A fascinating aspect of mathematics is the study of polyhedra, which are three dimensional geometric solids. There are several simple ones (cube, octahedron, dodecahedron, etc.) and there are many complex ones.
Dodecahedron
(12 sides) 
Icosahedron (20 sides)
Rhombicuboctahedron
Stellated
Icosidodecahedron
Stellated
dodecahedron 
Stellated
Dodecahedron
(marbled paint job)
Stellated Rhombicuboctahedron
Compound of 2 Tetrahedra
Compound of 5 Tetrahedra
Great
Dodecahedron 
Truncated
Icosahedron
Rhombicosidodecahedron
Faceted Cube.
Stellated Truncated Icosahedron
Great
Stellated Dodecahedron
Great Icosahedron (gold and black)
Truncated Octahedron
Compound of 10 Tetrahedra
Stellated Truncated
Icosahedron
Compound of Five Octahedra
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3-D model of Sierpinski's Triangle (not really a polyhedron but still WAY
cool!)
Watch for more polyhedra as they are added.
The number and variation of polyhedra is vast!
How do you make such polyhedra? The best resource book I've found is "Polyhedron Models" by Magnus J. Wenninger, published by the National Council of Teachers of Mathematics (NCTM).
I conduct workshops on these at various math conferences.
(last updated 4-18-2007)