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 .

See also Cuboctahedron ,

Dodecadodecahedron ,

Great Icosidodecahedron ,

Icosidodecahedron ,

Platonic Solid
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References Coxeter, H. S. M. "Quasi-Regular Polyhedra." §2-3 in Regular
Polytopes, 3rd ed. New York: Dover, pp. 17-20, 1973. Fejes
Tóth, L. Ch. 4 in Regular
Figures. Oxford, England: Pergamon Press, 1964. Hart, G. "Quasi-Regular
Polyhedra." http://www.georgehart.com/virtual-polyhedra/quasi-regular-info.html . Robertson,
S. A. and Carter, S. "On the Platonic and Archimedean Solids." J.
London Math. Soc. 2 , 125-132, 1970. Referenced on Wolfram|Alpha Quasiregular Polyhedron
Cite this as:
Weisstein, Eric W. "Quasiregular Polyhedron."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/QuasiregularPolyhedron.html

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