A picture in bands

The standard of Ur is in bands. So is the palace relief from Nineveh, so is Trajan’s column once it is unrolled, so is a great many manuscript pages, and so — without a break in the tradition anybody bothered to record — is every comic strip and every storyboard drawn since.

The convention divides the surface into horizontal strips, gives each strip its own ground line, and draws every figure at the same height. This essay puts that arrangement through the instrument the previous two built, and gets an unusual answer: not a bad reading, but no reading at all.

The measurement is an absence #

The horizon of a plane is where two parallel lines in that plane meet in the picture. For figures of equal height standing on a floor, the line through two feet and the line through the two heads are images of two parallel lines, so they meet on the horizon — which is the construction the same person, twice on one panel uses and a picture with nothing straight in it pushes to its limit.

In a register picture every figure in a band has the same drawn height and stands on the same ground line. So the feet line is horizontal, the heads line is horizontal, and the two are parallel.

They meet at 0.0 × 10⁰ degrees, which is a way of writing that they do not meet. There is no horizon point, so there is no horizon line, so there is nothing for the construction to find.

Not a bad perspective — a picture with the depth signal removed #

That contrast is the whole finding, and it is worth being careful about which way round it goes.

A picture drawn in registers is not a poor attempt at perspective that failed to converge. Convergence is the signal, and the convention has removed it deliberately: every figure the same height means every figure equally important and equally readable, which is the property a narrative frieze wants when it will be read at a distance in bad light by people who need to identify the figures rather than to place them.

What the geometry says is precisely how much has been removed. In perspective the drawn height of a figure at depth $d$ falls as $1/d$, so the ratio of two drawn heights is the reciprocal ratio of their depths — that is the whole of the depth information in a size cue, and a picture with no size–distance signal is where the collection prices its absence. A register picture sets every ratio to one, so it carries no depth information in size and the reciprocal cannot be inverted.

The depth scaling is one, which is what orthographic means #

There is a way of saying this that connects the convention to the parallel field rather than leaving it as a special case.

A parallel projection is the limit of a perspective as the eye goes to infinity, and its defining property is that the depth scaling is one: an object twice as far away is drawn the same size. The eye taken to infinity is where the collection takes that limit carefully, and two distances to infinity is where it distinguishes the two ways of getting there.

A register picture has a depth scaling of one within and between its bands, so it is an orthographic projection along the depth axis, stacked. That is not a metaphor. It is the same statement as “every figure is drawn the same height”, written in the parallel field’s vocabulary, and it means the whole of that field’s accounting applies: no station point, no horizon, no diminution, every length along the picture true to scale.

What a register adds to a plain orthographic drawing is the band index, and the band index is where the depth went.

What the band index is worth #

A band is an ordinal. Band three is further along than band two — later in the story, deeper in the room, higher up the hill — by an amount the picture does not state and has no way of stating.

So the ledger for a register picture reads: recovers the order along a band, the ratio of positions along one band, and which band is which. Withholds the depth of a band, the ratio of two bands’ depths, and any angle between directions lying in different bands.

That is a genuinely useful bargain and it is the same one a graph makes. A bar chart’s horizontal axis is often an ordinal too, and nobody complains that the gap between the third and fourth bars is not a quantity. What makes the register unusual is that its ordinal axis is drawn in the same picture space as its metric one, so a reader who is not told may take the vertical as a distance.

What the band index costs, counted #

The ledger above is a list of what survives and what does not, and the two halves can be put on one scale, because both are quantities of information about depth and both are countable.

A size cue carries about eight bits. A figure of real height $H$ at depth $Z$ is drawn at $fH/Z$, so a relative error in the drawn height is a relative error in the depth: $\delta Z/Z = \delta h/h$. A reader placing marks to a pixel on a figure drawn a hundred pixels tall therefore resolves depth to one per cent, and over a depth range of twenty to one that is $\ln 20 / 0.01 = 300$ distinguishable depths — about 8.2 bits.

A three-band register carries 1.6. The band index is an ordinal with three values, and $\log_{2} 3 = 1.58$. That is the whole of the depth information in the picture.

So the convention is spending roughly six and a half bits of depth per figure, and buying with them the property that every figure is drawn at a legible size. Put that way the trade is obviously sensible for the pictures it is used on: a procession’s depth ordering genuinely has three values, and paying eight bits to record which of three ranks a figure is in would be paying for precision nobody needs.

It also says what a register cannot be used for. Matching a size cue’s eight bits would take about three hundred bands — which is not a register, it is a raster — so any subject whose depth is genuinely continuous is a subject the convention cannot carry. A battle line, a river of tribute bearers and a sequence of episodes all have coarse depth; a landscape does not, and no tradition ever drew one in registers.

Which is the same conclusion the depth-ratio law reaches for the neighbouring convention, arrived at by counting rather than by measuring page: the subject’s own depth structure decides which system can carry it, and the deciding quantity is small and computable in both cases.

Where a reader goes wrong #

Two errors, and the second is the one that matters historically.

Reading the band separation as depth produces a room whose figures stand in ranks at regular intervals. Nothing in the picture supports that and nothing contradicts it; the picture is consistent with every assignment of depths that preserves the order, which is an infinite family. That is the same shape of statement as what a flat scene leaves free makes about a two-view reconstruction: not poorly determined, not determined.

Reading the equal heights as equal depths produces a room in which everything is at one distance, which is a plausible reading and is what a viewer does by default. It is also what makes the convention legible: the figures read as a procession seen from the side, which is very nearly what the picture is a projection of.

