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.
There are only two convex quasiregular polyhedra: 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 some examples on how to model* these two Archimedeans. These and other examples are included in my Doctoral Thesis.
Cuboctahedron, CO (from the cube; from the octahedron)
Icosidodecahedron, ID (from the regular dodecahedron)
[1] Coxeter, H. (1973).Regular Polytopes. New York: Dover Publications. p. 17.
* The software used is Rhinoceros (version 6.0)
