A grid stops a copy drifting

A grid on the wall is a scale without a projection read an Egyptian canon as a measuring instrument. Rule a wall into squares, count a figure’s height against the ruling, and any two figures on the wall can be compared with no horizon, no centre of projection and no error beyond the arithmetic’s own. That is what the grid gives a reader. It does not explain why anybody would rule a wall in the first place, since the workshops drawing those figures had no interest in being read by a modern measurement. A grid is a nuisance. It has to be snapped out with a cord, one line at a time, on every wall.

A long tradition of copying offers a possible reason. A figure in such a workshop was usually drawn from an earlier figure, by somebody who had learned from somebody else, over a very long time. Any process of that kind has a known failure. Each copy adds its small error to the errors already in the thing being copied. What survives being copied measured this in general, over traditions that have nothing to do with squares. What a grid does to that process turns out to be specific and rather surprising, because it is the same thing a modern digital signal does to noise.

A copy of a copy is a random walk #

Take five landmarks on a standing figure, as Gay Robins gives them for the eighteen-square grid of the classical period: the knee at six squares, the lower edge of the buttock at nine, the elbow at twelve, the junction of the neck and shoulder at sixteen and the hairline at eighteen. Put them as fractions of the height, and make copies of the figure, each from the one before it. Let the hand place each landmark with a small random scatter of $\sigma$ of the figure’s height.

With no grid, each copy starts from the previous copy’s landmarks with their errors, and adds another. The errors add up as independent steps, so after $k$ copies the root-mean-square error in a landmark is $\sigma\sqrt{k}$. With a hand of 0.8 per cent of the height, that is 5.7 per cent after fifty copies, and 6.2 per cent after sixty.

That is a large change in a figure. A landmark moving by 6 per cent of the height moves the waist of a metre-high figure by six centimetres. Enough lineages of that kind would produce figures with quite different proportions from where they began, and that is what the same kind of copying produces when nothing stops it.

Add the grid, and the same hand does something different. The copyist still places each landmark with the same scatter, and then, because the landmark belongs on a ruled line, brings it to the nearest one. If the scatter was small, it was the same line as before, and the error has gone. The next copy starts from the exact original value. Nothing accumulates, because each copy throws away what its hand contributed.

At a hand of 0.8 per cent the no-grid figure reaches 6.2 per cent after sixty copies. The eighteen-square grid reaches 1.0 per cent. This is not a slower walk. Its behaviour is different in kind. Most lineages have an error of exactly zero, and a few have a whole-square error, which comes to 5.6 per cent of the height at a stroke. The root-mean-square is reporting a very small chance of a large error, and not a small error everywhere.

It is a correcting code #

This should sound familiar to anyone who has worked with digital signals, since it is the principle they run on. A signal sent by a wire picks up noise. An analogue signal passed through ten repeaters arrives with ten amplifiers’ noise added to it. A digital signal arrives at each repeater, gets rounded to the nearest allowed level and leaves clean, so the noise never accumulates. The cost is that only certain values can be sent, and the benefit is that an unlimited number of repeaters can be used, as long as the noise stays small compared with the gap between levels.

A grid with lines does all of that with a piece of string. A landmark on a line is a discrete symbol, and the copyist’s rounding is a repeater. The same theorem applies, and it comes with the same two consequences. One is a threshold. The other is a trade between capacity and robustness. Neither of those is visible if a grid is thought of only as a scale.

The first is what happens when the hand is not small enough.

A threshold on the hand #

A landmark changes line when the hand’s scatter carries it more than half a square. On eighteen squares, half a square is 2.8 per cent of the height. The chance of that on one copy is the tail of a normal distribution at $0.5/(\sigma N)$ standard deviations, and these tails fall very fast indeed.

