The same person, twice on one panel

Trajan’s column shows the same emperor eighty-odd times. The Bayeux tapestry shows Harold arriving, swearing and dying. A predella panel shows a saint entering a city, kneeling in it, and leaving it, all within one frame and standing on one floor. The convention is called continuous narrative, it is older than any other in this field, and it is by a long way the least strange of them geometrically.

That is the finding, and it is worth stating plainly because the field’s other conventions are all departures from a projection through a centre. This one is not. There is one camera, one ground plane, one scale, and nothing about the projection is unusual at all.

What the picture claims #

What is unusual is the claim the panel makes, and the claim is not geometric. Four standing figures on a floor is a photograph of four people. The panel asserts that they are one person at four times — and no arrangement of marks can carry that assertion, because a projection maps a scene to a picture and says nothing about when the scene was.

So the essay divides in two. What can be measured is whether the four copies are consistent with one room, and they are, to the arithmetic floor. What cannot be measured is which order they go in, and the amount of that is countable.

One room, tested #

The test is the horizon, and it is the test the construction field already owns. Two figures of equal height standing on a plane give one point of the horizon: the line through their feet and the line through their heads meet there, because the two lines are images of two parallel lines in the world. The horizon, and the fraction makes that a construction and a picture with nothing straight in it is where the collection measures how well it works on a crowd.

Four copies give six such points, one for each pair, and if the copies really are one figure on one floor then all six lie on one line — the camera’s own horizon, which the construction never sees.

They do, to 8.5 × 10⁻¹⁴ pixels. And they are spread over several hundred pixels along that line, which matters: six points that all landed in the same place would be one measurement counted six times.

The instrument cannot tell a narrative from a colonnade #

The machinery that finds a horizon in a row of columns cannot distinguish a row of columns from a narrative panel, and that is the honest statement of what the convention costs a reader: nothing.

A picture with four copies of one saint and a picture with four different saints of the same height produce identical measurements. Every geometric quantity a reader can recover from the panel — the horizon, the camera’s height, the ground positions, the relative depths — comes back the same in both cases, because the geometry is the same in both cases.

That is why the convention worked for two thousand years without anybody having to be taught it. It costs the picture nothing it was using.

What the panel does not say, counted #

The geometry recovers where each copy stood. It says nothing about when, and the amount of nothing is a count.

With four copies there are twenty-four orderings and the picture is equally consistent with all of them. A reading convention — left to right, top to bottom, the order of a text beside the panel — picks exactly one, and picking one out of twenty-four is 4.58 bits.

That is the same shape of fact as the one the metrology field spends its length on. The one thing a single view cannot give says a photograph supplies every ratio and no size, so one length must come from outside; here the picture supplies every position and no sequence, so an order must come from outside. In both cases the outside supply is precise, small, and not optional.

When the order is in the marks after all #

The count above assumes the copies are interchangeable. Often they are not, and then the bits are in the picture and the reader is not supplying anything.

A figure that faces the direction it is walking carries the order in its own pose. A panel with a doorway on the left and a tomb on the right carries it in the setting. A tapestry with a text running along the top carries it in the text, which is outside the marks but inside the object. And a column whose narrative spirals upward carries it in the spiral.

So the honest form of the claim is conditional: a panel of four identical, symmetrical figures admits twenty-four readings, and every departure from identical or symmetrical takes some of them away. What the geometry can say is how many the marks would admit if nothing else were doing the work, and that is the number above.

How the bits scale, and why a layout is doing most of the work #

Four copies give 4.58 bits and seven give 12.3, and the growth is worth following out, because it says which part of the convention is carrying the load.

The count is $\log_{2} n!$, which by Stirling is about $n\log_{2} n$ — superlinear. At seven copies it is twelve bits; at eighty, which is roughly what Trajan’s column carries, it is about three hundred and ninety. So a single global rule — read upward along the spiral — supplies four hundred bits of sequence, and supplies them at no cost per figure.

Set that against what the marks can carry locally. A figure that faces the way it is walking supplies about one bit; a distinguishing attribute supplies another. So $n$ figures carry on the order of $n$ bits between them, against the $n\log_{2} n$ the ordering needs — and the gap grows without bound.

Poses cannot carry the order for a long narrative. At four copies the two counts are 4 against 4.6 and the marks nearly manage it; at eighty they are 80 against 390 and the marks are short by a factor of five. Something else has to supply the remainder, and the only candidate in the object is the layout: the fact that the figures are arranged along a line, a spiral or a band, so that a reader needs one rule about direction rather than one decision per pair.

That is the whole reason continuous narrative is always laid out along something. A panel of eighty copies scattered at random over a wall would need three hundred and ninety bits supplied figure-by-figure and would be unreadable; the same eighty along a spiral need one. The layout converts an $n\log n$ problem into an $n$ problem plus a constant, and it is the layout rather than the drawing that does it.

Which puts this convention beside the register rather than opposite it. A register spends its depth axis to buy legibility; a continuous narrative spends its layout to buy sequence. Both are using a spatial axis to carry an ordinal that the projection has no way to record, and in both cases the ordinal is cheap precisely because it is coarse.

The control: a panel that enlarges its protagonist #

A test that passes everything proves nothing, so the horizon test is asked to fail. Take the same panel and draw one figure larger than the others, which is what a painter does when one of them matters more.

Six per cent of extra height puts the six horizon points thirty-six pixels apart. Fifty per cent puts them six hundred and eighty-nine apart. So the instrument that passes a narrative panel catches an enlarged one, and the same test does both — which is what makes the two conventions comparable rather than two vocabularies standing side by side.

