Bands with no overlap name a height for the eye

A picture in bands measured a register picture with the horizon instrument and got no answer, and it took the emptiness of that answer as the finding. The feet line and the heads line of any pair of figures in a band are parallel, so no horizon can be found, and so the picture has no vanishing line. It is what a parallel projection does, and that essay read the depth scaling of one as meaning that the picture has been drawn orthographically, with the depth signal deliberately removed.

That leaves a question the absence of a horizon does not answer. A parallel projection has an elevation. An axonometric drawing, a military plan and an isometric game are all parallel projections, and what makes them different is the direction the parallel rays arrive from. So if a stack of bands is to be read as a parallel view of rows of figures on a floor, it should be possible to say from what direction. The bands have a spacing and the figures a size, so the picture itself offers something to go on. Asked that way, the convention turns out to carry a number nobody assigned to it, and the number exposes something strange about the arrangement.

Rows, and a parallel eye #

Take rows of figures of height $H$, set $D$ apart in depth, on a flat floor. A parallel eye at an elevation $e$ above the floor draws each figure with a page height of $H\cos e$ and sets one row above the next by $D\sin e$. The share of a band that each figure fills is therefore

$$s = \frac{H\cos e}{D\sin e} = \frac{H/D}{\tan e}.$$

A band has a share a person can measure on the wall. Egyptian and Assyrian registers commonly have figures nearly filling their bands, with a share somewhere between 0.8 and 1.

The figure shows where the condition comes from. A ray that just clears the head of a figure in one row continues down to the ground. If it lands short of the next row’s feet, that next row is wholly visible. If it lands beyond them, part of the next row is hidden behind the first. The share reaches one when a ray from a head meets the next row’s feet exactly, at

$$\tan e = \frac{H}{D}.$$

So a stack of bands in which no figure overlaps the row in front commits the eye to at least that elevation, whatever else is true of the picture. It is an unusual sort of number. No painter set it, and nothing in the picture marks it. Yet it is forced by the plain visual fact that no figure is hidden.

One drawing, a family of eyes #

A band’s share alone does not fix an elevation. The picture gives a ratio, and what it fixes is the product of the row spacing and the tangent of the elevation.

A figure filling its band exactly could be rows 5.7 heights apart seen from 10° up, 1.73 heights from 30°, or 0.58 from 60°. All three draw the same picture, and nothing on the page tells them apart. That is the same structure as the one thing a single view cannot give, in the different setting of a parallel projection: a single number on the page, two unknowns behind it, and a family of equally good answers.

So a reader has to bring the row spacing from outside, as with any other scale. Sometimes the subject supplies it. Soldiers in file, oarsmen on a bench, captives roped together, offering-bearers in procession: a person knows roughly how far apart such people stand. The bar chart gives the eye’s elevation for a set of such subjects.

Shoulder to shoulder at half a metre, 73.6°. A file of men at a metre, 59.5°. A procession at two metres, 40.4°. A field at five metres, 18.8°. A town at twenty metres, 4.9°. Open land at a hundred metres, 1.0°. So a single convention, repeated on the same wall, implies rays nearly overhead for its crowded scenes and hardly raised at all for its landscape ones. In the first case, the elevation is that of a bird. In the second, it is a person on a low mound. The elevation goes wherever the subject sends it, and it can differ from band to band of one wall, since nothing obliges two bands to be read from one direction. That is a further reason not to think of the wall as one view, even though each band is a projection on its own terms.

The wall itself can be the picture plane #

Reading a stack of bands as an ordinary axonometric view has a problem that shows immediately. An eye at 40° looking down at a standing figure, on a picture plane that faces the eye, draws the figure shortened by the cosine, at 77 per cent of its height, with the top of its head and its shoulders in view. No figure in a register looks like that. Each is drawn at full height, side-on, with its feet on a ground line. Is the convention then a mixture, with one eye for the rows and a different one for the people?

It looked that way when this measurement was first set up, and the conclusion was tempting. It is wrong, because it assumes a picture plane that faces the rays, and a parallel projection has no such obligation. Keep the picture plane upright, on the wall, and let parallel rays arrive at it from above at an angle $e$. A point at lateral position $x$, height $y$ and depth $z$ behind the wall lands on the page at

$$(x,\; y + z an e).$$

That is an oblique projection, a member of the family oblique is a shear measures under the name of the cavalier and cabinet views, with its depth axis running straight up the page. It keeps everything in the plane of the wall at full size, and it stacks the rows up the page by $D\tan e$. A figure in a frontal plane is drawn at its true height, at any depth, with no diminution. Its rows fill their bands by exactly the same share formula as before. So a register has one eye after all, at an elevation set by the rows, on a picture plane that is the wall. The figures’ full height and their stacked rows are both what that single projection draws.

The contradiction has moved, though, and it can still be stated in one number. An oblique view keeps frontal planes and shears everything else. Any solid figure has depth, and its depth would be sheared up the page by $\tan e$. A person whose depth is a fifth of their height would show a top, as a band above their head, of 17 per cent of their height at 40° and 35 per cent at 60°. A circle underfoot would appear as an ellipse of about the same depth. A register’s figures show no such thing: no top to any head, no footprint to any body. So the picture is an exact parallel view of rows of flat figures, with no depth, like paper cut-outs on stands.

That is a finding of a much cleaner kind than the mixture would have been. It lines up with no solid casts an aspective figure, which measured a single body drawn with a frontal torso and profile legs and found that no solid, from any direction, casts it as an outline. The body is a flat composition before it is ever placed in a band, so flatness is already where the figure starts. A stack of bands is then the natural arrangement for such figures: rows of flat things, and a shear that moves them apart without spoiling any of them. A shear does nothing at all to a frontal plane. That property, and no other, is what lets a register be read at all.

