A panel's creeps keep the order it was painted in, while the hand is steady

A panel’s pavements convict the creep one pavement hides settled a question of procedure. One crept pavement — a correct construction whose painter let each transversal gap grow a little on the one before — bows its diagonal exactly as the constant-ratio rule does. A panel with several pavements of different lengths does not let the creep hide: one steady creep bows a twelve-braccio pavement 1.72 times its rule bow and a four-braccio step 0.62 times, where the rule bows each by exactly its own, and even a creep tuned pavement by pavement is caught by the gaps’ curvature, whose sign on a correct pavement never changes. A painted interior with half a dozen receding grids tells the rule from any creep.

The essay ended by noticing that a creep says something about the hand as well as the procedure. If the creep is fatigue, its size on each pavement and how that size changes from pavement to pavement are a record of the painter’s hand moving through the panel. The question was whether a panel painted correctly by a hand whose creep grows as it works carries the order its pavements were painted in — whether the pavements’ fitted creeps, ranked by size, name that order, as a tiring panel keeps its order, not its direction found the strips of a divergent interior doing.

They do, often, and less often than the strips did, for a reason that sits in one pavement. And the reading has a blind assumption that the earlier essays did not need.

Fitting a pavement’s creep #

The panel is the earlier essay’s: a step four braccia deep, a table’s inlay of six, the floor of eight and a loggia’s coffers of twelve, each with its own ground line under one horizon. Once the panel has been convicted of a correct construction with a creep — which is what the earlier essay’s readings decide — each pavement’s correct construction is known from its ground line, its horizon and its distance point. What is not known is how much the painter crept on it. That is fitted: the creep whose crept pavement’s transversals lie nearest the painted ones, by least squares over every transversal after the ground line.

The fitted creep is only as good as the pavement’s transversals. On a pavement painted to a pixel of scatter, the coffers’ twelve gaps fix the creep to a tenth of a point a braccio, the floor’s eight to a fifth, the inlay’s six to half a point, and the step’s four short gaps to almost two points. A creep that grows by a point from one pavement to the next is a large signal on the coffers and the floor and a small one on the step, whose own uncertainty is twice the growth.

The figure paints one panel in a stated order — floor first, then the coffers, the step and the inlay — by a hand whose creep starts at two per cent a braccio and grows by a point with each pavement. The ticks are the creeps painted; the dots the creeps read; the bars their scatter under the same hand. The floor, the coffers and the inlay come back close to their creeps and in their order. The step, painted third at four per cent, reads 1.71, and the ranking puts it first. One pavement has carried the order wrong.

The slider sets how fast the hand tires. With no growth at all the four creeps are equal, and what the fit reads are their scatters, ranked; with growth of two points a pavement, even the step’s scatter is not enough to misplace it.

What a point of creep moves on the page #

A point of creep a braccio is not much to the eye. On the floor, eight braccia deep and 117 pixels from its ground line to its last transversal, a creep of three per cent instead of two moves the far transversal by 3.2 pixels; on the coffers, by 4.7; on the step, whose whole depth is 37 pixels, by half a pixel. A painter’s hand that places a transversal to a pixel scatters each row by about as much as a point of creep moves the floor’s last one, and twice what it moves the step’s.

That is why the fit, and not the eye, has to read the creep. The fit uses every transversal of a pavement at once and knows where each should fall; the creep is a pattern across all of them, growing towards the far end, and the hand’s scatter is not. On the long pavements the pattern is several pixels against a pixel of scatter spread over eight or twelve rows, and it is read well. On the step it is half a pixel against a pixel over four rows, and it is read barely at all — a stepped hand passes for a tiring one on a short strip found the same of a divergent strip with few bays, where an error has too few rows to show its shape.

How often the order comes back #

The four pavements can be painted in twenty-four orders, so a guess names the order once in twenty-four. A hand that does not tire is read at that rate — three per cent of panels, Kendall’s τ between the order read and the order painted indistinguishable from nought — which is the control: the reading names an order only when there is one to name.

A hand whose creep grows by a quarter of a point a pavement is read in exactly the right order sixteen times in a hundred at a pixel of scatter; by half a point, thirty; by a point, fifty-seven, with τ of 0.81; by two points, eighty-two. A steadier hand does much better — at half a pixel, a point a pavement gives eighty-one per cent and two points ninety-eight — and a rougher hand much worse: at two pixels, a point a pavement gives thirty per cent. The whole order is a demanding statistic, since one pair the wrong way round spoils it, and τ, which counts pairs, is the gentler measure: at a pixel and a point a pavement it is 0.81, which means about nine pairs in ten put the right way round.

The step decides #

Every pair of pavements without the step is put the right way round at least ninety-eight times in a hundred: the floor and the coffers every time, the inlay with either of them ninety-eight. Every pair with the step is put right only eighty-two or eighty-three times. The whole order fails mostly because the step lands in the wrong place.

That is the earlier essays’ arithmetic again, at a different question. A straightedge convicts the rule on five braccia found its test working on pavements of five braccia or more and not below, and the earlier essay found the step adding almost nothing to the pooled curvature: a few gaps fix little, whatever is being read from them. A creep is fitted from the same gaps, and four gaps fix a growth law poorly. A reader who wants a panel’s painting order should read it from the long pavements and treat the step’s place as a guess — or, where the panel has one, prefer a long tiled ceiling to a short tiled step.

More pavements, a longer record #

A panel with eight pavements — the four and four more of the same lengths a little higher on the wall — is read with a slightly higher τ at every growth: 0.88 against 0.83 at a point a pavement. Eight pavements stretch the creep twice as far from the first painted to the last, which separates the ends of the order well; they also hold twice as many short pavements whose creeps are barely known, which muddles the middle. Pair by pair the order is read a little better, but the whole order of eight, which needs twenty-eight pairs right, is much harder to read in full than the whole order of four, which needs six.

