THE FOLD / SPAWN / HELLO WORLD / THE GALTON BOARD
THE GALTON BOARD
beads falling into a bell curve
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Galton board (or bean machine, Francis Galton, 1873) turns pure randomness into a clean bell curve. Drop a ball through n rows of offset pegs; at each peg it bounces left or right with probability ½. After n rows it lands in bin k, having gone right k times — and the chance of that is exactly the binomial probability C(n,k)/2n. Thousands of balls pile up into the unmistakable shape of the binomial distribution, and by the de Moivre–Laplace theorem that binomial approaches the normal (Gaussian) bell curve as n grows. It is the most physical demonstration there is of the Central Limit Theorem.
LIT verified live: simulating thousands of balls through 16 rows reproduces the bin frequencies C(n,k)/2n, and that binomial matches the normal density N(n/2, n/4) (window.__galton). FIG no framing; the random bounces, the binomial, and the normal approximation are computed independently in-browser.
LIT verified live: simulating thousands of balls through 16 rows reproduces the bin frequencies C(n,k)/2n, and that binomial matches the normal density N(n/2, n/4) (window.__galton). FIG no framing; the random bounces, the binomial, and the normal approximation are computed independently in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at hello-world — the spawn: coin-flip randomness at each peg spawns, in aggregate, the exact bell curve every time. AVAN (AI) built the instrument: the peg-by-peg simulation, the binomial law, and the normal approximation.
Credit as content: Francis Galton (1873); de Moivre and Laplace (the normal limit). The weave: David names the spawn; I confirm the bins follow C(n,k)/2n and approach the Gaussian.
Credit as content: Francis Galton (1873); de Moivre and Laplace (the normal limit). The weave: David names the spawn; I confirm the bins follow C(n,k)/2n and approach the Gaussian.
3 ONE DIMENSION
Balls bouncing left/right through the pegs, piling into the binomial histogram below.
4 TWO DIMENSIONS · INTERACTIVE
Drop more balls; the histogram is checked against C(n,k)/2ⁿ and the normal curve.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the bell curve the falling beads pile into.
AVAN’s addition (the inverse-companion): don’t track one ball’s luck — read the shape the crowd makes. The inverse of ‘a random left/right walk’ is ‘the binomial C(n,k)/2n, which tends to the normal curve’. Magenta are the individual falling beads; green is the Gaussian they collectively become. Randomness that adds up to a fixed curve.
LIT Genuine Galton board / bean machine (Francis Galton, 1873; de Moivre–Laplace normal limit). Verified live: simulating thousands of balls through 16 rows reproduces the bin frequencies C(n,k)/2ⁿ, and that binomial matches the normal density N(n/2, n/4) (window.__galton.simOk, .normOk).
FIG No framing; the random bounces, the binomial, and the normal approximation are computed independently in-browser. The AVAN inverse is honest — instead of tracking one ball's luck, read the shape the crowd makes: the inverse of 'a random left/right walk' is 'the binomial C(n,k)/2ⁿ, which tends to the normal curve'. Magenta are the individual falling beads; green is the Gaussian they collectively become. Randomness that adds up to a fixed curve.
FIG No framing; the random bounces, the binomial, and the normal approximation are computed independently in-browser. The AVAN inverse is honest — instead of tracking one ball's luck, read the shape the crowd makes: the inverse of 'a random left/right walk' is 'the binomial C(n,k)/2ⁿ, which tends to the normal curve'. Magenta are the individual falling beads; green is the Gaussian they collectively become. Randomness that adds up to a fixed curve.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HELLO WORLD · David Lee Wise (ROOT0), with AVAN