What survives being copied

Most surviving pictures are copies. A workshop drawing, a pattern book, a manuscript illumination and a printed plate are all made by somebody looking at an earlier version rather than at the world, and the version they looked at was made the same way.

The construction field has already measured what one hand does to one drawing. One hand step each prices a slip at the scale of a single panel and the slip that leaves no trace shows which slips a reader can detect. This essay asks what happens over generations, and the answer depends entirely on what is being copied.

Two copyists #

Both of them are looking at the previous drawing and both of them have the same unsteady hand. The difference is what they take from it.

The mark copyist measures the drawing in front of them and reproduces it. Every transversal, every spacing, every proportion is read off the earlier sheet and drawn again, with whatever error the hand adds.

The method copyist looks at the drawing, recognises the construction it was made by, and redraws it from the construction. The earlier sheet tells them which recipe to use and nothing more.

Both are real. A pattern book is copied the first way; a workshop apprentice taught a rule and shown an example is working the second way. And the difference between them is not carefulness — the hand error is identical in what follows — but the object of the copying.

The parameter under study #

The quantity followed here is a pavement’s depth-spacing ratio — the ratio between successive gaps of the transversals in a tiled floor. It is one number, it is visible in any drawing of a pavement, and this collection already has an opinion about it.

A correct perspective has a ratio that changes down the floor, which is what makes a pavement a measurement. The rule taught for centuries makes each gap a fixed fraction of the last, and the rule that draws another room measures what that depicts: a room whose horizon sits a hundred and seventy pixels from the one the panel drew.

The distinction between a rule that is a fixed number and a construction that is not runs through the whole of the construction field — the distance point is the viewing distance, drawn is where a genuine construction’s spacing is derived rather than chosen, and stepped, or measured from the zero is where two ways of laying out the same spacing are shown to accumulate error differently.

So the taught ratio is a fixed number and the correct one is not, which makes the taught rule an attractor in the technical sense: a value the method copyist’s redrawing pulls toward, every generation, whatever the sheet in front of them says.

The mark copyist walks #

Each generation reads the last drawing and adds a hand error. The errors accumulate, and the accumulation is the textbook random walk: independent increments, so the variance adds and the spread grows as the square root of the number of steps.

Measured over nine hundred lineages of forty generations, the fitted exponent of the spread against the generation is 0.5001 against a predicted one half.

The exponent is fitted rather than asserted, and the fitting is the point. Drawing a curve that looks like a square root and agreeing with it is not a measurement — it is a picture of an expectation. Fitting a slope on a log-log plot and getting 0.5001 is a measurement, and it can fail.

It nearly did. The first version of this computation seeded its lineages at values thirty-seven apart in a linear congruential generator, which produces lineages whose first few steps are nearly the same — and the fitted exponent came out at 0.375. That is not a small deviation from a half; it is a different law, and it was a property of the seeding rather than of the copying. Spreading the seeds across the generator’s period fixed it.

The method copyist does not #

The other family’s fitted exponent is 0.012, which is no growth at all. Its spread settles at a constant and stays there for as long as the run continues.

The reason is that the hand error is thrown away every generation. The method copyist does not carry the previous sheet’s error forward, because they are not copying the previous sheet’s marks — they are redrawing from a recipe, and the recipe has no error in it. What survives from generation to generation is the recipe, and the sheet is only a reminder of which one.

So the two families differ not in magnitude but in kind: one has a growing spread and one has a stationary one. After forty generations they are 5.7 times apart, and after four hundred they would be eighteen times apart, and the ratio keeps growing because one of the two quantities is not growing.

The two families are the ends of one dial #

Real workshops are not one copyist or the other, and the mixture has a law with the two measured families at its ends.

Let a fraction $\alpha$ of each generation’s ratio come from redrawing the recipe and the rest from measuring the sheet. Then $x_{n+1} = (1-\alpha)x_{n} + \alpha r + \varepsilon$, which is an ordinary first-order recursion with a hand error added, and its spread settles at

$$\frac{\sigma_{\varepsilon}}{\sqrt{2\alpha - \alpha^{2}}}$$

rather than growing. Any admixture of method-copying at all makes the spread stationary. The random walk is the knife-edge case $\alpha = 0$ exactly, and one copyist in a hundred who redraws from the rule is enough to stop it.

