Each face in a quasiregular polyhedron is surrounded by faces of another type and all dihedral angles are equal. Quasiregular polyhedra are isogonal and isotoxal and their vertex figures are equal, cyclic, and equiangular [1]. Every quasiregular polyhedron admits a system of equatorial polygons, each of which is inscribed in a great circle of the circunsphere. Only two convex quasiregular polyhedra exist: the cuboctahedron and the icosidodecahedron, and their equatorial polygons are, respectively, regular hexagons and regular decagons. The cuboctahedron has 8 triangular and 6 square faces, while the icosidodecahedron has 20 triangular and 12 pentagonal faces. They are 2 of the 13 Archimedeans, and their names explain from which polyhedra they derive from rectification: the vertices of the cuboctahedron are the edges’ midpoints of the cube or the octahedron; the vertices of the icosidodecahedron are the edges’ midpoints of the dodecahedron or the icosahedron.

The tutorials below show examples of how to model* the two Archimedeans that are convex quasiregular polyhedra:

• the cuboctahedron (from the cube and from the regular octahedron)
• and the icosidodecahedron (here shown, from the regular dodecahedron).

These and other examples on how to model them are included in my Doctoral Thesis (in Portuguese).

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Cuboctahedron, CO

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Icosidodecahedron, ID

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[1] Coxeter, H. (1973).Regular Polytopes. New York: Dover Publications. p. 17.

* The software used is Rhinoceros (version 6.0)