For a hand of 0.8 per cent, a landmark on eighteen squares has a 97 per cent chance of coming through fifty copies untouched. At 1 per cent it is 76 per cent, at 1.2 per cent it is 35 per cent, and at 1.5 per cent it is 4 per cent. Over that very narrow range of hand, the behaviour of a whole tradition goes from stable to unstable. A workshop whose copyists have a scatter one third as large as a half-square has a proportional canon that can last for generations. One whose copyists have half that half-square does not, and the difference between those is not a large difference in anyone’s skill.

That raises a real question about what hand applies. The numbers in this essay give the same $\sigma$ to the copy with and without a grid, and that is a conservative assumption. A person placing a mark on or near a ruled line has a reference within a few centimetres of their brush, and they can place the mark much more exactly, in practice, than a person judging the proportions of a whole figure by eye. On a wall figure two metres high, 0.8 per cent is 16 mm. Marks on a ruled grid can probably be placed a good deal closer than that. So the grid has probably been far under its threshold in any competent hand, and these slip rates are upper limits. The no-grid hand might well be several times worse, in which case the difference is larger than this figure shows.

It also predicts the shape of how such a tradition would fail if it failed. A canon that is held by rounding would not gradually change its proportions. It would keep them exactly, and then have a whole-square step in one landmark, in one lineage, that would then stay. This is a checkable prediction, in principle, for anybody with a large enough set of figures and their grids: changes in such a system would come in units of a square, and a smooth trend in proportions would count against this reading. The historical record has been measured by people with far better data than this essay has, and whether it shows that pattern is a question this one does not attempt. A picture in bands offers a neighbouring instrument, since the figures in a band are drawn at one height and their heights can be compared within a wall.

Beside the other ways of not drifting #

Two other ways of stopping a copy’s drift are already known, and it is worth setting the grid beside them, since each has a different cost.

What survives being copied separated a copyist who copies the marks from one who copies the method: the first walks with a fitted exponent of 0.5001, and the second reconstructs from a rule and has an exponent of 0.012, with no growth at all. It keeps the recipe and loses the original. The workshop that throws drawings away added a selection rule that discards drawings that look wrong, and got a stationary process that settles at some distance from where it began. The rule corrects, but it does so toward whatever the eye expects.

A grid is different from both. It keeps a copy of the marks, as the first copyist does, and so it does not need anybody to understand a recipe. It uses no judgement about how anything should look, so it does not settle at a place of the eye’s choosing. It returns to the exact original each time, since the correction is arithmetic and the original is on a line. So it has one strong property the others lack. It is the only one of the three whose copy after fifty generations can be identical, not merely similar, to the first drawing.

How fine a grid is worth having #

A grid’s size is a real choice with a real optimum. A coarse grid has a large half-square, so it is very robust. But a real body’s proportions fall between its lines, so the first drawing already misses by up to half a square. A fine grid can say more, and it is much less robust. That is the trade every correcting code has between its capacity and its protection.

For a hand at 0.8 per cent the best size is about eighteen squares, for a total error of 1.84 per cent. For a hand at 1.2 per cent it is about twelve, at 2.75 per cent. For a hand at 1.8 per cent it is about eight, at 4.13 per cent. Coarser than the best size, the rounding dominates and the error falls off as one over the squares. Finer than the best size, slips dominate, and the error rises very quickly indeed.

It is rather striking that the steadiest of these three hands has eighteen squares as its optimum, since nothing in this calculation was told about Egyptian practice. That should not be pressed too far. The optimum depends on the number of copies, on the hand and on how heavily a slip is weighed. The grid’s actual size is also tied to a convenient unit, the fist, and to a figure’s standard parts, so its arrival at eighteen has several causes. The calculation shows only that eighteen is a reasonable size for a tradition of this sort: fine enough to state a figure’s proportions to a few per cent, and coarse enough not to invite a slip in the hands likely to be using it.

A change of grid is an edit #

Grids have changed, and the same arithmetic prices that too. In the late period the Egyptian canon went to a grid of twenty-one squares to the upper eyelid in place of eighteen to the hairline, and a figure moved to a new grid has to have each landmark placed on a line of the new one.

