A tiring hand draws a different habit

The rows count hands, not cameras found a number in the rows of a divergent picture that no lens can move. Every splayed strip — floor, footstool, table, book — has its interior rows placed where some power of the drawn width runs evenly from the near edge to the far one, and the power, called the habit, is read off the page with no focal length anywhere in the reading. A hand spacing its rows evenly down the page has a habit of 1; a camera drawing bays of equal depth has −1. One hand across four splays gave one number back on every strip, and two hands hidden in one picture gave back exactly two.

That essay’s scatter was the error of a steady hand: every row displaced by its own small slip, nothing carried from one row to the next. It closed on the error it was not protected against. A painter who tires as a strip is divided does not slip at random; the bays creep, each a little wider than the last, or a little narrower. A creep in the spacing is a change in the one quantity the habit is read from, and if the two cannot be told apart, every hand count in a picture drawn on a large panel may be counting fatigue.

This essay measures the creep, and the answer comes in three parts. On any one strip a creep is a change of habit, to a fraction of a pixel. Across a whole picture it is not, and the reason is arithmetic about the strips rather than anything about the painter. And once the picture is read as a whole, one habit and one creep can be recovered together, and a picture drawn by two steady hands refuses to be read as one tired one.

A creep of three per cent a bay #

The model of fatigue has to be stated before it can be measured. The painter rules the strip first — its near edge, its far edge and its two straight sides — and then divides it. Each bay is drawn by the painter’s habit, and then made a fixed fraction taller on the page than the habit alone would make it, relative to the bay before: 1 + d times, where d is the drift. The bays are rescaled together so that they still fill the ruled strip, because the edges were drawn first and do not move. A positive drift lets the far bays grow; a negative one crowds them.

The figure is the table strip — splay 1.32, six bays, drawn 99 px tall — divided at a habit of 1 by a hand that drifts three per cent a bay. The drift moves the rows by up to 2.2 px from where the steady habit would put them, which is a visible amount on a strip that size and more than four times the half-pixel slip the earlier reading allowed a steady hand.

And yet, read back from the page, the rows report a habit of 0.361, and the steady hand at 0.361 puts its rows within 0.014 px of the drifting hand’s. The three sets of rows in the figure are the drifting hand’s, the steady hand the reading returns, and the habit of 1 with no drift at all. The first two cannot be separated at the scale of the figure, and they cannot be separated at any scale a painted panel is measured at. The slider sweeps the drift from −4 to +5 per cent and the story does not change: whatever the drift, there is a steady habit that draws the same strip.

The reason is not that the fit is forgiving. It is that a steady habit and a steady creep make the same kind of change to a strip, and the next measurement makes that exact.

The habit a creep counterfeits #

Under a habit $q$, the bays of a strip shrink or grow down the page by very nearly a constant ratio. A strip of $n$ bays whose far edge is drawn $s$ times as wide as its near one has bays that change by $s^{(1-q)/n}$ from one to the next — exactly so at a habit of 0, where the drawn widths themselves form a geometric sequence, and to first order at every other habit. A drift multiplies that ratio by $1 + d$, and a ratio multiplied by $1 + d$ is the ratio of a different habit:

$$q' = q - \frac{n \ln(1 + d)}{\ln s}.$$

The figure draws each strip of the interior at a habit of 1, with drifts from −6 to +6 per cent a bay, and reads the habit back. The measured points sit on the closed form to 0.014 across the whole range, and the slope of each line is $n/\ln s$: 52.9 on the floor, 21.9 on the footstool, 21.6 on the table and 15.5 on the book.

Those slopes are the essay’s central number. A creep of one per cent a bay moves the book’s reported habit by 0.16 and the floor’s by 0.53. The floor is weakly splayed — its far edge is only 12 per cent wider than its near one — so the family of habits is packed tightly on it: every habit from a camera’s to a hand’s places the floor’s rows within a pixel or two of each other, and a small change in the spacing is a large change in the habit. That is the same fact the rows count hands, not cameras found as a scatter, when half a pixel of slip scattered the habit three times as much on weak splays as on the best strip. A creep is simply a slip that has a direction.

A drift of two per cent a bay, which is well inside what a person drawing freehand does without noticing, moves the floor from a habit of 1 to −0.05. That is the log placement, the habit halfway between a hand’s even rows and a camera’s equal bays. The creep has not merely perturbed the habit; it has taken the floor across half the whole range the family was built to describe.

