A walker who stops is ordered by the way it turns

Each copy’s facing reads the walk past the place it doubles back gave every copy of a walking figure a short arrow on the floor — the way it faces, undone from the picture through the horizon the copies themselves supply — and asked for the route through the copies whose legs agree best with both their lengths and the arrows at their ends. On walks turning 70° between stops that route returned the order of the moments 94 times in 100, where positions alone reached 59, and it returned it forwards. Every walk in that essay kept moving: a step between every pair of moments.

Storytellers often want the opposite. A walker arrives, stops, turns to look, and turns again before going on, and the painter draws a copy for each of those moments, so that three figures stand on nearly the same spot. That essay closed by predicting that its arrows would not help there, on the reasoning that a walker turning on the spot has no legs between its moments for the arrows to agree with. Half of that prediction turns out to be wrong, and the half that is right is the more interesting one: the legs into and out of a pause do say which of its copies came first and which last, and nothing about a leg can say anything about the copy in the middle. What orders the middle is the turn itself.

Three figures on one spot #

The walks are those of the earlier essays, drawn by a level eye 1.6 m above the floor, every copy 1.72 m tall, feet and heads marked to half a pixel. A walk now covers five places a step and a half apart, and at the middle place the walker stops for a stated number of moments while turning by a stated angle, in equal parts: arriving, it faces the leg it walked; leaving, the leg it is about to walk; and in between, a share of the way round. Each paused copy after the first has its feet moved from the spot by a normal 12 cm, the shuffle of a body turning in place. Everywhere else a copy faces halfway between the legs on either side of it, as before.

On the walk drawn here the walker stops at its third place and turns through 90°, so copies 3, 4 and 5 stand within a hand’s width of one another. Set back on the floor, they are three dots inside a ring a few centimetres across, and the picture’s own uncertainty about where each foot is — half a pixel at 10 m is several centimetres — is the same size as the distances between them. Positions cannot order them. The shortest route through the walk enters the pause at whichever copy is nearest the leg arriving and leaves at whichever is nearest the leg departing, and in between it takes them in the order their few centimetres of shuffle happen to suggest. Over three hundred walks, positions put the three in the order of their moments between 12 and 16 times in 100 — one chance in six, for any turn.

The arrows are another matter, and the magnified pause shows why. Copy 3 faces back along the leg from copy 2, because it has just walked it. Copy 5 faces on along the leg to copy 6, because it is about to walk it. Copy 4 faces halfway between. Those are three statements about the turn, and the walking reader of the earlier essay hears only two of them clearly.

What the walking reader hears #

That reader scores every directed order of the copies by the length of each leg and by how far each leg runs from the facings at its two ends, and keeps the cheapest. The legs from copy 2 into the pause and from the pause on to copy 6 are a step and a half long and have a definite direction, and the arrows at their ends either agree with it or do not. A route that enters the pause at copy 5 pays for it — copy 5 faces the leaving leg, nearly at right angles to the leg arriving — and so does a route that leaves at copy 3. The legs either side of a pause do place its first and last copies, which is the half of the earlier prediction that was wrong.

What they cannot place is anything in between. The legs inside the pause are a few centimetres long, and their directions are the shuffle’s: one points across, the next back, the next anywhere. The walking reader compares each of them with the facings at its ends exactly as it compares a leg a step and a half long, and every comparison is with noise.

So the walking reader orders a three-copy pause hardly better than positions: 15 times in 100 when the walker turns 15°, 25 at 90°, 34 at 150°. The slow rise is the end copies being placed better as the turn grows, since a sharper turn makes a wrong entry or exit more expensive. The middle copy stays a guess throughout.

For a pause of only two moments there is no middle, and the walking reader’s two clear statements are the whole of it. It reads two-copy pauses 54 times in 100 at a 15° turn — a coin toss, since an arriving and a leaving facing 15° apart are nearly the same arrow — 75 at 90° and 88 at 120°. That is what the earlier essay’s own reader can do with a pause, and it was more than that essay expected.

Charging the turn #

A walker standing still turns a little between moments. Between two of its copies on one spot there is no leg worth the name, but there is a turn — the angle between the two facings — and a walker sweeping from one facing to another passes every facing in between once, in order. Of all the orders through three copies on one spot, the one that sweeps from the arriving facing round to the leaving one turns least in total. Any other order turns back on itself somewhere, through more angle than the walker needed.

That suggests a second reader, the same as the first in two respects. It discounts the direction of a leg by how short the leg is: by $D^2/(D^2 + d^2)$, where $D$ is the leg’s length and $d$ about a step, so a leg of a few centimetres says nothing about direction and a leg of two steps says nearly everything. And it charges the same short legs for the turn between the facings they join, by the same weight it gives a leg running against an arrow. A long leg pays what it paid before; a short one pays for turning rather than for pointing.

