# Pseudo-uniform polytope

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

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

## 3D[edit | edit source]

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[edit | edit source]

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 duoprisms 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

So far, all known pseudo-uniform polychora are self-intersecting; no convex examples are known.

## External links[edit | edit source]

- Bowers, Jonathan. "Pseudo-Uniform Polychoron Categories".