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Given a vertex, edge, or face figure of a convex uniform polyhedron, is there a way to identify all of the partitions of space that result from stellation to infinity* of that polyhedron, without resorting to computational geometry?

This of course means I don't need to know their dimensions.

I've been able to determine that each vertex, edge, and face of the polyhedron has a corresponding partition, but for polyhedra with obtuse dihedral angles, you end up with partitions beyond that first shell and I don't see how to identify them in a discrete math kind of way.

I imagine the identification would take the form of an adjacency graph of the partitions, but whatever works is fine.

*Don't know if I'm using the term correctly. I'm talking about extending each face to the entire plane that includes it.

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  • $\begingroup$ Are you familiar with stellation diagrams? I'm not very, but I can't help but wonder how helpful they'll be. $\endgroup$
    – pjs36
    May 19, 2017 at 2:41
  • $\begingroup$ @pjs36 - I have the impression stellation diagrams are used to talk about stellations once you already have them figured out. Could be wrong though. $\endgroup$
    – MackTuesday
    May 19, 2017 at 16:03

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The stellation diagram is just a matter of projective or descriptive geometrical construction; choosing one reference face plane and extending the other faces until their planes intersect.

Once you have that, a method is described by Coxeter and du Val respectively in The Fifty-Nine Icosahedra, which has been republished a few times.

Coxeter identified faces of the stellations based on the combinatorics of the edges in the diagram, then du Val used that to identify the spatial cells thus created.

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