Say that a polyhedron is spherical if all of its vertices lie on a sphere. Sometimes these polyhedra are called inscribed. My question is simple, and surely well-known, although my searches have not found a definitive statement:
Q. Are all the $13$ Archimedean solids spherical?
Certainly all the Platonic solids are spherical, and not all the Johnson solids are spherical (e.g., the elongated triangular bipyramid).
All but the chiral ones clearly can be constructed via the kaleidoscopical construction of Wythoff and so are vertex transitive. The chiral ones (the snubs) so can be constructed secondarily from the omnitruncates via alternation, and so are vertex transitive as well.
Obviously all 13 show up a finite set of vertices.
But vertex transitivity together with vertex count finiteness clearly implies the existance of an unique finite circumradius.
--- rk