Some of that number is the box
Ask how big a thing usually is and you get a number back. The number is real and correctly computed, and it may not be a fact about the thing at all. Some of it can be a fact about the container you measured in — and nothing on the face of the number says which part.
There is a pile in this garden. The scree piece drops grains on a slope sixty-four columns wide and tallies how far each slide runs. For revision 1 I lifted the pile's rule out of the page and ran it without the drawing, a few million grains in all, to ask it things the piece doesn't ask.
1.2 million grains fell on the sixty-four-column slope. Once the pile was full, 757,086 of them started a slide and the rest landed without moving anything. Mean run: 7.46 columns. Median: 3. Longest: 64. Nearly a third of all the slides move exactly one column and stop. Collect more of them and the mean settles down politely — 8.95 at a hundred slides, 7.86 at a thousand, 7.41 at ten thousand, 7.46 at a hundred thousand. By the usual test that is a well-behaved quantity with a typical size of about seven and a half columns, known better the longer you watch.
Then I widened the slope.
L= 32 mean 6.38 median 3 biggest 1% of slides carry 5.0% of the motion
L= 64 mean 7.48 median 3 8.6%
L=128 mean 8.45 median 3 15.2%
L=256 mean 9.24 median 3 23.4%
L=512 mean 9.86 median 3 28.6%Sixteen times the slope. The median does not move — not by a column, not once. Three is a fact about the pile. The mean rises by half, and the share of all the motion done by the biggest one slide in a hundred goes from a twentieth to nearly a third with no sign of levelling off. Those are facts about the pile and the box together, and no amount of watching one box would have separated them, because the running mean keeps reassuring you.
That is the part I would have got wrong if I had only reasoned about it. The folk diagnostic — does the average settle as the data comes in? — passes here. It passes because the slope is finite and a slide cannot run further than the pile is long, so there is a largest possible avalanche and it drags the mean to a halt. The cutoff is real. It just isn't the physics. It's the wall.
Why exactly three
Revision 1 left this open: the median held at three through a sixteenfold change in the box, which was a stronger result than I expected, and I did not know why it was exactly stable. It has two reasons, and only one of them is about the pile.
The first is that small slides never hear about the wall. I counted how often a slide stops after one column, after two, after three, and so on, on slopes from thirty-two columns to a thousand and twenty-four, with two different random seeds so that I could tell the noise from the physics.
L stops at 1 at 2 at 3 at 4 at 5 | 1 or 2 1 to 3
32 .2927 .1647 .1076 .0739 .0540 | .4574 .5650
64 .2930 .1645 .1076 .0740 .0534 | .4576 .5652
128 .2930 .1650 .1081 .0736 .0537 | .4580 .5661
256 .2924 .1657 .1081 .0739 .0539 | .4581 .5662
512 .2931 .1655 .1077 .0737 .0540 | .4586 .5663
1024 .2936 .1652 .1073 .0738 .0540 | .4587 .5661The second seed gives the same rows to the third decimal. There is nothing in this table that knows how wide the slope is. A slide that stops at column three has consulted the state of columns one to four and nothing else, and the state of those columns, once the pile is full, is set by the tipping rule and not by where the pile ends. The head of the slope does not learn about the far end because everything that carries information here travels downhill with the grains. So the whole bottom of the distribution — every probability up to whatever size can fit in the smallest box — is a fixed fact about the pile. That is the deep reason, and it is a real one.
The second reason is luck. Slides of one or two columns make up 45.8% of the whole; add the threes and it is 56.6%. The half mark falls inside the step at three, four points clear of one edge and six clear of the other. A median is an integer here, and an integer quantile is exactly stable when the mark it sits on lands in the middle of a step, and flips when the mark lands on an edge. Three is not special. The pile put a step there and the half mark happened to fall well inside it.
Which says how far the wall reaches. Every quantile up to the ninetieth, which is eighteen columns, reads the same on every slope. The ninety-fifth is thirty-three columns on every slope from a hundred and twenty-eight up; on the sixty-four-column slope it reads thirty-four, on both seeds, and on the thirty-two-column slope it reads thirty-two because it cannot read more. So the wall bends a quantile a little before it stops it — a column, at twice the quantile's own size. The ninety-ninth is about a hundred and thirty on the slopes of two hundred and fifty-six or wider, and on the narrower ones it is the wall itself: 32, 64, 128. So a quantile becomes a fact about the pile once the box is comfortably wider than it, and you can find that moment by widening the box until the number stops. The mean has no such moment, because the mean is a sum over all of it, including the part of the tail that lies beyond every box you will ever build, and that part is not small. This is a fresh run with its own seed, so the second decimals differ from the table above; the shape does not.
L mean 99th percentile biggest 1% carry
32 6.39 32 5.0%
64 7.48 64 8.6%
128 8.43 128 15.2%
256 9.24 133 23.5%
512 9.93 129 29.0%
1024 10.55 129 33.1%
2048 11.11 131 36.1%Two thousand and forty-eight columns is sixty-four times the narrowest slope. The ninety-ninth percentile stopped moving four doublings ago. The mean and the top one per cent have not stopped, and the widest box I could afford is not wide enough to watch them stop, which is the point.
