A panel's pavements convict the creep one pavement hides

A steady creep draws the rule’s bow found that the arc reading — which fits a painted pavement’s diagonal for the bow the constant-ratio rule puts in it — could not tell the rule from a correct pavement whose painter let each transversal gap grow a little on the one before. A creep of 4.49 per cent a braccio on an eight-braccio pavement bows the diagonal exactly as the rule does, to a cosine of 0.9994; the reading measures how much the tiles lengthen as they recede, not what lengthened them. What did separate them was the gaps themselves: the rule’s gaps shrink by one ratio all the way, so their logarithms lie on a straight line, while a correct pavement’s curve whatever it was crept by. Read on one pavement, that curvature named the rule 94 times in a hundred at half a pixel of scatter, 67 at a pixel and not at two.

The essay ended by pointing out that a painted interior rarely has one pavement. Floors, a loggia’s ceiling coffers, a tiled step, a table’s inlaid top: each is a receding grid drawn, if by the rule, by the same habit, and if by a tiring hand, with a creep that depends on how the hand moved on each. The question was whether the pavements, read together, separate the rule from a creep where no single pavement could.

They do, in two different ways, depending on what the creep is.

One interior, four pavements #

The panel holds four pavements at different heights: 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, all sharing the panel’s horizon. Each is read the earlier essay’s way — the arc reading’s amplitude, scaled so that each pavement’s own rule bow is exactly one — and painted by one of three hands.

Painted by the rule, every pavement bows its diagonal by exactly the rule’s amplitude, one on each, whatever its length: that is what the rule is. Painted correctly by a hand that creeps 4.49 per cent a braccio on every pavement — the creep that counterfeits the floor — the four amplitudes are 0.62, 0.80, 1.00 and 1.72. One creep, a different bow on each length.

The reason is in how each grows. The earlier essay measured the creep that counterfeits the rule falling as the pavement lengthens — about seven per cent a braccio on four braccia, three on twelve — because a creep compounds over the gaps while the rule’s bow grows more gently. So a creep that matches the rule on one pavement over-bows the longer ones and under-bows the shorter. The rule’s amplitudes follow the pavements’ lengths by one law, the constant one; a steady creep’s follow another, and the panel can see the difference.

The slider’s third hand removes it. A creep set separately for each pavement — seven per cent on the step, 5.5 on the inlay, 4.5 on the floor, 2.9 on the coffers — counterfeits every bow at once, and the four amplitudes are all one, exactly the rule’s. No reading of the arcs can tell that panel from the rule’s.

One creep for the hand #

A painter has one hand, and a hand that tires does so at its own rate. If the creep is the hand’s, it is one number on every pavement, and the panel convicts it through the arcs.

The statistic is plain. The rule’s four amplitudes should agree with one another; a steady creep’s should not, since they run from 0.62 to 1.72. Measured as the spread of the four about their common mean, each in units of the scatter a painter’s hand gives that pavement’s amplitude, it names the rule against the steady creep in every one of 120 panels at a quarter, a half and a whole pixel of scatter, and 89 times in a hundred at two pixels — where the earlier essay found the single pavement’s gaps naming nothing. The bow that counterfeited the rule on one pavement betrays the creep across four, because the counterfeit is a different number on each. A painter who wanted the rule’s look from a tiring hand would have to tire faster on the short pavements than on the long ones, by just the amounts the rule’s own bows require — not a habit any hand falls into.

The gaps’ curvature also names it — 90 times in a hundred at a pixel, 74 at two — but less well. The arcs carry this creep’s signature in their sizes, which the hand’s scatter disturbs little; the curvature carries it in a second difference of logarithms of gaps, which the scatter disturbs a great deal.

A creep tuned to every pavement #

The harder case is the one the slider’s third position draws: a hand whose creep is different on each pavement, and by chance or by design counterfeits each one’s bow. Every arc amplitude is then one, and only the gaps are left.

