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 $\left\{{p\atop q}\right\}$, with

$$\left\{{p\atop q}\right\}=\left\{{q\atop p}\right\}.$$ (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

$${1\over p}+{1\over q}+{1\over r}>1.$$ (2)

But $p, q\geq 3$, so $r$ must be 2. This means that the possible quasiregular polyhedra have symbols $\left\{{3\atop 3}\right\}$, $\left\{{3\atop 4}\right\}$, and $\left\{{3\atop 5}\right\}$. Now

$$\left\{{3\atop 3}\right\}=\{3, 4\}$$ (3)

is the Octahedron, which is a regular Platonic Solid and not considered quasiregular. This leaves only two convex quasiregular polyhedra: the Cuboctahedron $\left\{{3\atop 4}\right\}$ and the Icosidodecahedron $\left\{{3\atop 5}\right\}$.

If nonconvex polyhedra are allowed, then additional quasiregular polyhedra are the Great Dodecahedron $\{5, {\textstyle{5\over 2}}\}$ and the Great Icosidodecahedron $\{3, {\textstyle{5\over 2}}\}$ (Hart).

For faces to be equatorial $\{h\}$,

$$h=\sqrt{4N_1+1}\,-1.$$ (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. ``Quasi-Regular Polyhedra.'' §2-3 in Regular Polytopes, 3rd ed. New York: Dover, pp. 17-20, 1973.

Hart, G. W. ``Quasi-Regular Polyhedra.'' http://www.li.net/~george/virtual-polyhedra/quasi-regular-info.html.