Pascal's Triangle

Pascal's triangle is the triangular array of binomial coefficients in which each number is the sum of the two directly above it. Though it is centuries old and named for Blaise Pascal, it was studied across multiple civilizations and is far more than a convenient bookkeeping device for combinatorics. When the odd entries are shaded, the triangle produces a discrete approximation of the Sierpinski triangle, revealing that the same scaling structure governs both arithmetic and geometry. The deeper significance of Pascal's triangle lies in its role as a bridge between discrete mathematics and the continuous phenomena of fractal geometry and probability theory. Its entries encode the coefficients of the binomial theorem, the rows converge to the Gaussian distribution, and its patterns have been rediscovered independently in Chinese mathematics centuries before Pascal.