For example, in 2-space, a regular hexagon's edges all have the same magnitude, but also share a magnitude with the radius of the circumcentre (intuitive by sticking a six equilateral triangles together). In 3-space, you get a convex regular icosahedron (sticking 20 regular tetrahedra together at a mutual point).
So, is there a classification of these kinds of polyhedra where all the edges are the same length as the radius of the circumsphere?