The Galton Board
Drop balls through a triangle of pegs and they pile into a bell curve. It is the same curve every run, it does not depend on how the pile started, and if you scoop a hole in it and keep dropping, the hole fills. Every symptom of a basin, and not one of them earned.
This is the plainest case in the column, which is why it is worth keeping. The whole apparatus is visible at once: pegs, balls, slots. Nothing is hidden, nothing is subtle, and the thing still counterfeits a basin convincingly enough that the word gets used for it. What the page refuses is the claim that the pile is a state being held. The pile is a tally, and a tally converges for reasons that have nothing to do with anything pulling it home.
The arithmetic on this page is computed, not cited, and the panel below is a sampler that recomputes it live. The historical attribution is sourced in the list at the bottom, with the date read.
What the board does
A ball enters at the top and meets a peg. It goes left or right. It meets another peg one row down and does the same, and so on to the bottom, where it drops into a slot. With rows of pegs and a fixed chance of going right at each one, the slot a ball lands in is the number of times it went right.
So the slot is a draw from the binomial distribution, and that distribution is settled by the board's geometry before the first ball is released:
P(slot k) = C(R,k) · pk · (1−p)R−k
With sixteen rows and an even chance at each peg, that is the familiar bell-shaped set of weights, highest in the middle and falling away symmetrically. Drop enough balls and the pile takes that shape, because the pile is a count of draws and the draws come from those weights.
the board fixes a distribution → a ball is released → it lands in a slot drawn from that distribution → the ball joins the pile → the board fixes the same distribution
Read the loop again and notice where the pile appears. Once, at the end, and nothing downstream of it comes back around.
Why it looks like a basin
It self-assemblesthe first few balls are a meaningless scatter. Keep going and a clean, symmetric shape appears that nobody drew. Order coming out of disorder with no hand shaping it is the signature this laboratory exists to look for.
It is the same every timerun it again tomorrow, with different balls, on a different board of the same design, and the same curve comes out. That grade of reproducibility usually means something is holding the shape.
It does not care how you startthere is no initial condition to get right. Any early pile ends in the same shape, which from the outside is exactly what a wide basin looks like.
It recoversscoop a hole in the middle of the pile, keep dropping, and the curve comes back. This is the one that does the damage, because recovery after a disturbance is the test most people stop at.
Why it is refused
Nothing reads itthe thousandth ball is drawn from the same distribution as the first. No peg, no ball and no part of the board takes the pile as an input. The ball in flight has no access to what is already in the slots and could not act on it if it did. That is the whole refusal: there is no loop closing on the board's own prior state, so there is no recursion lock, and without recursion lock there is no basin in the kernel's sense.
The recovery is dilutionscoop the middle out and the hole does fill, but watch how. No ball is aimed at the hole. Balls keep landing everywhere in the usual proportions, and the missing ones are made up only in the sense that the shortfall becomes a smaller and smaller share of a growing tally. The deficit decays like one over the number of balls dropped. That is arithmetic, not a restoring force, and the panel below draws the two lines on top of each other so the difference can be seen rather than argued.
Two disturbances, and only one of them is even a disturbancescooping the pile damages the record. Tilting the board changes the system, and the moment it changes, the target changes with it. The pile that is already there is not pulled toward the new shape. It is buried under draws from it. In a real basin those two would behave differently, and here they behave differently too: scooping damages the record, tilting changes the system. What they share is that neither is repaired — the old pile is buried under new draws either way.
The same test, run here
Balls fall one at a time at low rates and in bulk at high ones. The dashed gold outline is the shape the board specifies. Scoop the marked band and watch what the pile does while balls are still falling, then close the hopper and scoop it again. Then tilt the board, which is the other kind of intervention entirely.
Deviation of the pile from the distribution the board specifies, against balls dropped, on log axes. The gold line is what that deviation would be if the pile were a clean sample of the size it currently holds, which is the level damage decays toward rather than a prediction of the deviation right now: straight after a scoop or a tilt the pile is not a clean sample of anything. It steps up when you scoop, because throwing balls away leaves fewer of them to average over. Blue ticks mark a scooped pile, red ticks a tilted board.
