Pascal's Triangle — Hidden Depths

Blaise Pascal described his triangle in 1653, but it had appeared centuries earlier in China (Yang Hui's triangle, 1303), Persia, and India. Each number is the sum of the two above it. The triangle contains the binomial coefficients, the Fibonacci numbers (in diagonal sums), powers of 2 (row sums), powers of 11, and the Sierpiński triangle pattern when you shade all odd numbers. It is one of the most information-dense objects in elementary mathematics.