A tiring panel keeps its order, not its direction

A tiring hand draws a different habit found that a painter whose bays creep a steady two per cent wider down each strip has, on any one strip, drawn a different habit — and that across a whole picture the creep gives itself away, because it counterfeits a habit at a rate of the strip’s bays over the logarithm of its splay, which differs from strip to strip. One habit and one creep, fitted across the four strips of a divergent interior, came back exactly; and a panel drawn by two steady hands left a residual that no creep of one hand removed.

That essay’s creep restarted at each strip. A painter who tires across the panel as well as down each strip creeps more on the strip drawn last. The creep of each strip then records where in the order it was drawn, and the question left was whether a picture keeps that order — whether a divergent panel says which surface its painter divided first.

It keeps half of it. And keeping even that half costs the reading the one refusal it had.

An order and its reverse are one drawing #

The model adds one number. The creep on the strip drawn first is $d_0$; each strip drawn after it creeps by a further $r$. Since a creep of $d$ counterfeits a habit shift of $-c\ln(1+d)$ with $c = n/\ln s$, the habits the four strips report are

$$q_i = q_0 - c_i\,\delta_i, \qquad \delta_i = \ln(1 + d_i) = e_0 + \rho\,k_i,$$

where $k_i$ is the strip’s place in the order. For a given order that is linear in three numbers — the habit, the first strip’s creep and the rate of tiring — and each of the twenty-four orders of four strips can be fitted in one step.

The figure draws a panel painted floor, footstool, table, book, starting at one per cent a bay and tiring by a further one per cent on each strip. The strips report habits of 0.473, 0.565, 0.361 and 0.391. Every order is fitted, and each is drawn beside its reverse because they leave exactly the same residual: the order drawn and its reverse leave $1.6\times10^{-3}$ of habit — the first-order counterfeit law’s own floor, since the law is exact only for small creep — and the next best pair $1.9\times10^{-2}$.

The tie is not a coincidence of the fit; the two panels are the same drawing. A painter who starts on the floor at one per cent and tires by one per cent a strip puts 1, 2, 3 and 4 per cent on floor, footstool, table and book. A painter who starts on the book at four per cent and steadies by one per cent a strip puts exactly the same numbers on exactly the same strips. A drift that changes steadily along a sequence is fixed by the sequence and the size of the change, and says nothing about which end the change started from. So the panel keeps which strips were drawn next to which, and cannot keep which end its painter began at — not approximately, but in the same sense that a picture of a straight road does not say which way it was walked.

How often the order comes back #

A tie between an order and its reverse still leaves twelve distinct readings of four strips, and the fit has to pick one of them from four habits, each read with a hand’s own slip.

With half a pixel of slip on every interior row, four strips give back their order — up to reversal — 13 per cent of the time when the hand tires by a quarter of a per cent a strip, 23 per cent at one per cent and 25 per cent at two. Chance, among twelve readings, is 8 per cent. The order is there, but four strips read it barely better than a guess.

The reason is arithmetic about what four strips have to spend. Each strip reports one number. The habit, the first strip’s creep and the rate of tiring take three of the four, and the order is decided by the one that is left. 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. Here the fourth strip is the first that can say anything about the order, and one number cannot say much.

Two more strips change the answer. Add a step at a splay of 1.08 and a lectern at 1.26, drawn after the book, and the order comes back 63 per cent of the time at one per cent a strip and 95 per cent at two, against a chance of one in 360. Six strips spend three numbers on the hand and have three left for the order, and the weakly splayed step — whose habit moves furthest for a given creep, 52.9 to one on the interior’s floor and more on the step — is a sensitive gauge of where in the sequence it fell. A panel with half a dozen divided surfaces keeps the sequence it was painted in, if its painter tired steadily and was the only painter.

How well the rate of tiring is known #

The order is a discrete answer; the rate of tiring is a number, and a reader who is handed the order — from a workshop’s records, say, or from the way the paint overlaps — can ask how well the strips fix the rate alone.

Given the true order and half a pixel of slip, four strips return a hand tiring by one per cent a strip as tiring by 0.98 ± 0.61 per cent, and its habit as 0.98 ± 0.37. That is a rate known to within its own size and a habit known to within a third of the range between a hand’s even rows and a camera’s equal bays: four strips carrying a changing creep are nearly as uninformative about the hand as they were about the order. Six strips return 0.98 ± 0.07 per cent and a habit of 1.00 ± 0.10, a rate known to a tenth of itself. The difference is the same arithmetic as before, since the rate and the habit are two of the three numbers the strips pay for, and four strips have only one number of slack to average the slip against.

The comparison with the earlier reading is instructive. With the creep the same on every strip, a tiring hand draws a different habit recovered the habit as 1.007 ± 0.119 from the same four strips under the same slip. Letting the creep grow triples the habit’s uncertainty, before the order is even in question: one more number in the model of the hand, and the four strips’ readings are spread across it.

A line has no direction, and a curve would #

The tie between an order and its reverse is the most definite result here and the easiest to misread, so it is worth saying exactly what produces it.

A steady change of creep puts $d_0 + r\,k$ on the strip drawn $k$-th. Reverse the order, so the strip drawn $k$-th is now drawn $(3 - k)$-th, and the same creeps are produced by starting at $d_0 + 3r$ and changing by $-r$. Any straight line through four points can be walked from either end, and a creep that changes steadily is a straight line in the drawing order. So the tie is exact, and it holds for any number of strips, any splays and any slip.

