Self-Intersecting Truncated Quasi-Regular Polyhedra

A polyhedron is truncated quasi-regular if it is vertex-transitive with scalene triangular vertex figures. Vertex transitivity means that for any two vertices of the polyhedron, there exists a translation, rotation, and/or reflection that leaves the outward appearance of the polyhedron unchanged yet moves one vertex to the other. A vertex figure is the polygon produced by connecting the midpoints of the edges meeting at the vertex in the same order that the edges appear around the vertex. There are only two truncated quasi-regular polyhedra that are not self-intersecting, namely the Truncated Cuboctahedron and the Truncated Icosidodecahedron. When self-intersection is allowed, there are five other truncated quasi-regular polyhedra.

Albert Badoureau and Johann Pitsch each described all five in 1881.