What the admixture buys is a size rather than a kind. At $\alpha = 0.01$ the spread settles at seven times the hand’s own error; at $\alpha = 0.1$, at 2.3 times; at $\alpha = 0.5$, at 1.15. So a tradition that mostly copies marks still ends up with a bounded spread, merely a wide one, and the essay’s “growth against no growth” is the limiting statement rather than the general one.

The same recursion gives the forgetting. Memory of the original decays as $(1-\alpha)^{n}$, so its half-life is $\ln 2 / \alpha$ generations — sixty-nine at one per cent, seven at ten, one at a half. That is the honest form of the closing section’s convergence: two originals do not merely end up together, they do so on a timetable, and the timetable is set by how often somebody redraws rather than by how carefully anybody copies.

Which sharpens the whole finding into a single instruction for a reader of a surviving sheet. Its ratio is evidence about the recipe to the extent that $\alpha$ was large, and evidence about the original to the extent that it was small — and the two readings are exclusive, since the same mechanism that preserves one destroys the other.

Two originals, one destination #

The strongest form of the finding is a convergence rather than a spread.

Start one method-copying lineage from a drawing whose ratio is 0.55 and another from a drawing whose ratio is 0.92 — two originals as different as any two surviving pavements — and run both for forty generations. They end 5.0 × 10⁻¹⁵ apart. The lineages have forgotten which original they came from.

That is the useful and slightly melancholy statement of what a tradition does. A tradition transmitted as a method preserves the method with high fidelity and destroys the evidence of what was originally drawn; a tradition transmitted as marks preserves a degraded record of the original and never converges on anything.

What this means for reading a late copy #

The consequence for anybody looking at a surviving drawing is direct and it cuts both ways.

A late copy in a method-copying tradition is good evidence about the method and no evidence about the original. Its ratio is the taught ratio to within one hand’s error, whatever the first drawing did, so measuring it recovers the recipe and nothing else. That is why so many surviving pattern drawings agree with each other and with a rule, and why their agreement is not evidence that the rule was right.

A late copy in a mark-copying tradition — the situation four marks before anything is said sets out the reading discipline for — is weak evidence about the original whose weakness is quantifiable: the expected departure grows as the square root of the number of generations, so a copy at the tenth remove is about three times as far off as one at the first. That is a usable number, and it is why a stemma matters.

And a drawing that agrees with the taught rule exactly is, on this account, more likely to be a late method copy than an early one — which is the opposite of the intuition that agreement with a rule indicates a careful original.

The convention this belongs to #

Transmission is not usually thought of as a drawing convention, and filing it here rather than in the construction field is a decision worth defending.

Every other essay in this field measures a choice a picture makes about what to record: rank rather than distance, an ordering rather than a depth, lengths rather than dihedrals. Copying is a choice too — a workshop decides, usually without deciding, whether its drawings are records of earlier drawings or instances of a rule — and the choice has exactly the same shape of consequence: a quantity preserved, a quantity spent, and a count attached to each.

What a method-copying tradition preserves is the recipe, to one hand’s error, indefinitely. What it spends is the original, entirely. What a mark-copying tradition preserves is a degraded original, and what it spends is precision, at a rate the square-root law gives exactly.

Reading it that way also connects it to what perspective gave up, which is this field’s statement that every system is a trade. A tradition is a system for moving pictures through time, and it trades the same way.

Why this is not a claim about art history #

The model here is deliberately thin, and saying so is part of the measurement.

It has one parameter — a ratio — where a real drawing has hundreds. It has one kind of hand error, Gaussian and independent, where a real hand has systematic biases that a copyist inherits along with the marks. It has two pure copyists where a real workshop has people doing both at once, differently for different parts of a drawing. And it has no selection: no copy is discarded for looking wrong, which is the mechanism most likely to matter in practice and is not modelled here at all.

