A polyhedron is uniform if all of its faces are regular polygons, and it is transitive on its vertices ("all the vertices look the same"). It is possible for the faces to intersect and for three faces to cross at a point that is not a vertex. This Demonstration shows eleven nonconvex uniform polyhedra and their nets. The wireframe shows some of the edges of the figure.

In [5], Wenninger recommends making polyhedron models as a classroom activity. Among other polyhedra he chooses 15 nonconvex uniform polyhedra. This Demonstration shows 11 of them; five are in Uniform Polyhedron. He quotes Coxeter [1]: "The chief reason for studying regular polyhedra is still the same as in the time of the Pythagoreans, namely, that their symmetrical shapes appeal to one's artistic sense."

Coordinates of polyhedra were calculated using [3] and [4].

References

[1] H. S. M. Coxeter, Regular Polytopes, New York: Dover, 1973.

[2] P. R. Cromwell, Polyhedra, New York: Cambridge University Press, 1997.

[3] R. E. Maeder, The Mathematica Programmer II, San Diego: Academic Press, 1996.