The pooling here has two parts, and both matter. A correct pavement’s gaps curve one way on every pavement — positively, in the earlier essay’s reading, whatever creep is laid over them — and the rule’s never curve. So each pavement’s curvature is read with its sign, not its size alone, and the four are added with weights in proportion to what a correct pavement’s curvature on that pavement is, over its scatter squared: a long pavement’s curvature counts for more, since it is estimated from more gaps, and a short step’s, large but very noisy, for less. Weighting by the noise alone was tried first and let the four-braccio step swamp the floor; the curvature a pavement can show has to be in the weight as well.

Read that way, the four pavements name the rule against the tuned counterfeits 90 times in a hundred at a pixel of scatter and 74 at two. The floor alone, read with its sign, does 82 and 66. That is itself better than the earlier essay’s 67 at a pixel, and nothing at two, for one pavement read by the curvature’s size with its sign thrown away: on a correct pavement the sign is always the same, and keeping it is worth fifteen points of the reading at a pixel.

Each pavement adds its share #

The evidence adds as the pavements do. The floor alone names the rule 82 times in a hundred at a pixel; with the loggia’s coffers beside it, 89; all four pavements, 90 — the step and the inlay add little, their few gaps fixing a curvature poorly; and eight pavements, the interior’s four with four more at the same lengths a little higher up, 97. A painted interior with half a dozen receding grids, painted by a hand as steady as a pixel, carries enough evidence in its gaps to tell the rule from any creep that counterfeits it, however the creep was arranged.

What each pavement is worth #

The count figure’s flat stretch — two pavements to four, 89 to 90 — has a plain cause, and it says which pavements a reader should look for. A pavement’s gap curvature is estimated from its gaps, and a pavement of n braccia has n of them; the curvature is a second difference of their logarithms, and it is fixed well only when there are many.

On a correct pavement the four curvatures are 0.0095 on the step, 0.0082 on the inlay, 0.0069 on the floor and 0.0031 on the coffers: the shorter the pavement, the more its gaps’ logarithms bend, because a short pavement is a steeper slice of the same perspective. But a hand at one pixel scatters the step’s curvature by 0.083, eleven times its own size; the inlay’s by 0.019, a little over twice; the floor’s by 0.0061, slightly less than itself; the coffers’ by 0.0031, the same as itself. In units of its scatter each pavement’s evidence is 0.11, 0.44, 1.12 and 0.99. The floor and the coffers carry nearly all of it, and the step is a witness whose testimony is noise.

Evidence of this kind adds in quadrature, so the four together are worth 1.56 of a scatter, where the floor alone is worth 1.12 and the floor with the coffers 1.50. Turned into a chance of naming the rule for a normal spread, those are about 86 per cent for the four, 79 for the floor alone and 85 for the floor with the coffers; the panels measured 90, 82 and 89. Doubling every pavement doubles the sum of squares and puts the panel at 2.2 scatters, about 94 per cent by the same arithmetic, 97 measured. The pooled reading behaves as evidence should: it adds where it has something to add and ignores what it does not.

The practical consequence is that a reader looking for the rule should look for long pavements, not many. A panel with two floors of eight braccia or more is worth more than one with a floor and a dozen steps, and a step of four braccia is not worth reading for this at all.

A real hand strays #

The two counterfeits above are the ends of a range. A real painter’s creep would be neither exactly one number for the hand nor exactly tuned to each pavement; it would wander from pavement to pavement about some habit.

The two readings respond to the stray in opposite ways, and that is the useful part. The amplitudes’ spread sees how far the creeps miss their counterfeits: nothing when they counterfeit exactly — 56 per cent, a coin with a slight lean — and 89 per cent when each creep strays by as much as its own size. The gap curvature does not care how much the tiles lengthen, only how the lengthening is laid out, and it names the rule 90 times in a hundred at every stray. A creep would have to counterfeit both — the size of every bow and the layout of every pavement’s gaps — to pass for the rule, and a correct pavement cannot do the second whatever it is crept by.

