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 the dodecadodecahedron 
great icosidodecahedron 
,
as well as 12 others.
For faces to be equatorial 
,
 |
(4)
|
The polyhedron 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 rectangles. The polyhedron edges are also all equivalent, a property shared only with the completely regular Platonic solids.