Back on the drawn walk, the turning reader takes the pause 3–4–5 at every one of the slider’s turns, where the walking reader took 4–3–5. Over three hundred walks it reads a three-copy pause 24 times in 100 at a 15° turn, 61 at 45°, 92 at 90° and 97 at 120°. The dashed line above it is a reader told outright which three copies form the pause, ordering them by facing alone from the arriving leg’s direction round to the leaving one’s: 35, 71, 97 and 100. The turning reader is not told, and it comes within five walks in a hundred of that from 90° up — it finds the pause as well as ordering it, for a pause is the one place where several copies stand close together and face different ways.

At the far end both fall back a little, and for the same reason. A walker turning 150° or 180° could have turned either way round; the arriving and leaving facings are nearly opposite, and only the middle copy says which side the walker swung through. Told the pause, a reader that measures each facing from the arriving leg reads 180° pauses 50 times in 100, because half of them are measured the wrong way round. The turning reader does not measure from anything — it only asks which order turns least — and reads them 81.

Two charges that were the wrong size #

The turn is charged on short legs only, and that was a measurement rather than a choice. Charged on every leg, in a first version of the reader, it taxed the turns of a walk that keeps moving, whose copies differ by the walk’s own turn between stops, and walks turning 120° between stops — with no pause in them at all — were read 63 times in 100 against the walking reader’s 74. A walker on the move turns by its walk’s turn, which the arrows already account for. Only a walker standing still owes the charge.

And the charge is no sharper than the arrows are. Set from the facings’ reading error alone — as sharp for facings read to 5° as those facings are — it made a pause of large turn cheaper to split than to sweep: leave it partway through, walk on, come back. On pauses turning 150° with facings read to 5°, that version kept the pause’s copies together 33 times in 100. The turn between moments belongs to the walker, not to how well the reader sees it, and the same 20° floor that keeps a well-read arrow from being trusted absolutely keeps the charge from being so sharp that turning costs more than walking.

Where a short leg ends #

The one setting the turning reader adds is d, the length below which a leg’s direction stops counting, and the pause and the walk want opposite things of it.

A pause wants its legs ignored, because inside it they are the shuffle of a body turning, and the share of 90° pauses read rises from 64 in 100 at a quarter of a step to 92 at one step and 95 at two. A walk that doubles back wants every leg counted, because its legs are the only thing that says it went back — the earlier essay’s whole finding — and walks turning 120° fall from 75 in 100 at a quarter of a step to 72 at one and 61 at two. Walks turning 70° hardly mind, 95 against 92.

One step serves both: 92 in 100 for the pause, and 95 and 72 for walks turning 70° and 120°, against the walking reader’s 94 and 74. The turning reader costs a walk that keeps moving at most two walks in a hundred and buys a pause sixty-seven. A reader of a panel does not know whether it holds a pause before reading it, and this is the setting that can be used without knowing.

How far the walker must turn #

A facing is read off a drawn figure, and the earlier essay found a facing read to 15° about as precise as a painted figure honestly allows, with 30° and 45° the settings for a careless or stylised hand. The middle copy of a three-copy pause faces half the turn away from each of its neighbours, and whether it can be put between them depends on how that half compares with the error in reading each facing.

Facings read to 5° order a pause turning 45° 84 times in 100 and one turning 60° 93. Read to 15°, the walker must turn 90° for 92 in 100. Read to 30°, the reading tops out at three in four at 120° to 150°; read to 45°, it never passes 56. At a 15° turn all four are near chance, one in six, since three facings within 15° of one another are three versions of one arrow; only facings read to 5° get much past it by 30°, to 63.

So a painter who wants a pause’s order to be in the picture has to turn the figure a good deal between moments — the arrival, the look, the departure each facing clearly somewhere new — and a figure that merely shifts its weight from one foot to the other, as many paused figures do, has left the order of those moments to the convention of the page. That is not a failure of the facings. It is the same statement the same person, twice on one panel made about the whole of a continuous narrative: a panel supplies what its marks hold, and a reading convention supplies the rest.

Four moments on one spot #

The order is held by the turn between neighbouring moments, not by the whole turn, and drawing the pause in more moments makes each step of it smaller.

At 90° of turn, with facings read to 15°, the turning reader orders a two-copy pause every time, a three-copy pause 92 times in 100 and a four-copy pause 69. Four copies turning 90° face 30° apart, three copies turning 60° face 30° apart, and the two are read at about the same rate — 69 against 77 — which is the turn between moments doing the work. A four-copy pause needs about 120° before it is read 86 times in 100.

The walking reader falls away much faster with the count: a two-copy pause 75 times in 100 at 90°, three 25, four 10. Every copy added inside a pause is placed by nothing but the legs inside the pause, and the walking reader has nothing to place it with. The gap between the two readers is the size of the problem the earlier essay’s prediction named: the copies a pause holds between its first and its last.

How still the walker must stand #

The turning reader ignores the direction of legs shorter than about a step, and that is what lets it treat a pause as a pause. Feet that wander while the walker turns make the legs inside it longer, and at some point long enough to be given directions.