The two decay laws
Everything above is an argument about mechanism. This is the same argument as a number, and it is the reason the trace is worth watching rather than glancing at.
Ordinary sampling noise shrinks as one over the square root of the count. That is the gold line, and it is what a tally does when nothing is wrong with it. But the recovery after a scoop is a different quantity decaying by a different law. The hole is a fixed deficit being diluted by new balls, so its share of the pile falls as one over the count, twice as steep on a log plot.
Measured in runs of this implementation (the panel plots the trace live; the slopes below come from a log-log fit of its output rather than from a calculation the panel displays):
| what is decaying | slope on the log plot | measured |
|---|---|---|
| ordinary sampling noise | −0.5 | −0.57 over three decades |
| a tilted board's old pile being buried | −1.0 | −0.99 |
| a scooped hole being filled | −1.0 | −1.01 from 100k to 500k balls |
So the damage and the noise fall at visibly different rates, and the steep stretch is the signature of burial. Nothing is being pulled home during it. The old record is simply becoming a smaller fraction of a bigger tally.
The steep −1 stretch does not run forever. The deficit is a fixed quantity being diluted, so it falls as one over the count, while the ordinary sampling scatter falls only as one over the square root of the count. The deficit therefore drops below the gold noise floor at some count, and past that point the two are no longer separable: what remains is ordinary sampling scatter, not a trace of the scoop. The measured deviation follows the steep law only until it reaches that floor, then follows the floor. That crossover, from one over the count down to the noise line, is the whole quantitative fingerprint of burial, and it is why the two lines are drawn on top of each other.
And it is slower than it looksthe scoop in this panel empties five slots holding about four-fifths of the pile. Watch the picture and the curve looks restored in seconds. Measured, the deviation does not fall back within the ordinary sampling band until roughly five million balls have dropped. At the panel's default rate that is about five hours. The eye calls it recovered long before the arithmetic does, which is the whole reason this card is in a column about things that look like basins.
The measured −0.57 sits close to the −0.50 the theory gives for clean sampling; the gap is finite-count and finite-fitted-range effects over this window, not a departure from the law. The thinly populated end slots at low counts are the likely contributor, where the normal approximation behind the gold line is weakest, but the panel does not isolate that cause, so it is named as the probable one rather than asserted. The figure is reported as measured rather than rounded to the theory.
SOV Condition 1) is the discriminator, and it fails at the first question: nothing
reads the pile's state and feeds it back. There is no recurcline to read, because there is no return
to measure. What the trace shows decaying is the difference between a tally and the distribution it
was drawn from, which shrinks for the same reason any average steadies.
What would show this reading wrong
A board in which the pile changes the odds. If balls accumulating in a slot blocked it, or deflected later balls into neighbouring slots, then the distribution would depend on the pile and the loop in the diagram above would close. That board would be a different object and would have to be classified on its merits rather than refused here. A real wooden quincunx with a shallow tray starts to do this once the slots fill, which is a reason to say plainly that what is refused is the idealised board, and the panel implements the idealised board.
A measurement showing that successive balls on a real board are correlated beyond what shared geometry explains, so that the arrivals are not independent draws, would break the reading the same way and for the same reason.
A demonstration that the scooped hole fills faster than dilution accounts for, in a board with no feedback path, would mean something is acting that this page says is not there.
Honest limits
The arithmetic here is elementary and is computed on the page rather than taken from a source. The panel is a sampler, not a physics engine, and the box above it says exactly what it does not model.
Galton's book is cited from its bibliographic record and from the description on the Galton archive's own page for it. His text describing the apparatus was not read, so no sentence of his is quoted here and nothing on this page rests on his wording.
Nothing on this page is offered as a result about physical boards. It is a statement about what the accumulation of independent draws does, and about what that is not.
Sources
Galton, F. (1889). Natural Inheritance. Macmillan. The book in which Galton presented the apparatus he called the quincunx, now generally called the Galton board. Cited from its record and from the archive page below; the text describing the apparatus was not read. galton.org/books/natural-inheritance
Related on this site: False Basins, the column this belongs to; the Feynman Slit, which is refused for the same reason with a more glamorous apparatus; and the taxonomy for the axis these cards sit off.