Quasiregular Polyhedron - Nonconvex Examples

Nonconvex Examples

Coxeter, H.S.M. et al. (1954) also classify certain star polyhedra having the same characteristics as being quasiregular:

Two are based on dual pairs of regular Kepler–Poinsot solids, in the same way as for the convex examples.

The great icosidodecahedron and the dodecadodecahedron :

Regular Dual regular Quasiregular Vertex figure

great stellated dodecahedron
{5/2,3}


3 | 2 5/2
great icosahedron
{3,5/2}


5/2 | 2 3
Great icosidodecahedron


2 | 3 5/2
3.5/2.3.5/2

Small stellated dodecahedron
{5/2,5}


5 | 2 5/2
Great dodecahedron
{5,5/2}


5/2 | 2 5
Dodecadodecahedron


2 | 5 5/2
5.5/2.5.5/2

Lastly there are three ditrigonal forms, whose vertex figures contain three alternations of the two face types:

Polyhedron Vertex figure

Ditrigonal dodecadodecahedron
3 | 5/3 5
(5.5/3)3

Small ditrigonal icosidodecahedron
3 | 5/2 3
(3.5/2)3

Great ditrigonal icosidodecahedron
3/2 | 3 5
((3.5)3)/2

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