Where the bands come from, and why they are horizontal #

There is a structural reason the bands run across rather than up, and it is not decoration.

The one direction a register keeps metric is the direction along the band, so that direction has to carry whatever the picture is measuring — the length of a procession, the order of events, the extent of a battle line. Those are all things that happen along a ground rather than up a wall, so the band runs along the ground, which images as horizontal for a level view.

The direction that is spent is the one perpendicular to the band, and the convention spends it on the ordinal. Stacking upward rather than sideways then follows from the shape of the surfaces the convention was used on — a wall, a column, a page — all of which are taller than the thing being depicted is deep.

Two bands are a stereo pair that is not one #

There is a tempting reading of a register picture that is worth putting down carefully, because it is nearly right.

Two bands showing the same procession look like two views of one scene, and this collection has a great deal of machinery for pairs of views. The machinery does not apply, and the reason is instructive: two views of one scene are two projections of the same world points, and two bands of a register are projections of different world points. The picture is one projection with its results laid out in strips, not two projections compared.

The distinction has a test. Two views of one scene have correspondences — pairs of marks that are images of one point — and a register has none, because no figure appears in two bands. Where a figure does appear in two bands, which is what a continuous narrative laid out in registers does, the pair is a correspondence in time rather than in space, and the epipolar apparatus a point is a line over there builds has nothing to constrain.

So a register is a single picture with an unusual layout, and calling its bands views would import a whole apparatus that has nothing to work on. Naming what an arrangement is not is worth as much here as naming what it is, and it is why the ledger above lists what the convention withholds rather than leaving the reader to assume.

What the convention buys, priced #

The essays in this field measure what a system spends. It is fair to ask what this one buys, and the answer is a number rather than an aesthetic.

A perspective picture of a procession fifty figures long gives its far end almost no room: the drawn height falls as one over depth, so the last ten figures occupy a fraction of the page and are unreadable. A register picture gives every figure the same height, so the readable extent is the whole surface divided by the figure’s width — which for a fifty-figure procession on a wall is fifty legible figures rather than a dozen and a smear.

That is the same trade the landscape convention two essays along makes for a different reason, and it is a trade the geometry can price. What is bought is legibility, measured as the number of figures a surface will carry at a readable size; what is spent is the depth cue, measured as the ratio of drawn heights, which the convention sets to one.

A designer today makes the same trade under a different name. An isometric drawing of a factory floor keeps every machine the same size on the page for the same reason, and what isometric actually means is where the collection sets out what that costs.

The comic strip is the same object #

The convention did not stop. A comic page is a register picture with the bands separated by gutters and the ordinal made explicit, and every property measured above holds of it.

Figures within a panel are drawn at whatever size the panel wants; the sequence between panels is an ordering with no stated interval; a reader supplies the reading order from a convention learned in childhood, which is the same 4.6 bits per four panels the same person, twice on one panel counts for a narrative panel. The gutter is an admission that the interval is not a quantity — it is drawn as a gap rather than as a distance, and its width means nothing.

A storyboard is the same again, with the ordering supplied by numbering. So is a strip of film, where the ordinal is time and the interval genuinely is constant — which is the one member of the family that has a metre on its second axis, and it is supplied by the machine rather than by the picture.

The boundary, stated #

Bands of equal figures with their own ground lines. Real reliefs are less tidy in three ways and each changes the reading.

Figures that overlap carry depth information the sizes do not — an occlusion is a strict ordering of two objects and it survives any projection, which is why what a projection destroys lists occlusion among the survivors. A register picture with overlapping figures has depth information within a band even though it has none in the sizes.

Figures at different heights within a band — a mounted king among foot soldiers — restore a horizon of a sort, and a reader who finds one has found the ratio of two objects’ real heights rather than a camera.

And a band whose ground line is drawn as a landscape rather than as a rule is a band with its own internal depth, at which point the convention is a stack of small perspectives and the bands are frames rather than registers.

Why the answer had to be exact rather than small #

One property of the measurement deserves a sentence of its own, because it is unusual in this collection and it is the reason the essay is short on caveats.

Almost every number here is a residual: something that ought to be zero and is 10⁻¹³ instead, or something that ought to be small and is 28 millimetres. This one is not. The two lines are parallel because the drawn heights are equal by construction, so the angle between them is zero for the same reason two copies of the same number are equal — there is no arithmetic in it to leave a floor behind.

That makes the reading unusually robust. A residual of 10⁻¹³ can be pushed up by a badly conditioned intermediate step, and a reader has to be told what precision to expect. An exact zero cannot. Any departure from parallel in a real register picture is a departure in the drawing, and its size is a direct measure of how carefully the heights were kept equal — which turns the instrument from a classifier into a measure of workmanship, and puts it beside the hand-error measurements the construction field makes.

What is measured here #

Two numbers and a ledger.

The feet line and the heads line of every pair within a band are parallel to 0.0 × 10⁰ degrees, which is exact rather than small, because the drawn heights are equal by construction. The same instrument on a photograph of the same figures returns 15.4 degrees at the widest pair. And the ledger of what the convention recovers has three entries and what it withholds has three, with the division falling exactly along the axis the bands run.

The short version #

A register picture draws every figure at one height on its own ground line, so the construction that finds a horizon finds nothing — the feet line and the heads line are parallel, exactly, and the picture has no vanishing point anywhere in it.

That is an orthographic projection along the depth axis, stacked, with the band index carrying an ordering and no metre. What a reader recovers is position along a band and the order of the bands; what a reader cannot recover is how far apart the bands are, and the picture is equally consistent with every assignment that keeps the order.