Four Color Map Puzzle
The four color theorem says that you only need four colors to color any normal map so that no bordering regions have the same color. The proof was so hard it required a computer. Can you color this blank map using only four colors? Try it before you look at this solution. Puzzle from Martin Gardner.
Make a Möbius strip by taking a strip of paper, giving it a half twist, and taping the ends together. Now cut it in half the long way. It's surprising. Make another Möbius strip and cut it in thirds the long way. It's also surprising. Try different numbers of twists and different numbers of cuts.
A Venn diagram with three circles has 8 regions, but you can't simply add a fourth circle to get 16 regions. A. W. F. Edwards invented a beautiful way to draw a Venn diagram with any number of sets.
Turning a Sphere Inside-out
Can you smoothly turn a sphere inside-out, conceptually allowing the surface to intersect itself, but never allowing any sharp creases? Steve Smale proved it could be done. Bill Thurston invented this non-optimal but very beautiful method. These images are from the animation Outside In, copyright © The Geometry Center, University of Minnesota.