the concave quasiregular

introduction

Each face in a quasiregular polyhedron is surrounded by faces of another type and all dihedral angles are equal. Quasiregular uniform polyhedra are isogonal and isotoxal and their vertex figures are equal, cyclic, and equiangular [1: 17]. Some quasiregular polyhedra admit a system of equatorial polygons, each of which is inscribed in a great circle of the circunsphere.

Only 2 convex uniform quasiregular polyhedra exist: the cuboctahedron and the icosidodecahedron, and their equatorial polygons are, respectively, regular hexagons and regular decagons (this page shows how to model them). There are 14 concave quasiregular polyhedra, that may be grouped in the following way:

the great dodecadodecahedron and the great icosidodecahedron, that derive from the intersection of a dual pair of regular stellated polyhedra with the same reciprocation sphere:

the small ditrigonal icosidodecahedron, the great ditrigonal icosidodecahedron and the ditrigonal dodecadodecahedron. They are characterized as ditrigonal because the vertex figures are cyclic hexagons with equal alternate angles [2: 412-413]. In each vertex, six faces of two types intersect. Although quasiregular, they do not admit equatorial polygons:

the remaining quasiregular are hemipolyhedra because in each, a set of faces is diametral, meaning they contain the centroid of the polyhedron. There are 9 of these and their names are the following: tetrahemihexahedron, octahemioctahedron, cubohemioctahedron, small icosihemidodecahedron, small dodecahemidodecahedron, great dodecahemicosahedron, small dodecahemicosahedron, great dodecahemidodecahedron and the great icosihemidodecahedron:

great dodecadodecahedron

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 equatorial polygons of the great dodecadodecahedron are convex hexagons. The video below shows how the great dodecadodecahedron is obtained from the icosidodecahedron through faceting.

great dodecadodecahedron by vera viana on Sketchfab

great icosidodecahedron

the great icosidodecahedron has 12 pentagrammatic and 20 triangular faces and is the result of the intersection of the great stellated dodecahedron with the great icosahedron with a common reciprocation sphere. The equatorial polygons of the great icosidodecahedron are {10/3} decagrams. The video below shows how the great icosidodecahedron is obtained from the icosidodecahedron through faceting.

faces of the great icosidodecahedron from l-r: 32 | 12 | 20 | 10 | 10

great icosidodecahedron by vera viana on Sketchfab

small ditrigonal icosidodecahedron

the small ditrigonal icosidodecahedron has 12 pentagrammatic and 20 triangular faces.
in each vertex, 3 pentagrams and 3 triangles intersect. The video below shows how the small ditrigonal icosidodecahedron is obtained from the pentagonal dodecahedron through facetting.

the term ditrigonal means that the vertex figures are hexagons with equal alternated angles, in this case, convex and cyclic [2], as shown in yellow in the first frame of the video.

small ditrigonal icosidodecahedron by vera viana on Sketchfab

great ditrigonal icosidodecahedron

the great ditrigonal icosidodecahedron has 12 pentagonal and 20 triangular faces and, in each vertex, 6 pentagons and 3 triangles intersect. the video below shows how the great ditrigonal icosidodecahedron is obtained from the pentagonal dodecahedron through faceting.

the term ditrigonal means that the vertex figures are hexagons with equal alternated angles, in this case, isogonal, unicursal, and cyclic [2], as shown in yellow in the first frame of the video.

great ditrigonal icosidodecahedron by vera viana on Sketchfab

ditrigonal dodecadodecahedron

the ditrigonal dodecadodecahedron has 12 pentagonal and 12 pentagrammatic faces and, in each vertex, 9 pentagons and 3 pentagrams intersect. The video below shows how the ditrigonal dodecadodecahedron is obtained from the pentagonal dodecahedron through facetting.

the term ditrigonal means that the vertex figures are hexagons with equal alternated angles, in this case, isogonal, unicursal and cyclic [2], as shown in yellow in the first frame of the video.

ditrigonal dodecadodecahedron by vera viana on Sketchfab

tetrahemihexahedron or tetratrihedron

the tetrahemihexahedron or tetratrihedron is a hemipolyhedron with 4 triangular faces and 3 mutually perpendicular squared faces, that contain the centroid.

tetrahemitetrahedron by vera viana on Sketchfab

octahemioctahedron or octatetrahedron

the octahemioctahedron or octatetrahedron is a hemipolyhedron with 8 triangular and 4 hexagonal faces. the hexagonal faces contain the centroid.

octahemioctahedron by vera viana on Sketchfab

cubohemioctahedron or hexatetrahedron

the cubohemioctahedron or hexatetrahedron is a hemipolyhedron with 6 squared and 4 hexagonal faces. the hexagonal faces contain the centroid.

03 hemi by vera viana on Sketchfab

small icosihemidodecahedron

the small icosihemidodecaedron is a hemipolyhedron in which the vertices and the triangular faces (20) are the same as in the icosidodecahedron. besides those, there are 6 diametral decagonal faces.

Small Icosihemidodecahedron by vera viana on Sketchfab

small dodecahemidodecahedron or dodecahexahedron

the small dodecahemidodecahedron or dodecahexaedron is a hemipolyhedron in which the vertices and pentagonal faces (12) are the same as in the icosidodecahedron. besides these, there are 6 diametral decagonal faces.

small dodecahemidodecahedron by vera viana on Sketchfab

great dodecahemicosahedron

the great dodecahemicosahedron is a hemipolyhedron with 12 pentagonal and 10 regular hexagonal faces. In each vertex, 2 pentagons and 2 hexagons intersect. the hexagonal faces contain the centroid.

great icosihemidodecahedron 02 by vera viana on Sketchfab

the following 3D model shows some faces in a different colour:

great dodecahemicosahedron by vera viana on Sketchfab

small dodecahemicosahedron

the small dodecahemicosahedron is a hemipolyhedron with 12 pentagrammatic and 10 regular hexagonal faces. the hexagonal faces contain the centroid.

small dodecahemicosahedron by vera viana on Sketchfab

great dodecahemidodecahedron

the great dodecahemidodecahedron is a hemipolyhedron with 12 pentagrammatic and 6 decagrammatic faces. In in each vertex, 2 pentagrams and 2 decagrams intersect. the decagrammatic faces contain the centroid.

great dodecahemidodecahedron by vera viana on Sketchfab

great icosihemidodecahedron

the great icosihemidodecahedron is a hemipolyhedron with 20 triangular faces and 6 diametral decagrammatic faces.

great icosihemidodecahedron by vera viana on Sketchfab

Notable descriptions of the concave quasiregular polyhedra are found in:

  1. Coxeter, H. (1973). Regular Polytopes. New York: Dover Publications.
  2. 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.
  3. Cundy, H., and Rollett, A. (1989). Mathematical Models. Norfolk: Tarquin Pub.
  4. Wenninger, M. (1975). Polyhedron Models. Cambridge: University Press.

My doctoral thesis (in Portuguese) includes a subchapter on the quasiregular polyhedra, 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.