A doubled grid moves nothing: every old line is also a new line, so a landmark stays where it is. A grid of 21 shares only one line in six with a grid of 18, since their common factor is 3, and landmarks on the others must move. Doing that here changes the largest landmark’s position by 2.38 per cent of the height, in the simple version with a common height. The real change also redefined where the top of the grid fell, which is a further change that this calculation has no opinion on.

The important distinction concerns kinds of change, and it is more interesting than the numbers. An error on a copy, on a fine grid, is random: it is a slip that happens to a lineage at random. A change of grid is the same size or larger, but it is systematic, since it moves all figures at once, in the same direction, by a known rule, at a known moment. That is an edit. A copying process with a correcting code at its heart cannot drift, but it can be deliberately revised in discrete steps. Such a process would present to an archaeologist as long periods of near-constancy interrupted by sudden changes. That is roughly what the record of the canon’s proportions looks like at a large scale, though there are too many other reasons for that to bear much weight here.

Scaling is free #

One property of the grid should not be forgotten. A figure can be copied from a small sketch to a big wall by ruling both, and the size of the square has nothing to do with the correction. Nothing above depends on the absolute size of a square, as long as $\sigma$ is proportional to the figure’s height. If $\sigma$ is an absolute error in millimetres, as it more nearly is for a hand, then a larger wall has a smaller relative scatter, and a larger figure is more robust. The earlier essay’s point that the count survives being ruled differently is what makes this possible. A count has no scale, which is also what one view cannot give from a perspective picture, where a size has to be brought in from outside.

That has an unintended benefit in a workshop. A master’s sketch on a small ostracon, squared up, can go onto a wall of any size by anyone who can count. So a grid is also a means of separating design from execution, which may have been a bigger reason to use one than any concern for drift. The workshop that throws drawings away looked at the other side of this question, where copying practice shapes what survives. Whatever the reason a grid was adopted, its protective effect on proportions comes with it for free.

What falls between the lines gets no correction #

This treats landmarks as independent. A real figure is drawn as a whole, and a mistake at the shoulder would usually be corrected by reference to the head. Correlated errors would change the slip rates, likely reducing them for small slips and making some larger ones more likely.

The hand has a normal scatter here. A real hand has occasional gross errors, which do not depend on the grid’s fineness, and they would set a floor under each curve that does not fall with a better hand.

A real figure has more landmarks than five, and many do not fall on lines at all. A nose, a hand or a fold in a garment is placed between lines, by eye, after the grid has fixed the frame, and those have no correcting mechanism. So features between lines ought to drift in a way that features on lines do not, and they might be a useful control if anybody had the data to test it.

And nothing here compares with the Greek canon or with any other system that specifies a figure by ratios between its parts. A ratio-based canon has its own kind of correction, since a part is measured against a part, and it does not obviously fit this analysis.

Still open: whether a ratio is a code #

A grid corrects because a mark is rounded to an absolute line. A proportional rule, such as a head being an eighth of the height, is a different kind of instruction: the copyist measures one part of the current figure and sets another from it. That rule refers to the copy being made, not to a fixed ruling, and it might or might not correct anything. A figure whose head has grown by a slip would, by such a rule, have a body that grows to match. Its overall proportions would stay the same, while its size in the frame would change.

The measurement that settles what a ratio canon can do makes the copies under a stated set of ratio rules, with the same hand error, and asks whether a landmark’s error stays bounded or walks; whether the errors that do accumulate are in the proportions or only in the overall scale, which would not matter; and how many rules a canon needs before it fixes the figure at all, since too few leave free directions for a drift to run along, and too many over-determine the figure and leave the copyist with a choice of which to obey.

Rounding to a line discards the hand’s error before the next copy can inherit it. A grid is a correcting code made with a piece of string, and it has a code’s threshold and a code’s trade between what it can protect and what it can say.