So a larger panel records more of the painter’s progress, and records it in the same way: well at its long pavements, poorly at its short ones, and best as a statement about which pavements came early and which late rather than about the exact sequence.

A creep that depends on length reads as an order #

Everything so far assumed that the creep differs between pavements only because the painter tired. The earlier essay found a reason the creeps might differ otherwise: the creep that counterfeits the rule’s bow on each pavement is larger on short pavements and smaller on long ones, seven points a braccio on the step and three on the coffers, and a painter who had learnt to imitate the rule’s look would creep that way whatever the order.

A tiring hand’s creeps follow the order painted, τ of 0.81, and have nothing to do with the pavements’ lengths. A steady hand’s follow neither. A hand whose creep is set by each pavement’s length — the counterfeiting creeps — gives creeps that follow the lengths almost perfectly, τ of 0.91 against longest-first, and the order painted not at all. Read by ranking creeps, that panel is named as painted coffers first, then the floor, the inlay and the step, confidently and in nearly every painting, whatever order it was really painted in. A painter who both tired and crept by length gives a reading that follows the lengths more than the painting: τ of 0.59 against longest-first and 0.38 against the truth.

This is the reading’s real limit, and it is not about scatter. Ranking creeps can say that a panel’s creeps differ in an order; that the order is the painting’s is an assumption about why they differ. What a panel says about its maker found that a drawing names the class of error in it and not the recipe that produced it; the creeps here name an ordering and not its cause. Here the check is available within the panel. A creep that tracks length shows as creeps falling with each pavement’s braccia, which the four creeps read here can be tested for directly; a creep that tracks time shows no relation to length. With four pavements the test is coarse — creeps that fall exactly with length happen by chance once in twenty-four paintings, the same odds as a right order by chance — and it sharpens with every pavement added. A reader should look for a creep that tracks length before believing one that tracks time.

Why a pavement keeps the direction a strip lost #

A strip keeps its ratio, not the end it began found that a divergent strip’s rows cannot say which edge its painter started from, and so a tiring panel of strips knows the order of its strips only as a line: tiring from the floor to the book and steadying from the book to the floor leave the same creeps. A pavement does not have that symmetry. Its transversals are drawn from the ground line inward, the near gaps first and largest, and a creep compounds away from the ground line whichever way the painter thought of the work; there is no second edge to start from. The creep’s sign is therefore always the same, gaps growing beyond where they should, and a larger creep means a hand that has drifted further.

So the pavement panel carries what the strips could not, at one assumption’s price. Rank the creeps and the order comes out with an end: the least crept pavement first. Whether first means painted first depends only on whether the painter’s hand tired rather than loosened, and the rule is exact for a floor that lengthens is the reminder that a creep, read alone, is indistinguishable from a floor that really does lengthen; what makes it a hand’s is that it changes from pavement to pavement, as a floor’s lengthening would not.

What a panel records about its painter #

Put together, a panel of pavements painted correctly by a tiring hand does record the order of its painting in its creeps, and reads it back as well as its pavements’ lengths allow. With the creep growing a point a braccio from one pavement to the next and the hand placing transversals to a pixel, the whole order of four pavements comes back fifty-seven times in a hundred and pairs of long pavements come back right every time; the step, whose four gaps fix its creep to two points, is where the order goes wrong. A hand that does not tire is read at chance, as it must be.

The comparison with the strips of a divergent interior is instructive. A tiring panel keeps its order, not its direction found six strips naming their order nineteen times in twenty but unable to say which end of it came first, because a strip’s creep reverses with the edge it was divided from. A pavement’s creep does not reverse: its gaps grow away from the ground line whichever way the painter worked, so a pavement panel carries the direction of its order as well as the order — larger creeps are later, on the assumption that hands tire rather than warm up. What it loses against the strips is evenness: the strips were all about as long, and the pavements are not, so one short pavement carries most of the uncertainty.

A steady creep draws the rule’s bow began this part of the subject by finding a creep indistinguishable from a procedure on one pavement. The procedure question was settled by many pavements; the question about the hand is answered by the same many pavements, read for a different number.

What the reading assumes about the hand #

The correct construction is known. Each creep is fitted against the pavement’s correct transversals, which the panel’s horizon, ground lines and distance point fix once the panel has been read as correct. A misread distance point moves every pavement’s correct transversals together and adds the same error to every creep, which leaves their ranking alone.

The hand tires steadily. The creep grows by a fixed step with each pavement. A painter who rested between pavements, or worked one pavement in two sittings, would leave creeps that rise and fall, and the ranking would read the rests as reversals.

Hands tire rather than warm up. The direction of the order rests on larger creep meaning later. A painter whose hand loosened as the work went on, creeping less on later pavements, would be read in reverse.

The scatter is the same on every pavement. A painter who took more care over the floor than over the coffers places the floor’s transversals more precisely, which changes which pavements carry the order; the measurement here gives every pavement the same hand.

Still open: whether the creeps can tell two painters from one who tired #

A workshop panel may have been painted by two hands, each on its own pavements, each creeping at its own rate. Its creeps then fall into two groups that need not follow either painting order or length, and a ranking would read them as one painter’s progress.

The measurement that settles what the creeps can say about that paints a panel’s pavements with two hands — one steady at a low creep, one at a higher creep, assigned in a stated pattern — and asks whether the creeps, together with each pavement’s gap curvature and arc amplitude, separate two hands from one that tired, how often they put each pavement with the hand that painted it, and how many pavements a panel needs before two hands of creeps a point apart are told from one hand that tired by a point over the panel.