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
When self-intersection is allowed, there are five other truncated
Albert Badoureau and Johann Pitsch each described all five in 1881
Jean Paul Albert Badoureau, Mémoire sur les Figures Isocèles, Journal de l'École polytechnique49 (1881), 47-172.
Johann Pitsch, Über Halbreguläre Sternpolyeder, Zeitschrift für das Realschulwesen6 (1881), 9-24, 64-65, 72-89, 216.
H. S. M. Coxeter, M. S. Longuet-Higgins, and J. C. P. Miller, Uniform Polyhedra, Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences246 (1954), 401-450.