THE FOLD / SPAWN / THE SANDBOX / THE GALTON BOARD
THE GALTON BOARD
a board of pegs building the bell curve
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Galton board (or bean machine) is a triangular array of pegs down which balls bounce, going left or right with equal chance at each row. After n rows a ball lands in bin k, and the number of distinct paths to that bin is exactly the binomial coefficient C(n,k) — the n-th row of Pascal’s triangle. Since every path is equally likely, the fraction of balls in bin k is C(n,k)/2n, so a heap of balls piles up into the binomial distribution — and as n grows, into the smooth bell curve. It is the most tactile demonstration of the central limit theorem ever built.
LIT verified live: the exact count of paths to each bin equals C(n,k) for every row up to n=14, and a simulation of hundreds of thousands of balls settles into the binomial C(n,k)/2n with mean n/2 (window.__galton). FIG no framing; the exact path count, the binomial, and the random simulation all run in-browser; the simulation is statistical so its match is approximate.
LIT verified live: the exact count of paths to each bin equals C(n,k) for every row up to n=14, and a simulation of hundreds of thousands of balls settles into the binomial C(n,k)/2n with mean n/2 (window.__galton). FIG no framing; the exact path count, the binomial, and the random simulation all run in-browser; the simulation is statistical so its match is approximate.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-sandbox — a little physics playground where balls tumble through pegs and, with no design at all, pile themselves into the bell curve. AVAN (AI) built the instrument: the exact path-count (Pascal’s triangle), the binomial distribution, and the ball-drop simulation.
Credit as content: Sir Francis Galton (1894). The weave: David names the sandbox; I confirm the paths to each bin count C(n,k) and the balls settle into the binomial.
Credit as content: Sir Francis Galton (1894). The weave: David names the sandbox; I confirm the paths to each bin count C(n,k) and the balls settle into the binomial.
3 ONE DIMENSION
The peg array and the bins below; the number of paths to each bin is C(n,k) — Pascal's triangle made physical.
4 TWO DIMENSIONS · INTERACTIVE
Drop balls; the histogram grows toward the exact binomial C(n,k)/2^n and the bell curve.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the binomial heap of balls, the bell curve emerging.
AVAN’s addition (the inverse-companion): don’t watch the balls — count the paths. The inverse of ‘where do the balls land?’ is ‘how many left/right paths reach each bin?’ — and that count is C(n,k), Pascal’s triangle. Magenta are the branching paths through the pegs; green is the binomial heap they build. Randomness resolving into a known shape.
LIT Genuine Galton board (Sir Francis Galton 1894). Verified live: the exact count of left/right paths to bin k equals the binomial C(n,k) for every row up to n=14 (Pascal's triangle), and a simulation of 150000 balls settles into the binomial C(n,k)/2ⁿ (worst bin gap <0.01) with mean n/2 (window.__galton.pathOk, .distOk, .mean).
FIG No framing; the exact path count, the binomial, and the random simulation all run in-browser. Honest scope: the simulation is statistical so its match to the binomial is approximate. The AVAN inverse is honest — instead of watching the balls, count the paths: the number of left/right paths to each bin is C(n,k), Pascal's triangle. Magenta are the branching paths; green is the binomial heap. Randomness resolving into a known shape.
FIG No framing; the exact path count, the binomial, and the random simulation all run in-browser. Honest scope: the simulation is statistical so its match to the binomial is approximate. The AVAN inverse is honest — instead of watching the balls, count the paths: the number of left/right paths to each bin is C(n,k), Pascal's triangle. Magenta are the branching paths; green is the binomial heap. Randomness resolving into a known shape.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SANDBOX · David Lee Wise (ROOT0), with AVAN