A polyhedron is quasi-regular if it is vertex-transitive and edge-transitive
but not face-transitive. In this context, transitivity means that for any
two vertices (edges, faces) of the polyhedron, there exists a translation,
rotation, and/or reflection that leaves the outward appearance of the
polyhedron unchanged yet moves one vertex (edge, face) to the other.
This definition implies that a quasi-regular polyhedron must have two kinds
of regular faces, where each face of one type is surrounded on all sides by
faces of the other type. There are only two quasi-regular polyhedra that are
not self-intersecting, namely the Cuboctahedron
and the Icosidodecahedron.
When self-intersection is allowed, there are 14 other quasi-regular polyhedra.
Nine have faces that pass through their centers and are often subcategorized
as versi-regular polyhedra.
The remaining five are listed on this page.
Edmund Hess described the Dodecadodecahedron and the Great Icosidodecahedron
in 1878 .
Johann Pitsch described these two plus the Small Ditrigonal
Icosidodecahedron in 1881 .
Albert Badoureau described all five in 1881
Edmund Hess, Über vier Archimedeische Polyeder höherer Art, Schriften der Gesellschaft zur Beförderung der gesammten Naturwissenschaften zu Marburg11(4) (1878).
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.
Norman W. Johnson, Uniform Polytopes, unpublished manuscript.