Folding and cutting spheres.

Awesome symmetry balls include card constructions by George Hart, which you can learn about here (and others are on georgehart.com): http://youtu.be/YBUEYrzMijA
and the Gardner Ball by Oskar van Deventer with design by Scott Kim made in honor of Martin Gardner, which you can learn about here: http://youtu.be/rVHXlltXIlI
and the Wolfram rhombic hexacontahedron thingy, and others!