Feet that stray 10 cm leave the turning reader where it was, 92 times in 100. At 30 cm — a person turning with a step to one side — it reads 80, and finds the pause every time. At 70 cm, half a step, it finds the pause 86 times and reads it 47. Those legs are now long enough to be given directions, and the directions are the shuffle’s, not the walk’s. The walking reader, which never relied on the legs inside a pause being short, reads a quarter of them at every setting.

A wandering pause is not the only way for the copies to drift apart on the floor. Marks read to 2 px rather than half a pixel do the same thing at a distance: undoing the floor divides by each foot’s drop below the horizon, about 43 pixels for a stop 15 m out, and a two-pixel error there moves a foot by 70 cm along the line of sight — half a step. With the marks read that coarsely the turning reader reads a 90° pause 62 times in 100 and finds it 85, while a reader told which copies pause, ordering them by facing alone, still reads 96. The facings have not lost the order; the floor has lost the pause.

What the turn asks of the panel #

Three of the earlier essay’s conditions carry over. The angle of view is guessed — a focal length is not an angle is why a panel without a stated format does not give it up — and a wrong guess stretches the floor from front to back and turns every undone arrow by the same rule. Guessed at half the truth, the pause is still read 91 times in 100; guessed at twice it, 82 — a guess too long stretches the floor along the line of sight, and the paused copies’ few centimetres of shuffle are stretched with it towards the length at which a leg starts to count. The eye’s height decides how hard the page compresses the arrows, and raising it to 3 m lifts the turning reader from 92 walks in 100 to 95, with nothing further to gain at 6 or 12 m. The horizon at eye level is why: the page’s compression of direction is the tangent of the angle down to the stop, and a pause compared only with itself loses less to it than a route through stops at different depths. And a walk that turns hard elsewhere is still read past its turns by the arrows: with the walk turning 70° between places as well, the pause is read 91 times in 100 and the whole panel 89.

Two conditions are new.

The walker turns one way. The pause here sweeps from the arriving facing to the leaving one without going back. A walker who turns to look behind, then turns back further than it started before going on, draws three copies whose facings are not in sweep order, and the reader that charges each turn will put them in the order a walker who swept would have stood in. That is the pause’s version of the earlier essay’s false arrow — a figure who faces somewhere other than where the story is going — and nothing in the facings can tell it apart from a walker who simply swept.

A pause is a pause in the walk. The copies are moments of one walker, and the reader charges a short leg for turning because a person standing still turns between moments. A narrative that places two different episodes at one spot — Masaccio’s three Peters in the Tribute Money stand on three different grounds, but nothing stops a panel setting two scenes at one doorway — carries two facings with no turn between them, and the reader will order them as if the walker had turned. Three procedures, one panel warns against reading a procedure off its traces, and this is that warning at the scale of a single doorway.

What the copies on one spot keep #

A walk’s order is on the floor, not on the page found that a walker’s places hold its sequence, provided the floor is undone first. The essay on facings found that the way each copy faces holds the sequence past the places where the walk doubles back, and holds its direction. This one finds the last script a walking figure writes: where the places run out entirely, because the walker stood still, the turn between its facings holds the order of the moments it stood there.

Read together, the three are one statement about what a projection destroys in a continuous narrative and what it leaves. It destroys the order of the moments outright — the page holds the copies side by side, and four copies admit twenty-four readings — and it leaves three things from which that order can be rebuilt, as a tiring panel keeps its order, not its direction rebuilt the order a panel was painted in from the drift of the painter’s hand: where each copy stands, which way it faces, and, among copies on one spot, how far each has turned from the last. All three are written on a floor the page has compressed, and all three are undone by the same horizon. Each system answers its own question asks of every convention what its pictures keep rather than how close they come to another convention’s, and the answer for continuous narrative keeps growing: a good deal more of the story’s order than the convention admits, held in the bodies of the figures rather than in any device of the page. A carpet and the people on it is the painter’s own answer to the compression, a ground drawn from above so that the walk can be followed by eye; a pause on that ground is drawn as figures standing on one spot and facing round, which is a turn any viewer can read without undoing anything.

Still open: whether a walker who looks back can be told from one who turns on #

Every paused walker here turns one way, from the leg it arrived by to the leg it leaves by, and the reader that charges each turn reads the order a sweeping walker would have stood in. Storytellers also draw the other kind of pause: a figure who stops, looks back the way it came — at a companion, at what it has left — and then turns forward again before going on. Its three copies face the arriving leg, back along it, and the leaving leg, and the cheapest sweep through those facings is not the order of the moments.

A walker who looks back leaves a mark the sweeping walker does not: one of its copies faces against the leg it arrived by, a direction no walker sweeping from that leg to the next ever passes through on the short way round. The measurement that settles what that is worth gives a pause a look back of a stated angle in its middle moment, asks how often the turning reader then misreads it and how often a reader allowed one reversal inside a pause — charged for it, so that a walker who sweeps is still read as sweeping — returns the order instead; how far back the look must turn, against the error in reading each facing, before the reversal is told from a sweep; and whether the leg the walker arrived by, which the look back points along, is what makes that possible, since a look back at something off to one side would point along no leg at all.