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Under the extended definition of regularirty there are 9 regular polyhedra, 5 Platonic solids and 4 Kepler-Poinsot solids (where we assume them to have a finite volume to exclude honeycombes and other weird stuff). All the proofs that there exist only nine of these which I've read so far run as follows:

  1. They assume the proof that there are 5 Platonic solids (usual definition of regularity)
  2. Next they show that every star polyhedron must contain a Platonic solid inside, i.e. it must be a stellation of Platonic solid
  3. Finally they "exhaust" all possible stellations of Platonic solids. Namely they show that tetrahedron has no stellations, so does a cube; that octahedron stellates to stella octangula; that there are 3 stellations of dodecahedron and finally that there is only one stellation of icosahedron.

But, on the other hand, Coxeter et al. showed that there are 59 stellations of icosahedron. So I'm a bit confused.

My question is: does a complete proof that there are 4 Kepler-Poinsot solids require showing that none of the other 57 stellations of icosahedron is regular in the extended sense?

Edited Seems like the faces of the rest 57 Coxeter stellations do not count as polygons, although some of them are connected. But I'm not sure

serpens
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  • It looks as though the Coxeter book adopted a different definition of "stellation" to the one used for Kepler-Poinsot polyhedra (look up "Miller's Rules" for stellation). I wonder if this fact annoys you as much as it does me ;-). – Chris Lewis Jul 17 '23 at 15:27
  • I've read Miller's rules but I haven't thought they suppose different definition of stellation. Wouldn't it be too much to ask what's the essential difference between those two approaches then? For example, when I look at the final stellation, it seems to me that this object is 'of the same type' as other Kepler solids, and from the picture it's not at all obvious whether it is regular or not – serpens Jul 17 '23 at 15:46

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Regularity of a polyhedron also implies the regularity of its (single) face type. Although some of the stellations of the icosahedron indeed have connected faces, there are just 2 with regular ones.

Btw. there are also dual settings too, then considering facetings of the regular convex polyhedra instead. In fact it was originally Edmund Hess' theorem stating that regular starry polytopes always have coinciding vertex sets with according non-starry ones.

--- rk

Dr. Richard Klitzing
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