Pascal's Triangle (mod n)
Colour by divisibility — a fractal falls out
Write Pascal's triangle. Now colour every entry by its remainder modulo a prime p. For p = 2 (odd cells black, even cells white) the result is the Sierpiński triangle — exact, infinite, generated by counting. For p = 3, 5, 7 each you get a different self-similar gasket. The theorem behind it (Kummer, 1852) says C(n, k) is divisible by p exactly when the base-p addition k + (n − k) has at least one carry — so the fractal is, secretly, a picture of when carries happen.
