I've recently been interested in crocheted mathematical objects.
The easiest one to understand is the Möbius (or Moebius) strip. You can make a Möbius strip with a long thin rectangular piece of paper. Hold the two ends, give the paper a twist, then tape the two ends together. The cool thing about the Möbius strip is that it has no inside or outside. That is, the strip has only one side, even though it looks like it has two. Using a pencil, start at any point on the strip, follow the strip around, and you'll touch each side of the strip and end up back at your starting point. A good illustration of a Möbius strip is M.C. Escher's Ants. A Möbius strip makes a great scarf or shawl. It has a natural drape that allows it to rest nicely on your shoulders. I've made this pattern, although I used Lion Brand Homespun.
Another mathematical concept that illustrates the "no inside or outside" principle is the Klein bottle. Here's a pattern for a crocheted Klein bottle. This knit Klein hat gives a good explanation of a Klein bottle, but of course we need a crocheted Klein hat. While you're looking at the crocheted Klein hat, just for fun, look at the rest of this cute guy's website.
Fractals turn up in crochet, too. This pattern makes the Sierpinski triangle.
Crochet can also be used to construct a hyperbolic plane. A pattern for a hyperbolic plane is here.
And for a seriously dedicated mathematician/crocheter, check out the Lorenz equations crocheted by Dr. Hinke Osinga!