The habit was never only a description of where the lines went. The rows under a splay measure the bays, not the lean and its sequel found that a habit of $q$ depicts a near bay deeper than the far one by about the splay raised to the power $1 + q$, so a change of habit is a change in the furniture depicted. On the floor, a habit of 1 depicts a near bay 25 per cent deeper than the far one; the habit the tired hand reports, −0.05, depicts one 11 per cent deeper. A reader who takes the reported habit at its word reconstructs a floor whose boards shorten half as fast as the painter’s even rows implied — a different floor, drawn by nobody, produced by a hand getting tired.

Why the rows cannot see it #

A counterfeit that left a mark would be harmless, since the mark would give it away. The measurement that matters is what the drifting hand’s rows leave behind once the best steady habit has been fitted to them.

On the table strip, drawn at a habit of 1 with a drift of two per cent, the worst row sits 0.015 px from the fitted steady habit, though the drift moved the rows 1.47 px from where the habit would have put them. At a habit of 0 the residual is 0.040 px and at −1 it is 0.107. Across the three habits, the residual stays under half a pixel — the slip a steady hand was already allowed — for every drift up to six per cent a bay in either direction.

So there is nothing in a single strip to find, and the reason is structural rather than a matter of precision. One hand step each found that every construction leaves a hand exactly one step to perform and that the finished drawing records only that step’s mistakes; the division of a ruled strip is such a step, and its one-parameter family of habits already absorbs the one-parameter family of creeps. The rows are six numbers; a steady habit is one parameter that places all six, and a steady creep is one parameter that moves all six; and the two parameters move them in shapes so nearly alike that what one leaves the other takes up to a fiftieth of a pixel. The slip that leaves no trace found the same structure in a construction: a distance point put wrong moved the pavement and left the projective test reading exactly correct, because the slip landed back on the set of correct drawings. A drift lands back on the set of steady habits. The reading does not detect it because, strip by strip, it is not an error in the family’s terms at all.

One tired hand reads as two #

The per-strip slopes differ, and that difference is what turns a harmless confusion into a false attribution.

Draw the whole interior with one hand, at one habit, drifting two per cent a bay on every strip — the most uniform fatigue there could be. Read strip by strip, as the earlier essay showed a picture must be read if a split between hands is not to be hidden, the four strips report −0.048 on the floor, 0.565 on the footstool, 0.572 on the table and 0.692 on the book.

A reader looking at those four numbers sees a floor drawn at the log placement and three pieces of furniture drawn at a habit a little above a half: two hands, and a clean split between the floor and everything on it. The earlier essay’s half-pixel scatter was 0.08 to 0.23 on strips like these, so the split of more than 0.6 is well outside it. The attribution would be confident, specific and wrong. Nothing about the picture is inconsistent; the painter did exactly one thing throughout, and the strips reported different numbers because each converts a creep into a habit at its own rate.

A reader who pooled the strips instead — one habit for the whole picture — would get 0.466, leaving 0.48 px of page departure, which is under the two-pixel brush a camera count needs a tolerance charged redrawing against. That reading calls the picture one hand, which is right, and calls its habit 0.47, which is not what the hand did.

The splays are the handle #

The same fact that makes the per-strip reading lie is what lets the whole picture tell the truth. A steady creep of $d$ shifts each strip’s habit by $n\ln(1+d)/\ln s$, and the four strips have four different values of $n/\ln s$. So the four per-strip habits of one tired hand are not arbitrary: they lie on a line against $n/\ln s$, with the hand’s true habit where that line meets zero and the creep as its slope.

A reading that says so — one habit and one creep for the whole picture, fitted to every row of every strip at once — gives back 1.0000 and 0.0200 from the tired interior above, leaving $3 \times 10^{-14}$ px. That is exact, and exactness on a constructed picture is only the control. What matters is whether the creep survives a hand that also slips.

With half a pixel of slip on every interior row, a steady hand’s picture returns a drift of 0.0002 ± 0.0037, and the same hand drifting two per cent a bay returns 0.0202 ± 0.0038. The two are more than five of their standard deviations apart, so a creep of two per cent — the creep that moved the floor across half the family — is detectable across the picture at the same slip that the earlier reading could tolerate strip by strip.

