A strip keeps its ratio, not the end it began

A tiring hand draws a different habit found that a painter whose bays creep a little larger down a strip has drawn, to within a fiftieth of a pixel, what a steady painter with a different habit of spacing would draw: one strip cannot tell fatigue from habit. A whole picture can, because the creep counterfeits a change of habit in proportion to the number of bays over the logarithm of the strip’s splay, and that is different on every strip. A tiring panel keeps its order, not its direction then let the creep grow from strip to strip and found that the picture keeps the order the strips were drawn in, but only as a line: tiring from the floor to the book and steadying from the book to the floor are the same drawing.

Both essays made the creep start at each strip’s near edge, which assumes every strip was divided from its near edge outward. The second closed by asking about the other possibility. A strip divided from its far edge inward creeps the other way: its far bays are drawn first and steadiest, its near bays carry the drift. Within a strip the creep compounds bay by bay, rather than growing in a straight line as it did across strips, and that seemed like the kind of structure that might record which end the painter started from.

It does not, and the reason is exact. What does record it is the picture again.

A far start is a near start, steadying #

The hero figure draws the table strip of the interior — splay 1.32, six bays — at a habit of 1, four ways.

Divided from its far edge by a hand whose bays grow five per cent each as it tires, the strip’s rows lie exactly on those of the same strip divided from its near edge by a hand whose bays shrink 4.76 per cent each — the two drawings agree to $3\times10^{-14}$ px. Divided from the near edge by the same tiring hand, the rows are 7.25 px away. So which end the painter tired from is written into the rows, but only as the sign of a ratio, and a tiring hand at one end and a steadying hand at the other write the same sign.

The algebra is one line. A compounding creep makes each bay a fixed multiple of the bay before, counted from where the painter started: $1+d$ times as tall, each time. Counted from the other end, the same bays are each $1/(1+d)$ times the one before — a fixed multiple again, and $1/(1+d)$ is $1 - d/(1+d)$. The strip’s two edges are ruled first and the bays are fitted between them, so only the ratio matters, and a ratio read backwards is a ratio. The rows record the ratio and not which end it was counted from.

The same holds for a creep that grows in a straight line, each bay a fixed step taller than the last. A sequence growing by a fixed step, read from its other end, shrinks by a fixed step, and after the bays are rescaled between the ruled edges it is exactly a near-started creep of the opposite sign. The earlier essay’s expectation — that compounding rather than linear growth might carry the direction — is refused for both.

A straight line has no start #

The figure below shows the refusal as a picture, by plotting each bay’s height against what the steady habit would draw, on a logarithmic scale.

A compounding creep is a straight line on this plot: each bay a fixed ratio of the last is a fixed step in the logarithm. From the near edge the line rises away from the viewer; from the far edge, it falls. A straight line falling is a straight line rising with the opposite slope, and nothing on a straight line says which end it was drawn from. That is the whole of the refusal, and it is why the hand’s direction and its sign are one unknown and not two.

The same plot shows what would break it. A hand that works steadily for a while and then begins to tire draws a bent line: flat for the first bays from wherever the painter started, rising after. The bend sits near the starting edge. Start from the other edge and the bend moves to the other end of the strip, and a bent line flat at the near end and rising is not the same as one flat at the far end and rising. A creep with an onset is not closed under reversal, and so, in principle, it carries a direction.

A bend that is too small to see #

The principle turns out to be worth very little on a real strip. The measurement below takes the floor strip, divides it from its near edge with a creep that begins after one, two or three bays, and searches every habit, creep and onset a painter starting from the far edge could have used for the nearest drawing.

With no onset, the gap is nothing at every creep: every compounding creep has an exact twin divided from the other end. With an onset the gap is no longer nothing, but at a creep of six per cent a bay — already more than any steady painter shows — it is five hundredths of a pixel for an onset of one bay and two hundredths for two or three. Only at twenty per cent a bay, a hand that doubles its bays over four of them, does the gap reach a pixel.

The reason is that a painter starting from the far edge has three free choices — habit, creep and onset — and a six-bay strip has only five interior rows to constrain them. A bend near one end can be closely imitated by a slightly different habit, a slightly different creep and a bend near the other end. So a single strip’s direction, even when its creep has an onset, is carried in a few hundredths of a pixel at any plausible rate of tiring, below what the hand’s own slip hides.

The picture keeps every strip’s end #

The earlier essays’ lesson was that a picture can see what a strip cannot, because one hand shares something across strips that respond to it differently. The same lesson answers this question.

Take the interior’s four strips — floor, footstool, table and book — and let one hand paint them all at a habit of 1, tiring five per cent a bay, dividing the floor and the table from their near edges and the footstool and the book from their far ones. Each strip, read alone, reports a habit shifted by the creep in proportion to its number of bays over the logarithm of its splay, as the first essay found; a near-started strip’s shift is downward, and a far-started strip’s is the same size upward. Plotted against that ratio, the near-started strips lie on a line falling from the true habit and the far-started ones on its mirror image, rising.

Fitting one habit and one creep with each strip’s direction free — sixteen patterns — and keeping those whose fitted creep is a tiring one rather than a steadying one, the best is the true pattern, at a creep of 0.0501 a bay against a true 0.05, with a residual of $2.6\times10^{-3}$. The next best pattern leaves a residual of 1.315, five hundred times larger. No strip said which end it was divided from. The picture says it for all four, because the one creep they share has to be read with the right sign on each or it does not fit.

The one ambiguity left is the global one the second essay found for the order of strips: every strip reversed, with a steadying hand instead of a tiring one, draws the same picture. A hand that tires rather than steadies — which is what fatigue is — settles it.