That comparison is the whole of the next essay, where the enlargement is not a slip but the point of the picture.

What the convention actually spends #

It is worth being precise about what continuous narrative gives up, because the answer is unusual for this field.

It does not give up the centre: there is one. It does not give up straightness, ratio, diminution or boundedness — the five properties what each system keeps tabulates for the parallel systems. It keeps all of them, because it is a photograph.

What it spends is time, and time was never in the picture to begin with. A photograph of four moments is not available: a camera takes one, and a long exposure of a moving figure gives a smear rather than four copies. So the convention is not paying a geometric price at all; it is representing something the medium has no way to represent, and doing it in the one way that costs the geometry nothing.

Where the reading goes wrong #

There is a failure mode, and it is a reader’s rather than a painter’s.

A panel of four copies read as four people gives a perfectly consistent scene: four figures of one height, standing where the geometry says, at one moment. Every measurement holds. What changes is what the picture is of, and no measurement can recover that.

This is the same boundary counting the eyes needs the room runs into from the other side. A picture can be measured for consistency with a room, and consistency does not identify the room — the panel is consistent with a room containing four people and with a room containing one person four times, and the marks do not choose.

The panel as evidence about a room #

There is a use for all this that is not about art history, and it is the reason the horizon test is worth running on a painting at all.

A panel that passes the test is a panel whose figures were placed by somebody who had a consistent room in mind — or who copied one. A panel that fails it by a large amount was made by placing figures where they looked right, which is a different procedure and leaves a different signature. Three procedures, one panel sets out how the collection distinguishes procedures from their traces, and this test is one more entry in that list.

The signature has a useful property: it is scale-free in the right way. A panel photographed badly, cropped, printed at any size, or reproduced with a lens distortion still has its horizon points on one line, because collinearity survives any projection — which is the whole content of what a projection destroys read from its surviving side. So the test can be run on a photograph of a photograph of a fresco, which is generally what a reader has.

What it cannot survive is the panel not being flat. A fresco on a curved apse is a picture on a curved surface, the composition through it is not a projectivity, and the collinearity that the test depends on is gone — which undoing a picture made on a curve measures and repairs.

Six points, and why not two #

The construction needs at least three copies and works better with more, and the reason is worth a paragraph because it is the same reason the metrology field gives for preferring one reference over another.

Two copies give one horizon point. A point is not a line, so two copies constrain the horizon to pass through somewhere and leave its direction entirely free — which is a one-parameter family of horizons, all consistent with the marks. Three copies give three points and the first genuine test: three points are collinear or they are not, and the residual is a number.

Four give six points and a real fit. The improvement from three to four is much larger than the improvement from four to five, which is the ordinary shape of a fit going from barely determined to comfortably determined, and it is the shape the ladder of assumptions is a ladder of conditioning draws for the general case.

There is also a placement effect. Two copies close together in depth give a horizon point far off the paper, computed as the meet of two nearly parallel lines, and a small error in either mark moves it a long way. So the pairs that carry the evidence are the ones spread in depth, and a panel with all its figures at one distance gives six points that are all badly determined — which is a panel with figures standing in a row, and rows of figures are common.

The boundary, stated #

One ground plane, one camera, and figures of equal height. Each can go.

Figures at different heights are not a failure of the convention but a failure of the test: real people differ, and a picture with nothing straight in it prices what a scatter of heights does to a recovered horizon. A panel whose copies stand on stairs or on separate terraces has several ground planes and the six meets scatter accordingly, which is the register convention arriving early.

And a panel whose copies are at different scales on purpose — a saint large at the altar and small in the distance — is either a perspective or a rank-scaled picture, and telling those apart is the next rung’s whole subject.

Why this convention is filed here at all #

A reader who has followed the field this far might reasonably ask why continuous narrative belongs in a collection about projection, given that the essay’s own finding is that it does nothing unusual to the projection.

The answer is that this field’s method is to measure what a system preserves rather than to rank it against a photograph, and a system that preserves everything is a row in that table rather than an absence from it. Each system answers its own question is where the collection sets that principle out, and the principle cuts both ways: a convention that spends nothing has to be measured to establish that it spends nothing, and the establishing is this essay.

There is a second reason and it is the more useful one. The instrument built here — six horizon points from four figures, and their scatter — is the instrument the next three essays use. The rank-scaled picture fails it in a particular way. The register picture fails it in a different way, by having no horizon at all. And a picture assembled from several stations fails it in a third. Having one test that all three are measured against is worth more than three descriptions, and the test had to be calibrated on the case that passes before it could be trusted on the cases that do not.

That calibration is the whole of the control section above. A test that has never passed anything is as useless as one that has never failed anything, and this convention is what it passes.

What is measured here #

Three numbers.

The six horizon points a four-copy panel gives lie on the camera’s own horizon to 8.5 × 10⁻¹⁴ pixels, and they are spread along it rather than piled up. Four copies admit twenty-four orderings, which is 4.58 bits a reading convention supplies from outside the marks. And the same horizon test, run on a panel whose second figure is drawn six per cent taller, returns thirty-six pixels — which is what says the first number is a measurement rather than a property of the instrument.

The short version #

Continuous narrative is the one convention in this field that departs from a projection through a centre in no way at all. One camera, one floor, one scale, and the horizon test that finds a colonnade’s vanishing line finds the panel’s to the arithmetic floor.

What the picture withholds is the order, and the amount is countable: four copies admit twenty-four readings, and a convention supplies four and a half bits from outside the marks. That is the same shape of statement as the one length a measurement needs from outside a photograph, arrived at from the other end of the collection.