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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}.
(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

 1/p+1/q+1/r>1.
(2)

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}
(3)

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},

 h=sqrt(4N_1+1)-1.
(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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