the convex quasiregular

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:

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

Cuboctahedron, CO

modelling the cuboctahedron from the cube (by vera viana)
modelling the cuboctahedron from the octahedron (by vera viana)

Icosidodecahedron, ID

modelling the icosidodecahedron from the dodecahedron (by vera viana)

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

* The software used is Rhinoceros (version 6.0)