What the model does establish is that the two mechanisms have different laws — one grows and one does not — and that the difference is large and easy to test. A tradition whose surviving copies agree with each other more closely as they get later is a method-copying tradition; one whose copies disagree more is a mark-copying one; and the exponent of the disagreement against the remove is a measurable quantity rather than an impression.

Where the two mechanisms leave different traces #

The spreads above are statements about a population of copies, which is not what a reader normally has. Usually there is one drawing. So it is worth asking what a single sheet can be made to say.

A mark-copying descendant carries correlated departures: the hand errors of every ancestor, all of them still in the marks, so a drawing at the tenth remove has ten hands in it and its departures from any construction are spread over every part of the sheet in proportion to how much of the sheet each generation redrew. A method-copying descendant carries one hand’s error, its own, because everything earlier was discarded when the drawing was re-derived.

That is a difference a single sheet can show, and the instrument for it already exists. The slip that leaves no trace measures how far a drawing’s implied horizon moves when a hand slips by a stated amount, and finds that a single hand at a reader’s own ruler moves it one to twelve pixels while the taught rule moves it a hundred and seventy. A sheet with ten accumulated hands in it should sit between those, at roughly the square root of ten times a single hand — which is about four to forty pixels, still well below the rule’s signature.

So the three cases separate on one measurement: a drawing made from a construction, a drawing copied many times from marks, and a drawing made by the taught rule, at roughly one, four to forty, and a hundred and seventy pixels of horizon displacement. That is a usable ordering, and it comes out of the accumulation law rather than out of a new instrument.

The selection nobody models #

The omitted mechanism deserves its own paragraph, because it is the one that would change the answer.

A workshop that discards drawings which look wrong is applying a filter, and a filter on a random walk is not a random walk. If the discard threshold is a fixed departure from an accepted appearance, the mark-copying family stops spreading at that threshold and becomes stationary too — at which point the two mechanisms are indistinguishable from their spreads and can only be told apart by where they settle.

That is a testable difference and it is a good one. A selected mark-copying tradition settles wherever the acceptable band is centred, which is on the original; a method-copying tradition settles on the recipe. So the question “did this tradition copy marks under selection, or copy a method” becomes “does the settled value agree with the earliest surviving drawing, or with the taught rule” — which is a question about two numbers rather than about a mechanism.

Recording that here rather than modelling it is deliberate. The model in this essay measures what it contains and the extension is named rather than gestured at, which is this collection’s habit whenever a simplification is doing work.

The boundary, stated #

Forty generations, nine hundred lineages, one parameter, one hand error, and no selection. The first three are choices about the computation and the last two are the model’s limits.

Forty generations is more than most surviving traditions and fewer than a manuscript tradition with a long tail; the law is a power law, so it extrapolates without trouble in either direction. Nine hundred lineages is enough that the fitted exponent is stable to four figures, which was checked by running fewer and watching it move. And one parameter is the honest scope: this is a measurement of a mechanism, on a quantity chosen because the collection already knows what it means.

What is measured here #

Three numbers and a rejected one.

The spread across nine hundred mark-copying lineages grows with a fitted exponent of 0.5001 against a predicted 0.5. The method-copying family’s exponent is 0.012, which is no growth. After forty generations the two families’ spreads are 5.7 times apart, and two method-copying lineages started from originals 0.55 and 0.92 apart end 5.0 × 10⁻¹⁵ apart.

And the rejected number is 0.375, which is what the same computation returned when its lineages were seeded thirty-seven apart in a linear congruential generator — lineages whose first steps were correlated, giving a spread that was not the spread of independent walks. A wrong exponent that looks like a plausible law is the kind of result this collection is written to catch, and it was caught by the prediction being sharp enough to fail against.

The short version #

Copying the marks is a random walk: the spread across lineages grows as the square root of the generation, fitted at 0.5001, and the copies wander away from the original and from each other without limit.

Copying the method is not a walk at all: the spread does not grow, fitted at 0.012, and lineages started from different originals converge on the recipe and forget where they began. So a late copy is good evidence about a method and poor evidence about a drawing, and a copy that agrees with a taught rule exactly is more likely to be late than careful.