A walk's order is on the floor, not on the page

The same person, twice on one panel measured continuous narrative and found it the least strange convention in its field: one camera, one floor, one scale, and four copies of a figure that meet the horizon pair by pair to the arithmetic floor. What the panel withholds is the order. Four interchangeable copies admit twenty-four readings, a reading convention picks one, and the bits it picks with come from outside the marks. That essay went on to argue that a long narrative cannot carry its own order in its figures — a pose supplies about a bit, and eighty copies need nearly four hundred — and that the remainder is carried by the layout: the copies stand along a line, a band or a spiral, so a reader needs one rule about direction rather than a decision for every pair.

That argument treats the layout as a convention the painter adds. There is a case in which it is not added at all, and it is the commonest subject of the convention: a person walking. Between one moment of a story and the next a walker moves a step or two and does not leap. So the moment after any copy stands next to it on the floor, and the moments that are far apart in time are, for a walk that does not double back, far apart in space. The order is then in the places, which the panel does record. The question is whether a reader can get it out, and the answer turns on a distinction the first essay did not need to make: next to it on the floor is not next to it on the page.

Nearest on the floor is next in time #

The rule a reader applies is the one a child applies to a dot-to-dot with the numbers rubbed off: join every copy to its nearest neighbours, in the single route through all of them that is shortest. For a walk whose steps are all of a length and whose turns are gentle, that route is the walk, because any other route has to replace at least one short step with a longer jump. The rule returns an order up to reversal — a route has two ends and no arrow — so for six copies it narrows the 720 possible orders to two, and leaves a reader one question to settle: which end came first.

The walks measured here are made to be the commonest kind. Six stops 1.4 m apart, a stride and a half, each heading turned from the last by an angle drawn from a normal distribution with a standard deviation of 30°; every stop inside the angle of view and between 7.5 and 18 m from an eye 1.6 m above the floor, which is standing height; every copy 1.72 m tall. The panel is upright, as panels are, so the camera is level and the horizon runs through the copies’ heads, which is the arrangement painters call isocephaly and which an eye at standing height makes automatic.

From the standing eye the shortest route on the page runs 4–5–3–6–2–1 — it keeps only two of the five true neighbours. The walk crosses the near part of the floor from right to left (1, 2, 3), turns at 4, the stop furthest left, and goes away from the eye (5, 6). On the floor that is a hook 3.8 m across and 3.5 m deep; drawn at the scale of a panel 690 px wide by a standing eye, the same hook occupies a patch about 185 px wide and 26 px deep. Every step away from the eye has been drawn a few pixels long and every step across thirty or forty, and the shortest route on the page prefers to stitch up and down through the depth, where the drawn distances are small, rather than follow the walk along its width.

Why the page misreports which copies are close #

The page does not merely shrink the far end of the walk; it shrinks depth more than breadth, and by a factor that is easy to state. A level eye at height $E$ draws a point of the floor at depth $Z$ and across-distance $X$ at

$$x = c + \frac{fX}{Z}, \qquad y = y_h + \frac{fE}{Z},$$

so a small step across moves the drawn foot by $f/Z$ per metre and a small step away moves it by $fE/Z^2$. The ratio is $E/Z$, which is the tangent of the angle the eye looks down at the stop. From 1.6 m up, a stop 8 m away is seen 11° down and a stop 11 m away 8° down, and a step away from the eye is drawn at a fifth to a seventh of the length of the same step across. A reader measuring nearness on the page is using a ruler whose depth graduation has been compressed fivefold, and nearness is exactly the quantity the shortest-route rule depends on.

This is the horizon at eye level seen from its floor side. That essay sets out why an upright picture puts the horizon at the height of the eye; the same upright picture plane is what makes the compression a tangent rather than a sine, and it is why every panel of this kind, painted on a wall or a board and seen square, compresses its floor the same way.

Undoing the floor needs the horizon and one guess #

To read nearness on the floor, the floor has to be put back, and the panel holds almost everything that takes. The horizon it holds twice over. The first essay found it as the line through the meets of the copies’ feet and heads; here a form that degrades more gently is used. Every copy is the same person standing on the same floor, so each foot sits below the horizon by the same multiple of that copy’s drawn height:

$$y_{\text{foot}} = y_h + k\,(y_{\text{foot}} - y_{\text{head}}), \qquad k = E/H,$$

and a straight-line fit of the feet’s rows against the drawn heights returns the horizon as its intercept and the eye’s height, in figure heights, as its slope. On the panel above, with its marks exact, it returns the horizon to the last digit and the eye at 0.930 figure heights, which is 1.6 m over 1.72.

