Each copy's facing reads the walk past the place it doubles back

A walk’s order is on the floor, not on the page found that a continuous narrative of one walker keeps the order of its moments in the places its copies stand. A walker moves a step between moments and does not leap, so the shortest route through the copies is the walk — provided the route is drawn on the floor, undone from the picture with the horizon the copies themselves supply, and not on the page, where an upright panel draws every step away from the eye at a fraction of its length. The reading reached the walks’ own ceiling and stopped there. A walk that turns back on itself puts a later stop nearer an earlier one than a step, and then the floor holds the places without their sequence. At a turn of 70° between stops the floor reader returned 58 walks in 100; at 120°, 19.

That essay closed on what a walking figure carries besides its position. Drawn in profile or three-quarter, as walking figures nearly always are, each copy faces the way it is going. The direction it faces is a short arrow on the floor at every stop, and an arrow is exactly what a walk that doubles back needs: when the nearest stop is the wrong one, the arrow at the stop can say that the walker was not heading there.

Three things decide whether the arrows earn their place. How far above the walk’s own ceiling they lift the reading; how coarsely a facing can be read and still help, since a drawn figure’s facing is placed far less precisely than its feet; and whether the arrows suffer on the page what the distances suffer — since an arrow is itself a short stretch of floor, and an upright panel compresses it like any other.

A route that has to agree with the arrows #

The walks are those of the earlier essay: six stops 1.4 m apart, a stride and a half, kept between 7.5 and 18 m from a level eye 1.6 m above the floor, every copy 1.72 m tall, feet and heads marked to half a pixel. Each heading turns from the last by a normally distributed angle, and the turn is now the quantity varied. A copy is given the direction a walker faces mid-stride, which is the direction of travel at that moment: halfway between the leg it arrived by and the leg it is about to walk, with the first copy facing its first leg and the last its last. That is the honest arrow. It does not point at the next stop; on a walk turning 70° it points at it give or take half the turn.

The reader gets the floor back exactly as before — the horizon and the eye’s height from a straight-line fit of the feet against the copies’ drawn heights, and depth from each foot’s drop below the horizon — and now gets each copy’s facing as well, read off the drawn figure with an error of a stated number of degrees. It then scores every directed order of the six copies, all 720, by two things together. The length of each leg, in units of the typical distance from a stop to its nearest neighbour, which is the earlier reader’s whole criterion. And, for each leg, how far its direction runs from the facing at the stop it leaves and the facing at the stop it reaches. The order with the lowest score is the reading.

On the walk drawn here the positions alone run 4–3–5–2–6–1, keeping one true neighbour of five: the walk comes back past ground it has already crossed, and once it has, the nearest stop is seldom the next one. With the facings read to 15° the route that agrees with the arrows runs 1–2–3–4–5–6. It keeps reading it with the facings read to 30° and to 45°, and gives up at 60°, where it returns the positions’ own wrong route. And it reads the walk forwards. A route through positions has two ends and no arrow, so the earlier reader always left one bit to the viewer: which end came first. A route that agrees with the arrows has a direction, because the arrows point along it.

Past the ceiling of positions #

One walk is an illustration. The question is how often, over the walks a painter might draw, and how that depends on how sharply the walk turns.

Below 50° of turn there is nothing to lift: the walks are their own shortest route and positions read them all. From there the two readers separate. At 50° positions read 90 walks in 100 and positions with arrows read every one. At 70° positions read 59 — exactly the share of walks that are their own shortest route at all, the most any reading from positions can achieve — and the arrows read 94. At 90° the figures are 39 and 87; at 120°, where a walker is about as likely to turn back as to go on, 20 and 74. Everything above the dashed ceiling is order no reading of places can extract, because it is not in the places.

The arrows have a ceiling of their own. With exact facings on the true floor the same reader returns 97 walks in 100 at 70° and 80 at 120°. What defeats it is a walk that loops back across an earlier leg heading the same way: then a later stop sits beside an earlier one and faces along the same line, and neither distance nor direction says which came first. That is rare at 70° and one walk in five at 120°, and it is a property of the walk, not of how the arrows were read.

Every correct reading with the arrows is correct forwards. Of the 9.5 bits that order six copies, the earlier essay found the floor supplies 8.5 for a walk that does not double back and the viewer the last. The facings supply that last bit, and on a walk that doubles back they supply most of what the floor could not.