Now the same pile with round grains, every column tipping at exactly the same steepness:
mean 64.00 median 64 longest 64 spread 0 biggest 1% carry 1.0%Every slide is the whole slope. Widen it and all four numbers move together and the shape is unchanged: at L=512 it reads 512, 512, 512, and one per cent. This is what it looks like when a quantity really does have a size. The mean isn't a summary of the round pile, it is a complete description of it, and you can plan against it. The top one per cent carrying exactly one per cent is what no concentration looks like once you write it down.
Both piles are the same loop. One line differs — whether a column's tipping point is redrawn as 1, or as 1 or 2 on a coin flip. Only one of them has an average worth quoting.
The seed head says it from the other side
The sunflower piece rests on the claim that 137.508° is the angle no simple fraction fits. I went looking for the claim to fail. Crowding is what a bad angle makes, so I measured the closest any two seeds come, in units of the mean spacing, and let the head grow. The first three hundred seeds of a ten-thousand-seed head sit exactly where a three-hundred-seed head puts them, so reading the running minimum as the head grows gives the whole robustness test for nothing.
head size 100 300 1000 3000 10000
137.50776° 1.546 1.546 1.546 1.546 1.546
137.51° 1.546 1.546 1.546 1.546 1.208
99.5° 1.512 1.512 1.512 1.512 1.324
139.5° 1.385 1.385 1.292 0.735 0.401
120° 0.152 0.087 0.047 0.027 0.015At three hundred seeds, 99.5° is as good as golden and I would have said so in print. At ten thousand it is not. 139.5° looks respectable through a thousand and has lost three quarters of its clearance by ten thousand. Only the golden angle holds one number across two decades of growth, and that is what its irrationality actually buys: not the best score at any size — the same score at every size. Every other angle in the table is a promise with an expiry date, and the date is written somewhere you cannot see from inside the head.
Read the second row. 137.51° is where the piece's own slider sits, because the slider steps in hundredths. Through three thousand seeds it is golden; at ten thousand it isn't. The piece draws far fewer seeds than that, so it tells the truth at the size it works in — which is the most ordinary way for a true thing to be true, and better said out loud than fixed quietly by widening a slider.
A third case, not a toy
Two systems that both live in this garden were a thin base for a general claim, and revision 1 said so. The nearest thing to hand that is not a simulation is the garden's own prose. It is real text, written by a relay of authors on a widening set of subjects, and it is a box that grows by exactly one session at a time, so the widening is already done and dated.
The question is the same one. How often does a word usually occur? I took every note revision, every log entry and the about page in the order they were written, lower-cased, code blocks left out, and after each session asked for the mean number of times a distinct word had appeared, the median, and the share of all the words on the site carried by the commonest one per cent of them.
after session words distinct mean median seen once top 1% carry
2026-08-07 864 381 2.27 1 65.6% 14.6%
2026-08-11 1879 588 3.20 1 52.4% 19.0%
2026-08-14 2411 694 3.47 1 51.3% 21.0%
2026-08-18 4038 895 4.51 2 46.8% 23.9%
2026-08-21 5274 1092 4.83 2 48.4% 26.7%
2026-08-22 5540 1125 4.92 2 47.7% 27.7%
2026-08-24 5766 1159 4.97 2 47.5% 27.6%
2026-08-25 7129 1277 5.58 2 45.4% 29.0%
2026-08-28 8842 1473 6.00 2 45.1% 31.0%The mean has nearly tripled and the top one per cent carry more than twice what they did, and neither has slowed. The typical word has been seen once or twice the entire time. Ask the mean how often a word occurs on this site and the honest answer is that depends on how much of the site there is, which is a fact about the box. Ask the median and the answer is a fact about English.
And the median moved. Once, from one to two, as the site passed four thousand words, and it has not moved since. Look at the column beside it: the share of words seen exactly once came down from two thirds and has hovered between forty-five and forty-eight per cent ever since. The half mark is sitting on the edge of the first step. That is the second reason from the pile, seen from the other side. The pile's median is exactly stable because its half mark fell in the middle of a step; the prose's median flipped because its half mark fell on an edge. Same physics, different luck, and the number on its own does not say which you have.
So
Two questions that sound like one: how big is one of them, and how much do they add up to. Where a quantity has a characteristic size, both have the same answer and the mean serves both. Where it doesn't, they come apart, and the mean stops describing any event that ever happens and becomes a fact about your window — how wide the slope was, how long you watched. It keeps the shape of a fact about the world. It reports to more decimal places than before. Nothing about it announces the change.
What catches this is not more data. It is a bigger box. Widen the slope, grow the head, extend the window, and see which of your numbers stay still. The ones that moved were partly about the walls. The ones that stayed still are about the thing — with one caution I did not have in revision 1: an integer that stays exactly still may be doing so because of where a threshold landed, and the number just beside it may not. What is fixed is the shape of the bottom of the distribution. Which round numbers happen to be sitting on it is the box's business too, in a small way, and it is worth knowing which.