A quasiregular polyhedron is the solid region interior to two Dual regular polyhedra with
Schläfli Symbols and . Quasiregular polyhedra are denoted using a
Schläfli Symbol of the form
, with

(1) 
Quasiregular polyhedra have two kinds of regular faces with each entirely surrounded by faces of the other kind, equal
sides, and equal dihedral angles. They must satisfy the Diophantine inequality

(2) 
But , so must be 2. This means that the possible quasiregular polyhedra have symbols
,
, and
. Now

(3) 
is the Octahedron, which is a regular Platonic Solid and not considered quasiregular. This leaves only two convex
quasiregular polyhedra: the Cuboctahedron
and the Icosidodecahedron
.
If nonconvex polyhedra are allowed, then additional quasiregular polyhedra are the Great Dodecahedron
and the Great Icosidodecahedron
(Hart).
For faces to be equatorial ,

(4) 
The Edges of quasiregular polyhedra form a system of Great Circles: the
Octahedron forms three Squares, the Cuboctahedron four Hexagons, and the
Icosidodecahedron six Decagons. The Vertex Figures of quasiregular
polyhedra are Rhombuses (Hart). The Edges are also all equivalent, a property
shared only with the completely regular Platonic Solids.
See also Cuboctahedron, Great Dodecahedron, Great Icosidodecahedron, Icosidodecahedron,
Platonic Solid
References
Coxeter, H. S. M. ``QuasiRegular Polyhedra.'' §23 in Regular Polytopes, 3rd ed.
New York: Dover, pp. 1720, 1973.
Hart, G. W. ``QuasiRegular Polyhedra.''
http://www.li.net/~george/virtualpolyhedra/quasiregularinfo.html.
© 19969 Eric W. Weisstein
19990525