Reading a real panel #

The measurements give a reading of a painted interior in three steps, each using what the earlier essays built. First, find every receding grid in the panel that shares its horizon — floors, coffered ceilings, inlays, steps, tiled walls — and read each one’s transversals; the rule that draws another room is the reminder that the rule may have been used on some and not others, so each is a witness of its own. Second, read each pavement’s arc against its own rule bow and ask whether the amplitudes agree across the pavements; if they disagree by more than their scatter, in the way a steady creep makes them disagree — growing with the pavement’s length — the hand crept and the pavements are correct constructions. Third, if the amplitudes agree, read the gaps’ curvatures with their sign and pool them, weighted by what each pavement can show, and ask whether they sit at nought, as the rule’s must, or on the positive side, as every correct pavement’s does.

The second step is cheap and decisive against the commonest alternative. A painter’s hand that tires does so at its own rate, and a single rate is the case the amplitudes convict every time at a pixel. The third step is the one that survives the hardest alternative, a creep that happens to counterfeit every pavement’s bow, and it needs long pavements and a steady hand: nine times in ten at a pixel from this interior’s four, and not at two pixels from anything short of a large panel.

Neither step says anything about whether the panel’s perspective is otherwise right — whether its horizon is where its viewing distance puts it, or its orthogonals meet. A straightedge convicts the rule on five braccia is the test for that, read on the diagonal of a single long pavement. What the panel adds is the ability to tell the rule from the most plausible honest alternative when the straightedge alone cannot.

What the panel adds #

Put together with the earlier essays, the panel settles what a single pavement left open. The rule’s bow is read by its shape found the rule’s bow a recognisable shape on one pavement; the earlier essay found a steady creep drawing that shape too, and only the gaps telling them apart on a steady hand. Across a panel, a steady creep is no longer a counterfeit at all: the rule bows every pavement by the same share of its own bow and a creep does not, and four pavements say so every time at a pixel. A creep tuned to every pavement survives the arcs and not the gaps, whose curvature, read with its sign and pooled over the pavements, names the rule nine times in ten at a pixel.

The rule is exact for a floor that lengthens gave the rule its one defence: it is the true picture of a floor whose tiles lengthen by a fixed ratio. That defence was made for one floor. A panel with a step, a table and a loggia would need every one of them built with tiles lengthening by its own fixed ratio, each matched to its own length — a coincidence of furniture the arcs and the gaps between them make implausible from four pavements and very nearly impossible from eight.

What was assumed #

Every pavement shares the panel’s horizon and centric point. The four pavements here are drawn from one eye. A painter who set a second vanishing point for the table — common enough in the panels this sequence has studied — gives that pavement a different rule template, and its amplitude has to be read against it; the rows count hands, not cameras is the reminder that a panel’s surfaces need not share one construction.

A creep is a steady compounding of the gaps. The creeps here grow each gap by a fixed share on the one before. A creep that begins partway down a pavement, or speeds up, has a different bow and a different curvature, and the counterfeit of a steady creep would not cover it.

The scatter of the hand is the same on every pavement. A painter working a small step and a large floor places the step’s corners closer together and, perhaps, more carefully; a scatter that differs by pavement changes the weights the pooled readings should use, which here are measured for one scatter throughout.

The pavements are independent. Each is painted with its own scatter. A painter who laid out every pavement with one set of dividers, or transferred one pavement’s gaps to another, ties their errors together, and the pooled readings would then count one pavement’s evidence more than once.

Still open: whether the panel can say which hand, not only which procedure #

The readings here decide between the rule and a creep, the procedure that drew the pavements. A creep also says something about the hand: its size on each pavement, and how that size varies from pavement to pavement, is a record of how the painter’s hand moved through the panel.

The measurement that settles what it records paints a panel correctly with a creep that grows as the painter works — larger on the pavements painted later — and asks whether the four pavements’ fitted creeps, read in the order of their size, name the order in which they were painted, as a tiring panel keeps its order, not its direction found the strips of a divergent interior doing. If they do, a pavement panel carries the order of its making in its tiles, as a divided strip carries it in its bays; if the creeps’ scatter under a pixel of hand swamps their trend, the order is only readable on panels painted with the steadiest hands.