So the convention’s choice can be put in the terms of a projection. Choose the wall as the picture plane; pick the shear so that no row hides another; draw only what lies in each row’s frontal plane. Each of these is an ordinary thing to do in a parallel drawing, and the same person twice on one panel makes the corresponding choice for time.

What overlap would have cost #

The convention’s reason can be priced by comparing it with what a single, consistent, low eye does to the same subject.

Rows one figure height apart, seen from 15° up, lose 73 per cent of each rear figure. From 20°, 64 per cent. From 30°, 42 per cent, and only at the critical 45° is the row completely clear. In a real group, an artist drawing a crowd from a natural viewpoint gets the first few rows and then a mass of heads, which is an honest picture of a crowd and a very poor way to record who was there, what each was holding, and in what order they went. For a procession of named tribute-bearers, whose whole value is in each being shown holding the proper object, a hidden figure is not a figure at all.

So the register’s choice is clear when it is put as a trade. It gives up each figure’s depth, and in return every figure is whole. A view of solid people from the same elevation would have shown the tops of heads and the shoulders from above, so that the figures would be recognisable only with difficulty. Treating them as flat and shearing only the rows is the combination that gets both: the rows are separated as if from high up and the person is recognised as if from level.

There is a known intermediate. Many later traditions overlap their rows a little and raise the eye a moderate amount, as with the blown-off roof of a Japanese narrative scroll, and there a figure typically shows some of its top. Such pictures sit at a point on the same family of curves with a low share and a moderate elevation, with figures that have regained their depth, and that is a real change in kind rather than a difference of style.

What the bands buy on a wall #

The arithmetic of a wall says something more direct about why a culture with a great deal to record might adopt them. A wall 3 m tall and 5 m wide can have one band, or many. With $n$ bands and a figure filling 90 per cent of its band, each figure is $2.7/n$ metres tall and, with some breadth in proportion to height, the wall holds a number of figures that goes as $n^2$.

One band holds 4 figures 2.7 m tall, which would be colossi. Three bands hold 36 at 0.90 m. Six hold 144 at 0.45 m. Twelve hold 588 at 0.23 m, and at four metres each of those subtends about 3.2°, which is still many times what an eye needs to tell a king from a captive. So a monumental wall has room for a great many bands before its figures become illegible, and the number of figures rises very steeply as the bands divide. Doubling the bands gives four times the figures at half the height. A narrative wall in bands has an information capacity that a single scene in perspective cannot approach, for any choice of viewpoint.

A single view of a large crowd in perspective would diminish its rear figures by distance and hide them by overlap. Its information falls away as the depth grows. A set of bands, having no diminution at all, can hold its count without loss at any depth into the story. A picture in bands priced what that essay called the band index, the count of a figure’s register, and this is the same economy from a different direction: having no depth means having no depth-related loss.

An objection, and what it does not reach #

The objection is obvious, and it is right as far as it goes. Nobody who drew a register thought of it as a parallel view from an elevation. It was a system with its own logic, closer to writing than to seeing. Lines of figures in a band read as lines of signs do in a text. Trying to find the eye behind it is a category error, a historian might say, and a fair number have.

That is correct as history, and nothing above claims otherwise. The value of the exercise is in the other direction. Reading a convention as a view does not show what its makers meant. It shows what would have to be true of the world for the picture to be a record of it. For this convention that is now stated exactly: rows of cut-outs, seen in oblique projection, from rays at an elevation that varies with the subject. It is a very odd world, and a modern viewer is being asked to accept it without noticing. Viewers do not read a register as a strange aerial picture of paper people. They read a set of lines, because the picture’s own structure tells them to.

The fact that a picture can sustain such an oddity with no sense of error is what the essays on copying and on what is assembled from several views keep arriving at. A convention is held together by reading habits. Geometry can state how much it has been asked to carry, and it does not decide whether the carrying works.

Larger kings and bands out of reach #

A real register has more in it than this. Some have overlapping figures in a band, some have ground lines drawn as landscape, and some late ones have rows staggered within a band to suggest depth. Each is a local departure, and each moves part of the picture toward a different point on the same family of curves.

No account is taken of figure size within a band. Importance often sets it. A king is drawn larger than his attendants on the same ground line, and read as a view that would imply that the king is nearer, by a calculation a picture in bands refused for a good reason. It is a further instance of two conventions on one wall, and nothing here has priced it.

And the wall is treated as a flat rectangle seen from a stated seat. Real walls have doors, and the tallest bands are often out of reach of the eye, so that a viewer sees them at a slant. At the height of a pharaonic temple, that makes the upper bands’ figures appear shorter again from the ground, which may have encouraged enlarging them.

Still open: whether a band’s depth can be read at all #

Everything here treated a stack of bands as one spacing and one share. In a real register the bands are not all the same, and a higher band’s figures are sometimes drawn a little smaller or set with a little less room. That might be an accident of the wall, or a working practice, or it might be the first appearance of a depth cue. A band that diminishes consistently going up the wall has a vanishing line in the sense a picture in bands found absent, however faint.

The measurement that settles it takes a stack of bands with a stated small diminution from band to band, reads their ground lines and figure heights with a stated hand error, and asks how large that diminution has to be before the horizon instrument returns a finite answer, and how far above the wall that horizon would stand. Since figures vary anyway, it also asks how many bands, and how many figures in each, a real wall would need before such a trend could be told from the ordinary scatter of a hand.

Read as a view, a register is one exact oblique projection, from rays high enough to clear each row, of people with no depth at all. The convention is sustained by never being read as a view.