Then each stop drawn $d$ pixels below the horizon is $fE/d$ deep and $(x - c)E/d$ across. The centre column $c$ only slides the plan sideways and does not change a single distance between stops. The one number the panel does not hold is $f$, its angle of view, and a focal length is not an angle is the reminder that a picture without a stated format does not give it up.

The guess turns out to matter very little. Guessing the angle of view wrong stretches every depth by the same factor, so the recovered plan is the true one drawn longer or shorter from front to back. A stretch can change which stop is nearest to which, but only for stops at nearly equal distances in two directions, and a gently turning walk has very few of those: each stop’s two neighbours in time are a step away and everything else is further. Across two hundred walks, a reader who guesses half the true focal length reads the order 99 times in 100, one who guesses it right 100 times, and one who guesses twice the true value 97 or 98. A painter’s angle of view does not need to be known; it needs to be guessed to within a factor of two, which the proportions of a room or the size of a doorway nearly always allow.

Every copy added is another chance to go wrong on the page #

The single panel is one walk. The question is how often each reading succeeds over the walks a painter might draw, and how that changes as the panel holds more moments.

The floor’s route returns the order walked in 99 or 100 walks of every hundred, from three copies to eight, and the dashed line it sits on is the share of walks that are their own shortest route on the true floor — every one of these, at a turn of 30°. The page’s route returns it 91 times in 100 for three copies and 54 for eight. Reading left to right, the convention the first essay counted, does worse still, 90 down to 48, because it is right only when the walk happens to cross the panel without turning back.

The decline on the page has a plain cause. Each copy added is one more stop that might lie a step away in depth from one neighbour and a little further across from a non-neighbour, and on a page that draws depth at a fifth of its length, any such stop is joined wrongly. With three copies there are few chances; with eight, nearly half the panels have used one. On the floor the chances do not arise, because the distances are the walk’s own.

So the first essay’s arithmetic needs a correction for walkers. It said that $n$ copies need $\log_2 n!$ bits and that their poses carry about $n$, and that the layout must carry the rest by being a line or a spiral. For a walk the layout is not a rule laid over the copies; it is the copies’ positions on the floor, and read there it carries $\log_2 n! - 1$ bits by itself — 8.5 of the 9.5 that six copies need. The bit that remains is the direction, and a single copy facing the way it is going supplies it. A walking figure drawn in profile, which is how most walking figures are drawn, therefore carries the whole order of its panel in its stance and its footprints, with no reading convention at all.

From high enough, the page is the floor #

The compression that misleads the page reader is a tangent, and a tangent is one at 45°. An eye that looks down on the walk at about that angle draws a step away and a step across at nearly the same length, and then the page is a plan of the floor at one scale, needing no undoing.

Over three hundred walks the page’s route returns the order 65 times in 100 from an eye a metre up, 69 from standing height, 78 from 3 m, 97 from 6 m and all 300 from 12 m — an eye about as high as the walks are far. Higher still the page stretches depth instead of squeezing it, and the route degrades again, to 94 at 24 m and 80 at 48. The floor’s route is right from every height, and reading left to right is right 63 times in 100 at every height, because raising the eye moves no copy sideways.

That curve has a painter’s history attached to it. The narrative panels of the fourteenth and early fifteenth centuries very often put the horizon high, or leave it off the top of the picture altogether, and set the story on a road that winds down a hillside seen from above. Sassetta’s Meeting of Saint Anthony and Saint Paul shows the hermit three times on such a road. A road seen from high above is drawn nearly as a plan, and on a plan the nearest copy is the next moment — so these panels can be read along the road by eye, with no geometry. A carpet and the people on it measures the same choice in Persian painting, where the ground is drawn from overhead and the figures from in front; the overhead ground is exactly the one that keeps distances on the page what they are on the floor. And measuring a room off the page finds the parallel version of the fact, that an oblique drawing of a floor is already rectified.

The panels that lower the horizon to eye level — the Renaissance arrangement, with heads level along it — are the ones that need the floor undone. The convention did not stop being readable when the horizon came down; the reading stopped being a matter of looking along a road and became a matter of construction.

The control: a walk that doubles back defeats every reader #

A rule that returns the right answer every time might be returning it for a reason other than the one claimed. The claim here is that the floor holds the order because a walker moves a step between moments; if that is the reason, then a walk that turns back on itself — whose later stops land nearer earlier ones than a step — should defeat the floor reader too, and exactly as often as the walk stops being its own shortest route.