An arrow is a short stretch of floor #

The facing a reader sees is drawn on the page, and the page does to it what it does to every distance on the floor. The horizon at eye level is the reason. An upright panel draws a small step away from the eye shorter than a step across by the tangent of the angle the eye looks down at the stop, and a direction is the ratio of a step away to a step across.

From a standing eye the tangent is 0.20 at a stop 8 m out, 0.13 at 12 m and 0.09 at 18 m. A copy facing 45° away from straight across is drawn with an arrow 10.9° above horizontal at 8 m, 7.3° at 12 m, 4.9° at 18 m. Only an arrow pointing exactly across or exactly away keeps its direction; every other one is pressed towards the horizontal, and the further the stop the harder. A walker heading diagonally away and a walker heading straight across are drawn facing nearly the same way. And because the press grows with depth, two copies facing the same way on the floor at different distances are drawn facing different ways on the page — so arrows read off the page disagree with one another even where the walker did not turn.

The cure is the one the positions already needed. An arrow is the foot and a point a little way along the facing, and both can be undone through the same horizon and the same guessed angle of view that undo the stops. With exact marks and the true angle of view, the undone arrow is the arrow on the floor to the last digit.

What is lost by reading the arrows off the page #

So there are four readers, and only one of them is consistent: positions and arrows both undone. The other three take something off the page as drawn.

From a standing eye at 70° of turn, positions and arrows both undone read 94 walks in 100. Positions undone and arrows taken at the direction they are drawn read 82: the arrows still help, because they still point roughly across when the walker crosses the panel and they still separate left from right, but every arrow pointing into the depth has been flattened towards the horizontal, and the reader is told that a walker heading away was heading across. Positions and arrows both off the page read 62 — barely above the floor reader with no arrows at all, 59 — because the page’s distances and the page’s directions are wrong in the same way and agree with each other in being wrong. At 120° the four are 74, 48, 31 and 20.

Raise the eye and the readers converge. From 12 m, about as high as the walk is far, the stops are seen between 34° and 56° down and the page is near a plan; the three readers that use arrows return 95, 94 and 94 at 70°, and the undoing has nothing left to correct. That is the earlier essay’s finding about the high horizon of early narrative painting, extended to the figures standing on its road. A carpet and the people on it is the same choice made by drawing the ground from above and the figures from in front: an overhead ground keeps both the distances and the directions true on the page, and a reader can follow the walk by eye.

How coarse a facing can be #

A drawn figure’s facing is not read to a degree. A profile is a profile over thirty degrees or more, three-quarter view is a broad band, and a figure mid-stride may twist its shoulders against its feet. The useful question is how badly a facing can be read before the arrow stops carrying anything.

Facings read to 5° do what exact ones do: 98 walks in 100 at 70°, 80 at 120°. Read to 15° they give 94 and 74, to 30° 86 and 55, to 45° 76 and 43. A facing read to within a quarter turn either way, which is barely more than knowing whether a figure faces left, right, in or out, still lifts a 70° walk from 59 to 76. Only at 90° of error — a facing that could be anything in a half circle — does the arrow stop helping, and there the reading falls back to 58 and 21, where positions alone put it. It falls back rather than below because the arrows are weighed by how well they were read: an arrow read to 90° is given almost no say. Weighed as if it were sharp, an arrow that coarse would steer the reader wrong and do worse than no arrows at all.

So the facings do not need a careful painter. The far copies’ feet did: the earlier essay found that a floor undone from far stops marked to a few pixels was better left on the page. A facing is undone through the same horizon, but it is a direction, and a direction is read from the whole drawn figure rather than from one pixel of its foot.

How much to trust an arrow #

The weight is the reader’s one free choice, and the natural setting turns out to be wrong. A copy’s facing is half the turn away from the leg it is about to walk, so on a walk turning 120° the honest spread of a leg about its facing is 60° and more. A reader who weighed the arrows by that spread would be weighing them by what they truly are.

With facings read to 15° the reading sits at its best — 96 walks in 100 at 70°, 77 at 120° — for any allowed spread from 5° to 20°, and falls away as the spread is widened: to 88 and 51 at 80°, to 73 and 28 at 160°. Weighed by the honest spread, a 120° walk is read about 58 times in 100 rather than 77 — a quarter of what the arrows can do, thrown away. With facings read to 45° the best spread moves out to between 40° and 80°, giving 76 and 42 at its best, and narrowing it then costs nearly as much as widening it does for sharp arrows.

