Five points: four corners and a center — the dots on a die, the pegs of a board. Drop enough beans through it and chaos becomes a curve.
Each bead meets a peg, goes left or right on the toss of a fair coin, and forgets it ever happened. No bead can see the bins; none is trying to reach the middle. Yet the only way to land at an edge is to flip the same way every single time, while the center is the overwhelmingly common muddle of roughly-half-and-half — so the center fills first and fastest. The curve drawn over the heap was never aimed at. It is simply what a crowd of independent coin flips cannot help but become. Click the board to pour ten more.
The Roman as was divided into twelve unciae. A quincunx was worth five of them — quinque plus uncia — and stamped with five pellets in the now-familiar pattern: four at the corners, one in the middle. The same five you see on a die. The word was a quantity before it was ever a shape.
Romans planted orchards in the quincunx so trees packed tight and lined up along every diagonal. In The Garden of Cyrus, Sir Thomas Browne chases the pattern through gardens, plants, and the heavens in a book of magnificent obsession — the five-point lattice as a secret signature of nature.
Galton wanted to show the normal distribution as a physical fact, so he built a board of staggered pegs — a quincunx — and poured shot through it. Each pellet bounces left or right at every peg and lands in a bin. The pile that forms is a bell curve, every single time. A statistical law you can hold.
Each peg is an independent fair choice; a bin is just the running total of rights minus lefts. The Central Limit Theorem says sums of many independent little randomnesses converge on the normal distribution — and the board is that theorem, animated. No bean is steered. The shape is not optional.
Watching beans, then heights of fathers and sons, Galton named regression to the mean: the children of very tall parents tend to be tall, but less so. The pull toward the center that the quincunx makes obvious turned out to govern heredity, test scores, and every season's breakout performer.
The game-show drop board is a quincunx. So is a pachinko machine, and the five-pip face of a die, and the corner-plus-center of a domino. Every time you have seen four-around-one, you have seen the pattern the Romans counted and Galton poured shot through.
Simulations sum random draws and trust the bell to appear. Instruments scatter their readings into one. The normal distribution is the ground floor of statistics and machine learning alike — and the oldest demonstration of it is still a frame of pegs, doing nothing but letting beans fall.