It does. At a turn of 10° the walks are nearly straight and both readers succeed, 100 and 98 times in 100. At 50° the floor reader’s 89 matches a ceiling of 90; at 70° its 58 matches 59; at 120°, where a walker is as likely to turn back as to go on, it reads 19 against 20. The page reader is below it everywhere and falls faster, 27 at 70° and 6 at 120°. The two curves for the floor are one curve, to within the scatter of two hundred walks, and the difference between them is only the half pixel of marking error. The floor reader returns the order when the order is on the floor and only then. When the walk crosses its own path the panel no longer holds the sequence in any form a reader can extract from positions, and that is a property of the walk, not of how the picture was read.

Marks too coarse for the far end of the walk #

The floor reader’s advantage costs something, and the cost is in the far copies’ marks. Undoing the floor divides by each foot’s drop below the horizon, which is large for a near stop and small for a far one; on this panel a stop 40 m away is drawn 16 px below the horizon, and a foot misplaced by a pixel there moves the stop 2.5 m in depth — more than a whole step. The page reader never divides by anything, so its errors stay the size they were drawn.

Walks kept within 12 m are read from the floor 96 times in 100 with feet marked to 2 px, and 71 with marks to 4 px. Allow them to wander to 40 m and those figures fall to 34 and 17, while the page reader, whatever the marking, holds between 60 and 73. So a panel painted deep and coarsely — a crowd of small copies far up a valley, placed to a few pixels of a panel 690 wide — is better read straight off the page, and only a panel whose far copies are placed with real care is better undone first. The crossing lies between 1 and 2 px for walks reaching 28 m and between 2 and 4 px for walks reaching 18 m — a few tenths of a per cent of the panel’s width. That is a precision a careful painter achieves and a hurried one does not.

What is assumed about the walker #

Four things are taken as given and each could fail.

The copies are one height and stand on one floor. Both are needed to read the horizon and the eye’s height from the copies, and both are the assumptions a picture with nothing straight in it prices for a crowd of different people; here they are a single person, so the first holds by the convention’s own claim. The second fails on stairs and terraces, where the copies stand on several floors and the plan is several plans.

The moments are evenly spaced along the walk. A walker who pauses for three moments in one place and then strides off leaves a cluster and a gap, and the shortest route through a cluster can take its copies in any order. The rule recovers the order of places visited, not of moments within a place.

The camera is level, because the panel is upright. A panel seen from above its centre — a ceiling, a tilted lectern — draws its floor through a pitched camera, and the horizon fit then needs the full set of meets the first essay used rather than one regression.

And, most importantly, the copies are moments of a walk rather than episodes of a story. In Masaccio’s Tribute Money Peter appears three times: at the centre with the other apostles, at the left drawing the coin from the fish at the lake’s edge, and at the right paying the collector. The order is centre, left, right, and it is the order of where the story happened, not of a walk between them; the shortest route through the three would put him left, centre, right and be wrong. A narrative whose copies are placed by the geography of the story carries the order of that geography, and only a reader who knows the story can say how it was traversed. Three procedures, one panel is the collection’s general warning about reading a procedure off its traces, and this is a case of it.

What the order costs, counted #

The one thing a single view cannot give is that a photograph supplies every ratio and no size; the first essay on narrative panels put beside it the statement that a panel supplies every place and no sequence. For a walker the second statement is too strong. The panel supplies every place, and the places supply the sequence up to its direction, provided three conditions hold — the walk does not double back, the floor is undone rather than read off the page (or the eye was high enough that the page already is the floor), and the far copies are placed finely enough to survive the undoing.

Under those conditions the reader’s contribution shrinks from $\log_2 n!$ bits to one. That is not a small change in kind. Each system answers its own question sets out this collection’s method of asking what a system keeps rather than ranking it against a photograph, and on that accounting continuous narrative keeps more than it was credited with: its projection is a photograph’s, and the sequence it seemed to spend is mostly still in the picture, written on the floor in a script the page distorts and the horizon undoes.

Still open: whether the way each copy faces reads a walk that doubles back #

The floor reader fails exactly where the walk fails it — a walker who turns back puts a later stop beside an earlier one, and the floor then holds the places without their sequence. But a walking figure carries more than its position. Drawn in profile, or three-quarter, each copy faces the way it is going, and the direction it faces is a short arrow on the floor pointing at the next stop.

The measurement that settles what those arrows are worth gives each copy a heading — the direction of travel at that moment, read from the drawn figure with an error of a stated number of degrees — and replaces the shortest route with the route that best agrees with both the distances and the headings: every leg should leave a stop roughly along that stop’s arrow. It then asks how far above the walk’s own ceiling that reading climbs on walks that turn by 70° and more, where the floor alone reads barely half; how coarse a heading still helps, since a figure’s facing is drawn far less precisely than its feet; and whether headings read from the page, before the floor is undone, suffer the same tangent compression as distances do — since an arrow pointing away from the eye is drawn foreshortened and its direction on the page is not its direction on the floor.