What is the complete list of radially equilateral polyhedra?

Question

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:

• The cubohemioctahedron
• The octahemioctahedron
• The dodecadodecahedron
• The small dodecahemicosahedron

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).

Answer 1

• At least the great dodecahemicosahedron comes to mind as well. It even is uniform.

For other solids with edge length = circumradius we'd have additionally:

• The triangular orthobicupola, which at least is a Johnson solid.
• And you even could count the triangular cupola itself, so it exposes the centerpoint at the midpoint of the hexagon.

And for a full series of convex Wythoffian polytopes throughout all dimensions with this very property, you could refer to one of the replies to this SE-question.

--- rk

$\ $

PS: you even might look here

• for edge facetings of the cuboctahedron resp.
• for edge facetings of the dodecadodecahedron.