Pseudo-uniform polytope

A pseudo-uniform polytope is a polytope whose facets are all uniform and whose vertex figures are congruent (i.e. the same configuration of facets meets at each vertex and has the same dihedral angles), but is not itself vertex-transitive and therefore fails to be uniform.

Formally, a polytope's vertex figures are congruent if, for any two vertices, there is an that transforms all the facets adjacent to one into the facets adjacent to the other. It is vertex transitive if all such isometries are also symmetries of the polytope.

3D
There are two known pseudo-uniform polyhedra: the elongated square gyrobicupola (or pseudorhombicuboctahedron, a Johnson solid) and the great pseudorhombicuboctahedron. It is not known if there are any others.

4D
Besides the prisms of the pseudo-uniform polyhedra, several categories of pseudo-uniform polychora were discovered in the early 2020s. These include:


 * some members of the tetrasidpith regiment
 * some blends of members of the sishi regiment with icositetrachoron compounds
 * some blends of 12 10-10/3 duoprism s with decagon- or decagram-containing members of the rissidtixhi regiment, altogether possessing the symmetry of the small swirlprism
 * some blends of 12 of the above, similar to members of the idcossid regiment