The habit is recovered too, at 1.007 ± 0.119 from both pictures, and the two clouds lean. Within each, a larger fitted drift comes with a larger fitted habit, correlated at 0.91, because on every strip a drift is paid for with a habit: the fit can buy a little more creep with a little more habit, and the four splays are what stop it buying much. The book and the floor disagree most about the exchange rate, and so it is the spread of the picture’s splays, not the number of rows on any one strip, that decides how well the creep is known.

Two steady hands are not one tired one #

A reading that can explain a picture with an extra parameter must also be able to fail, or it explains everything. The failure it has to show is the one the drift was invented to excuse: a picture genuinely drawn by two hands.

Draw the floor and the footstool at a habit of 1 and the table and the book at 0.4, with no drift at all, and fit one habit and one creep. With no slip the fit settles on a habit of 0.31 and a creep of −1.3 per cent, and it leaves 0.46 px of page departure where a single hand — steady or tired — leaves nothing. The split is the wrong shape to be a creep. A creep moves the floor furthest and the book least, in the fixed proportion of $n/\ln s$; two hands split the picture between the footstool and the table, and no single slope against $n/\ln s$ puts the footstool with the floor and the table with the book.

With half a pixel of slip, the joint fit leaves a median 0.36 px on a steady hand’s picture and on a tired hand’s — the same, because the fit has a creep to spend and spends it — and 0.59 px on the two-hand picture. Seventy per cent of the two-hand pictures leave more than the 95th percentile of the tired hand’s, so a reader who refuses the one-hand reading above that line refuses two hands seven times in ten and wrongly refuses one tired hand once in twenty.

That is the reading the earlier essay could not give. Strip by strip, two steady hands and one tired hand look the same — two groups of numbers. Read as a whole, with the creep allowed for, one of them is a line against $n/\ln s$ and the other is not.

What a painter’s order does not enter #

The model of fatigue here is the plainest one: the same creep on every strip, starting afresh at each strip’s near edge. That is the right first model because it is what the family of habits can be confused with at all. A creep that changed from strip to strip, growing as the painter worked across the panel, would add one number per strip, and four strips would then have as many numbers to explain as they have habits to report.

That limit matters for the verdict. The recovery above works because four strips give four habits and the model spends only two parameters on them, leaving two to test. A picture of two strips gives two habits and two parameters, and nothing is left to test; one habit and one creep then explain any two strips exactly, whatever drew them. Four marks before anything is said found the same counting in a pavement — a correct perspective has three numbers to choose, so the fourth transversal is the first that can disagree — and the same rule applies here one level up: with a creep allowed, the third strip is the first that can refuse.

It also matters what the creep is not. Stepped, or measured from the zero measured the error of a hand laying off equal lengths by walking dividers from the last mark: independent errors that accumulate, growing as the square root of the count. That is a random walk, and its mean is zero. The creep here has a direction, and it is the direction, not the size, that makes it indistinguishable from a habit.

What this does not settle #

Whether real hands drift, and by how much. Two per cent a bay was chosen because it is small and because the earlier essay’s slip was half a pixel; neither number is a measurement of a painter. The result is conditional: a creep of that size is invisible on one strip and visible across four.

A creep that is not geometric. The model multiplies each bay by the same factor. A hand that drifts in steps, or whose drift runs one way for the first bays and the other way for the last, is not in the model, and on one strip it would be read as whatever steady habit is nearest.

Strips that share a hand but not a date. A panel painted over weeks may carry a habit that genuinely changed. The joint fit refuses two habits; it cannot say whether they belong to two people or to one person at two times. What a panel says about its maker drew the same line for convergent pictures: a drawing names the class of error in it, not the person who made it.

Still open: whether a picture keeps the order its strips were drawn in #

The creep here restarts at each strip. A painter who tires across the panel as well as down each strip would drift more on the strip drawn last, and the strips’ creeps would then carry the order in which the picture was made — floor first and book last, or the reverse.

The measurement that settles it lets the creep grow by a stated amount from one strip to the next in a stated order, fits a habit, a starting creep and a rate of tiring across the whole picture, and asks two things: whether four strips — three parameters against four readings — can still refuse two steady hands, and whether the fitted order is recovered when the true order is one of the twenty-four the four strips could have been drawn in. If it is, a divergent picture records not only how its painter divided a surface but which surface was divided first.