Each face in a quasiregular polyhedron is surrounded by faces of another type and all the dihedral angles are equal. Quasiregular uniform polyhedra are isogonal and isotoxal and their vertex figures are equal, cyclic, and equiangular . 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 uniform quasiregular polyhedra: the cuboctahedron and the icosidodecahedron, and their equatorial polygons are, respectively, regular hexagons and regular decagons. This page shows how to model them.
The following shows 3D models and a short description of the 14 concave uniform quasiregular polyhedra (the missing models will be available asap).
The great dodecadodecahedron has 12 pentagrammatic and 12 pentagonal faces and results from the intersection of the small stellated dodecahedron and the great dodecahedron with a common reciprocation sphere. The suffix dodeca + dodeca derives from the fact that it has 12 concave pentagonal faces + 12 convex pentagonal faces. The edges of the 24 faces of the great dodecadodecahedron are shown (as pipes) in a different colour, and the remaining intersections are not edges. The convex hull of the great dodecadodecahedron is the icosidodecahedron.
The great icosidodecahedron has 12 pentagrammatic and 20 triangular faces and results from the intersection of the great stellated dodecahedron and the great icosahedron with a common reciprocation sphere. The convex hull of the great icosidodecahedron is the icosidodecahedron.
The following images show how its faces are distributed. From left to right:
- 12 pentagrammatic faces (in green) and 20 triangular faces (in gray)
- 12 pentagrammatic faces
- 20 triangular faces
- one set of 10 triangular faces
- another set of 10 triangular faces:
Small Ditrigonal Icosidodecahedron
The small ditrigonal icosidodecahedron has 12 pentagrammatic and 20 triangular faces.
In each vertex, 3 pentagrams and 3 triangles intersect and the term ditrigonal means that the vertex figures are hexagons with equal alternated angles, in this case, irregular and cyclic .
Great Ditrigonal Icosidodecahedron
Tetrahemihexahedron or Tetratrihedron
Octahemioctahedron or Octatetrahedron
Cubohemioctahedron or Hexatetrahedron
Small Icosihemidodecahedron or Icosihexahedron
Small Dodecahemidodecahedron or Dodecahexahedron
Notable descriptions of concave quasiregular polyhedra are found in:
- Coxeter, H. (1973). Regular Polytopes. New York: Dover Publications.
- Cundy, H., and Rollett, A. (1989). Mathematical models. Norfolk: Tarquin Pub.
- Wenninger, M. (1975). Polyhedron models. Cambridge: University Press.
My doctoral thesis includes a subchapter on the quasiregular polyhedra (in Portuguese) between pages 42-48.
- Viana, V (2020). Aplicações didácticas sobre poliedros para o ensino da geometria / Didactic Applications on Polyhedra for the teaching of Geometry (http://hdl.handle.net/10348/10337) [Tese de Doutoramento, Universidade de Trás-os-Montes e Alto Douro]. Repositório Científico da UTAD.
- Coxeter, H. (1973).Regular Polytopes. New York: Dover Publications. p. 17.
- Coxeter, H., Longuet-Higgins, M., & Miller, J. (1954). Uniform Polyhedra. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 246(916), 401-450.