The reason is that the two terms in the score are not alike. The distances are a coarse guide: a doubled-back leg is often only a little longer than the right one, which is the whole difficulty. A facing is a sharp one: a leg that runs against the arrows at both its ends is wrong almost however short it is. A reader does best to let the arrows decide whenever they were read well enough to decide, and to discount them only by how badly they were read — never by how much the walk turns, which a reader of a picture does not know. The weight used everywhere else in this essay follows that rule: never below 20°, growing with the reading error.

What the arrows ask of the panel #

Three of the earlier essay’s conditions still hold, and the arrows ease two of them.

The angle of view is guessed. Undoing the floor needs the picture’s focal length, which the panel does not hold, and a focal length is not an angle is why a picture without a stated format does not give it up. Guessed at half the truth or twice it, the positions alone fall from 59 walks in 100 at 70° of turn to 49 and 45; with the arrows the reading falls from 94 to 90 and 89. A wrong guess stretches the floor from front to back, which turns every arrow towards or away from the depth by the same rule, and a route that has to agree with arrows bent together is harder to mislead than one that rests on distances alone.

More copies are more chances to go wrong. At 70° of turn the positions read 77 walks in 100 with four copies and 45 with eight; the arrows read 98 and 90. Each copy added is one more place a walk might double back, and each one brings its own arrow to say so.

The far feet are still divided by a small number. The arrows do not change how the stops are undone, so a deep panel marked coarsely still loses its positions to the undoing. The arrows survive that better than the positions do, being directions, but a walk whose stops are misplaced by a step cannot be rescued by knowing which way each copy faced.

And two conditions are new.

A figure faces where it is going. The copies here face along the walk, halfway between their legs. A storyteller’s figures do not always: a walker who stops to look back, to speak to a companion or to point at the next scene faces somewhere else, and that copy’s arrow is a false witness. The weighting assumes every arrow errs a little, not that one errs entirely, and what a single false arrow among six costs was not measured; a reader who suspected one could drop each arrow in turn and keep the reading the others agree on.

The moments are a walk. As before, a narrative whose copies are placed by the geography of the story — three procedures, one panel warns against reading a procedure off its traces, and Masaccio’s three Peters are the standing example — carries the order of where the story happened, and arrows only make a confident reader more confident about the wrong thing. In the Tribute Money Peter at the lake’s edge faces the water and Peter at the right faces the collector; both arrows are true to their scenes and neither points along a walk.

The order, counted again #

The same person, twice on one panel began this line of reading by saying a panel supplies every place and no sequence, and the one thing a single view cannot give puts the photograph’s version of that beside it: every ratio and no size. The earlier essay corrected the first statement for walkers whose walk does not double back. This one corrects it further. A walker’s copies carry their sequence in two scripts, the places and the facings, and the page distorts both the same way and the horizon undoes both the same way. Read together and on the floor, they return the order and its direction for nineteen walks in twenty that turn 70° between stops, and three in four that turn 120°.

That is the measurement each system answers its own question asks for: not how close continuous narrative comes to a strip cartoon, but what its pictures keep. They keep a good deal more of the story’s order than its convention admits, and almost all of it is written on a floor the page has squeezed — measuring a room off the page is the general case of getting a floor back, and this is the case of getting a walk back from it.

Still open: whether the facings order the copies of a walker who stops and turns #

Every walk here keeps moving: a step between every pair of moments. A storyteller often wants the opposite — a walker who arrives, stops, turns to look, and turns again before going on, so that three copies stand on nearly the same spot. The shortest route through a cluster can take its copies in any order, as the earlier essay noted, and the arrows here would not help, because a walker turning on the spot has no legs between its moments for the arrows to agree with.

What the copies of a turning walker do carry is the turn itself. Arriving, the walker faces the leg just walked; leaving, the leg to come; and between, a sweep from one to the other. The measurement that settles what that is worth gives a walk one stop at which the walker pauses for two or three moments while turning by a stated angle, draws each moment as a copy at that spot facing the way it then faced, and asks whether ordering the cluster’s copies by their facings — from the one that agrees with the leg arrived by to the one that agrees with the leg leaving — returns the moments in sequence; how large the turn must be against the facings’ reading error before the order within the cluster is read; and whether a reader that does not know which copies form a cluster finds the pause from the arrows as well, since a cluster is the one place where several copies stand close together and face different ways.