Sierpinski's gasket is a fractal, obtained by repeatedly deleting the middle section of a triangle. It shows up in a number of different areas of mathematics, with surprising cross-connections. The talk will describe four occurrences of the gasket: 1. What Sierpinski originally invented it for; 2. Parity of binomial coefficients; 3. The Tower of Hanoi puzzle; 4. Michael Barnsley's Chaos Game.