Pseudo-uniform polytope

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The elongated square gyrobicupola, perhaps the best-known pseudo-uniform. The green-square vertices and the blue-square vertices cannot be transformed into one another by the polyhedron's symmetries.

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 isometryisometry 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:

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

External links[edit | edit source]