is also specifically interested in non-regular polytopes. Sep 18, 2021 at 7:54

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.