I've been reading old works by Coxeter, thinking foundationally about his ideas concerning the Wythoff construction and the abstract definition of regularity as a polyhedron whose symmetry group acts transitively on the set of flags.
The other day, when attempting to use these ideas I realised the existence of star polygons (by no means an original thought) and noticed they were regular. I then realised there were apparently regular star polyhedra listed but my question is this:
What about the shape (polyhedron?) formed when you replace each face of a dodecahedron by a pentagram?
This appears to be regular, by the definition of transitivity of symmetry group acting on flags, but is not listed. Is this not a polyhedron? I understand it is not convex, but neither are the Kepler-Poinsot solids...
If you keep the pentagrams inscribed in the original pentagonal faces, then this is not any kind of polyhedron, since the faces do not connect to each other: every edge lies only on a single face.
If you scale up the pentagrams such that these edges do meet, then you can obtain two different Kepler-Poinsot solids depending on the scaling factor: the small stellated dodecahedron and the great stellated dodecahedron.