The most beautiful formula not enough people understand

A lecture-style exploration of the formula for the volume of an n-dimensional ball. Starting from intuitive probability puzzles (what's the chance that x²+y² ≤ 1 for random x,y?), it builds up to the general formula V_n = π^(n/2) / (n/2)! · r^n. Key ideas include: the relationship between a ball's volume and its surface area via derivatives/integrals, an Archimedean projection argument generalized to higher dimensions via a 'knight's move' recurrence, the gamma function extending factorials to half-integers, and the surprising fact that unit ball volumes peak around 5 dimensions and then shrink toward zero in higher dimensions — a counterintuitive result relevant to machine learning, cryptography, and quantum mechanics.