Fig. 1: I made these 60 uniform polyhedra in the later 90's. These are in my parents' house in Portugal. This set includes the
models of the regular and quasi-regular polyhedra displayed in my office, but on a smaller scale.
Fig. 2: Here is a 3-D printed edge model of a complex uniform polyhedron, the Great dirhombicosidodecahedron, also known as "Miller's Monster".
This was the last entry in the 1954 list of uniform polyhedra compiled by J. C. P. Miller, H. S. M. Coxeter and M. S. Longuet-Higgins.
The other remarkable polyhedron that shares this set of edges is the Great
dis-nub dirhombidodecahedron. In 1975, John Skilling proved that the 1954
list of uniform polyhedra is complete. In that paper he generalized the concept
of Uniform polyhedron: instead of two faces per edge, there can be any even
number (or, equivalently, edges can be multiple). This relaxation yields only
one additional polyhedron, the Great dis-nub dirhombidodecahedron.
Finally, the edges also belong to two different uniform
polyhedron compounds: The Compound of
twenty octahedra and the Compound
of twenty tetrahemihexahedra.
Model printed by Shapeways,
design by Mathematical
Creations.
Some of the complex, non-convex polyhedra with icosahedral symmetry.
The image above shows mainly Archimedean
polyhedra with icosahedral symmetry. Among these are two regular
polyhedra: the icosahedron (all in red) and the dodecahedron (all in
dark blue).
The convex Uniform polyhedra: the five regular Platonic polyhedra and the
13 Uniform Archimedean polyhedra.
A severe polyhedral storm!