Quasiregular Polyhedron

A quasiregular polyhedron is the solid region interior to two dual regular polyhedra with Schläfli symbols {p,q} and {q,p}. Quasiregular polyhedra are denoted using a Schläfli symbol of the form {p; q}, with

 {p; q}={q; p}.

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


But p,q>=3, so r must be 2. This means that the possible quasiregular polyhedra have symbols {3; 3}, {3; 4}, and {3; 5}. Now

 {3; 3}={3,4}

is the octahedron, which is a regular Platonic solid and not considered quasiregular. This leaves only two convex quasiregular polyhedra: the cuboctahedron {3; 4} and the icosidodecahedron {3; 5}.

If nonconvex polyhedra are allowed, then additional quasiregular polyhedra the dodecadodecahedron {5,5/2} great icosidodecahedron {3,5/2}, as well as 12 others.

For faces to be equatorial {h},


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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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.", 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.

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