Why a wrong direction cannot hide #

It is worth seeing why the right pattern wins by so much, because the same reason sets how far the reading can be trusted. The rows count hands, not cameras established that a strip’s rows report one number, its habit, and the rows under a splay measure the bays that the number is read from the page alone. A picture of four strips therefore offers four numbers. One hand, one habit and one creep account for them with two, and the direction of each strip decides only the sign with which the creep enters that strip’s number.

A wrong direction on one strip puts that strip’s reported habit on the mirror line instead of its own. For the other strips to accommodate it, the fitted habit and creep would have to move — but they are pinned by the other three strips, which all sit on their correct lines. The one misassigned strip is then left off the fitted line by twice its own counterfeit shift, which for the floor at a five per cent creep is more than three units of habit. That is why the wrong patterns leave residuals of order one while the right one leaves thousandths, and why a strip with a small ratio of bays to splay — whose counterfeit shift is small — is the first to be misread when the hand slips: its two possible lines lie close together.

The same arithmetic says which strips carry the reading. A strongly splayed strip with few bays barely responds to the creep and barely distinguishes its two directions; a weakly splayed strip with many bays responds strongly and distinguishes them well. An inverse perspective is a leaning plane found that the splay of a divergent strip is the lean of the plane it depicts; here the lean decides how much each strip testifies about the hand, and a floor seen nearly flat testifies most.

As the hand slips #

That reading was made on rows drawn exactly. A painted picture’s rows slip, and the question is how much slip the reading survives.

For four strips tiring two per cent a bay — the drift the first essay used — every strip’s end comes back in 92 per cent of pictures at half a pixel of slip, and 65 per cent at a pixel; guessing would name all four right one time in eight. At five per cent a bay the reading holds 92 per cent even at a pixel. Six strips tiring two per cent give 94 per cent at half a pixel and 66 at a pixel, against a chance of one in thirty-two: more strips are more ends to get right, and they are also more evidence for the one creep, and the two nearly balance.

The figure measures something stricter than it needs to. It counts a picture as read only if every strip’s end is right; a picture in which three of four ends come back is counted as a failure. The rate at which each individual strip’s end is named correctly is higher at every slip.

What a strip keeps, and what a picture keeps #

The three essays in this sequence now say the same thing three ways. A strip keeps the ratio of its bays and nothing that ratio does not contain: not whether the ratio is fatigue or habit, and not which end it was counted from. A picture keeps whatever its strips share and express differently: the habit and the creep, because the creep’s counterfeit scales with each strip’s own ratio of bays to splay; the order of the strips, as a line; and now each strip’s direction, because the shared creep must be read with each strip’s sign for the strips to agree on one hand.

What no picture keeps is the direction of anything shared by all its strips at once. The order of strips comes back only up to reversal, and the directions of all strips come back only up to reversing every one and the hand’s tiring together. One camera means one horizon, not one point found the same shape of limit in the divergent pictures themselves, and four surfaces and no one camera that draws them the same shape again: a picture fixes what its parts disagree about, not what they agree about.

The exclusion is two conditions, not ten rows settled a different line of questions about these pictures by finding that what looked like a table of cases was two conditions. The questions here have settled in a similar way: each question about what a painter did has been answered by asking what the drawing is invariant to, and each answer has been a symmetry — a reversal of the order, a reversal of the strip — that the rows cannot see and the other strips can.

What the directions would say about a panel #

It is worth being careful about what a recovered direction means. It says from which edge a strip’s bays were divided — which edge the dividers, or the eye, started from — and nothing about why. A painter dividing a floor from its far edge might have been working from a line ruled first at the back of the room, or copying a pattern book that started there, or simply turning the panel. What the reading adds is that the choice is recorded at all: not in any one strip, where it is invisible, but in the agreement of all of them about one hand’s fatigue. On a panel where every strip turns out to have been divided from the same edge, the finding is a habit of the workshop; on one where they differ, it is a sign the strips were laid out at different sittings, or by different rules, which two grounds and what the second costs is the reminder a divergent picture’s parts often were.

What was assumed #

One hand, one habit, one creep. Every picture here is painted by one painter whose habit and rate of tiring are the same on every strip. A workshop picture painted by several hands, or one whose painter’s rate of tiring changed from strip to strip, gives the fit more freedom than it has patterns to test, and the directions become as uncertain as the second essay found the order of strips to be once the rate was free.

The edges are ruled first. Each strip’s near and far edges are drawn before it is divided and hold still, so only the ratio of the bays can carry anything. A painter who marked bays from one edge without ruling the other first would leave the last bay’s height free, and a free last bay is a statement about where the painter stopped.

Slips are independent and even. A real hand’s slip grows as it tires, which would put more scatter on the bays drawn last and so, in principle, mark the end drawn last. That would be a second direction signal, of a different kind, and it is not measured here.

Still open: whether the hand’s slip marks the end it drew last #

The creep is closed under reversal, so the size of the bays cannot say which end a strip was divided from. The scatter of the bays is another matter. A tiring hand is not only less accurate on average by the end of a strip; it is less consistent, and the bays it draws last should scatter more about the creep than the bays it drew first. Scatter that grows along a strip is not symmetric under reversal: read from the wrong end, it shrinks.

The measurement that settles whether that is readable paints strips whose slip grows by a stated factor from the first bay drawn to the last, fits the habit and the creep as before, and asks whether the residuals’ spread along the strip — larger at one end than the other — names the end the painter began from on a single strip, or whether a strip of five or six bays leaves too few residuals for a trend in their spread to be told from chance, so that only a picture, pooling the trend over every strip, can read it.