It would not hold for a creep that changes unevenly. A painter whose fatigue accelerates — little difference between the first two strips, more between the last two — puts a curve on the strips, and a curve has a direction: its steep end is the late end. Whether the strips could read a curve’s direction is a separate measurement and was not made here; a quadratic creep costs one more number, and the count of strips it would need follows from the same arithmetic as the rest of this essay. What can be said is that the direction of a painter’s work is not in a divergent panel unless the painter tired unevenly, and that the most natural model of fatigue, the steady one, is exactly the one that erases it.

Stepped, or measured from the zero found a different kind of accumulation in a construction: dividers walked from the last mark accumulate independent errors, growing as the square root of the count, while marks set from a common origin do not accumulate at all. That is a random walk, whose spread grows in the order the marks were made; a steady creep is a drift with no randomness at all. The two leave different traces, and neither leaves the direction.

What a two-hand panel is read as #

The last condition is the one that matters, because the same freedom that lets a fit find the order lets it explain a panel that was never drawn by one tiring hand.

Draw the interior with two steady hands — the floor and footstool at a habit of 1, the table and book at 0.4, no creep at all — and fit it as one tiring hand in the best of the twenty-four orders. The best reading is a hand at a habit of 2.08 that drew floor, footstool, table, book and tired from a 2.1 per cent creep to 11.3 per cent, and it leaves 0.019 of habit unexplained. That is less than the scatter half a pixel of slip puts on a single strip. Every strip’s reading is reproduced; the hand the reading describes never drew anything.

In a tiring hand draws a different habit the same two-hand panel was refused: one habit and one creep the same on every strip left 0.46 px of page residual, where one hand left none. The refusal worked because a constant creep shifts each strip’s habit in a fixed proportion, $n/\ln s$, and two hands split the strips in a pattern that proportion cannot make. A creep that grows with the order has an extra number and twenty-four orders to try it in, and a two-level step in the habits is well within what a growing creep and a chosen order can imitate.

Under a hand’s slip the loss is plain. With four strips the best-order residual has a median of 0.011 on panels one tiring hand drew and 0.025 on panels two steady hands drew, and the ranges overlap: only one two-hand panel in five leaves more than the tiring hand’s ninetieth percentile. With six strips the medians are 0.135 and 0.130 — the two-hand panels fit slightly better than the tiring ones — and three in a hundred exceed the line. More strips found the order; they did not bring the refusal back.

Two questions a panel cannot answer together #

The two essays on this hand now pull in opposite directions, and the tension between them is the finding.

With the creep the same on every strip, the reading had one number to spare, and it spent it on a test: one tiring hand or two steady ones. With the creep growing across the panel, the reading has an order to find and a rate of tiring to fit, and it spends its spare numbers on those instead, leaving nothing to test the hand with. A picture can be asked the order in which one hand worked it, or whether one hand worked it at all, and the answer to each question assumes the answer to the other.

The rows count hands, not cameras separated what the rows of a divergent picture say about the painter from what the sides say about the viewpoint, and found the rows’ reading free of any lens. It is still free of any lens. What it is not free of is the model of the hand, and the model’s freedom has to be paid for in strips. Four strips pay for one habit and one steady creep with a number left over. They do not pay for a creep that changes with the order.

The same shape has turned up before on the other side of this subject, in the count of cameras rather than hands. A camera count needs a tolerance found that asking how few viewpoints a splayed picture needs has no answer until the reader says how many pixels of redrawing are allowed, and that the answer then belongs to the hand as much as to the picture. The hand count has now reached the same place by a different route: how many hands drew a panel has no answer until the reader says which model of a hand is allowed, and a model generous enough to include fatigue that grows is generous enough to absorb a second hand. In both cases the picture is not being evasive. It holds a fixed number of readings, and every freedom the reader grants the model is paid for out of them.

The slip that leaves no trace put the general form of this in one sentence about constructions: an error that lands back on the set of correct-looking results cannot be seen by any test that asks only whether the result looks correct. A two-hand panel read as one tiring hand lands back on the set of one-hand panels, once that set has been widened to include tiring in any order.

That is a practical limit on attribution from rows, and it sits beside the one what a panel says about its maker found for convergent pictures: a drawing names the class of error in it, not the person who made it. Here the class of error itself — steady fatigue, fatigue that grows, or a second hand — is decidable only if the picture has more divided surfaces than the model of the hand has numbers.

What the order model leaves out #

A creep that does not grow steadily. The model makes the creep a straight line in the drawing order. A painter who tires and then rests, or who takes the strips in two sessions, puts a different pattern on them; the tie between an order and its reverse is a property of the straight line and would not survive a curved one.

Large creep. The counterfeit law is first order in the creep, and the six-strip panel’s weakly splayed step reports a habit of −2.17 at a tiring rate of one per cent a strip. Much past two per cent a strip, the step’s reading leaves the range a strip’s habit can be read over, and the order figures stop there for that reason.

Strips of the same splay. Two strips with the same splay and bay count have the same $n/\ln s$ and so respond identically to creep; the order between them cannot be read at all, and the fit has one fewer useful number than it has strips.

Still open: whether the rows of one strip carry the order within it #

Everything above reads one habit per strip and asks about the order of whole strips. A painter’s creep also happens within a strip, row by row, and the first essay’s model made each strip’s creep start afresh at its near edge — which assumes each strip was divided from its near edge outward.

A strip divided from the far edge inward creeps the other way: its far bays are the first drawn and the steadiest, and its near bays carry the drift. The measurement that settles whether the rows keep that direction draws one strip with its creep running from the far edge, fits the model with the creep’s direction free, and asks whether the direction comes back — whether a strip’s own rows, unlike the panel’s strips, record which end the painter started at, because within a strip the creep compounds bay by bay rather than growing in a straight line.