On the Wikipedia article for cuboctahedron, it states at follows.
The cuboctahedron is the unique convex polyhedron in which the long radius (center to vertex) is the same as the edge length[...] This [is known as] radial equilateral symmetry.
Polygons in radially equilateral polyhedra have determined spherical angles, so perhaps this could be proven by enumerating all possible vertex figures and trying to put them together.
What I’m curious about is the non-convex radially symmetric polyhedra. Due to the possibility of retrograde faces, and the much greater variety of them, a naive computer enumeration is not possible. After digging through the Wikipedia list of uniform polyhedra, I found four more examples:
My question is: is this list complete? If not, what is the complete list?
Addendum:
As of 2023, Wikipedia has dropped the uniqueness claim cited above. The relevant passafe now reads:
[R]adial equilateral symmetry is a property of only a few uniform polytopes, including the two-dimensional hexagon, the three-dimensional cuboctahedron, and the four-dimensional 24-cell and 8-cell (tesseract).