One flight, two pictures

A stair does not use all its faces finds that a flight of nine steps is eighteen planes and that no eye reaches more than fifteen of them. It leaves a consequence undrawn, and the consequence is the more interesting half.

If the faces one reader can use and the faces another can use are different faces, then one object can carry two pictures.

The two ends of one flight #

A flight has treads and risers, and which of them an eye can reach is decided entirely by which end of the flight it is at.

From the top, coming down, every riser faces down the stairs and away from the reader. Not some of them, and not at some eye heights — at every eye height between 1.4 and 5.0 metres, not one riser is reachable. The design lands entirely on treads, and 68 per cent of it lands on the first one.

From the foot, going up, both kinds are reachable. The risers face the reader squarely and the treads are seen at a shallow angle, and the design divides 68 per cent to the risers and 32 to the treads.

So the two readers do not compete for the risers at all: one of them cannot use them. What they do compete for is the treads.

The arrangement that works #

Give the reader at the top the treads and the reader at the foot the risers, and no face carries two pictures.

That is not a compromise: it is the allocation each reader’s geometry would have chosen anyway. The risers are where the ascending picture is comfortable — a median stretch of 1.41 and a worst case of 2.1, so the marks are nearly undistorted everywhere — and the treads are where the descending picture is comfortable, at a median of 1.64 and a worst case of 3.3.

Both pictures are therefore painted on the faces that suit them, and each reader’s own picture is the well-behaved one.

The cost is that the ascending reader gives up 32 per cent of the design they could have had. Their picture occupies the riser strips only, with the treads between them carrying somebody else’s — which is exactly what a real stair mural is, and this is the geometric reason for it rather than a convention.

The asymmetry, which is the interesting part #

The arrangement is not symmetric, and the way it fails to be is the finding.

The reader at the top cannot see the risers at all. Every one of them faces away, so the ascending reader’s picture is invisible from the descending reader’s position — completely, not faintly.

The reader at the foot can see the treads, and the descending reader’s picture is on them. So the ascending reader sees both pictures, interleaved: their own on the risers, and a stranger’s on the treads between.

What saves the arrangement is the angle. From the foot, the treads are seen at a shallow angle — a median stretch of 2.59 and a worst case of 12 — so the picture painted on them for somebody standing at the top arrives at the ascending reader compressed into slivers, up to a dozen times narrower than it was drawn. It reads as texture between the riser strips rather than as a competing picture.

That is a satisfying resolution and it should be stated as a measurement rather than as a reassurance: a third of the ascending reader’s field is tread, and the content on it is squashed by a median factor of two and a half. Whether that is unobtrusive is a question about the pictures, and a bold flat colour on the treads would be extremely obtrusive.

The allocation rule, and why it needs no negotiation #

The arrangement is described above as the one each reader’s geometry would have chosen anyway, and that is a general rule rather than a fact about stairs. It is worth stating because it says when two readers can share an object and when they cannot.

Each face has a stretch for each reader: $1/\cos\iota$, the reciprocal of how squarely that reader’s rays meet it. Allocate every face to whichever reader sees it more squarely. On this flight the allocation is not close: the risers are at 1.41 for the ascending reader and unreachable — infinite — for the descending one, and the treads are at 1.64 for the descending reader against 2.59 for the ascending, a margin of 1.58. Neither reader would trade, so there is nothing to negotiate and no compromise to price.

That also explains the protection the essay reports, and gives it a formula. Both readers see a horizontal tread at $1/\sin\delta$ for their own depression $\delta$, so a picture painted for one and looked at by the other arrives compressed by

$$\frac{\sin\delta_{\text{ascending}}}{\sin\delta_{\text{descending}}},$$

the ratio of the two stretches, which is exactly the 2.59-against-1.64 the measurement gives as a median and reaches twelve at the top of the flight. The squashing is not a lucky feature of the arrangement; it is the same number as the allocation margin, so an allocation that is comfortably right is automatically an allocation whose loser’s picture is hard to read — the two statements cannot come apart.

Which is the general form of the result and it is stronger than the stair. Two readers can share any folded object without conflict wherever the faces they see squarely are different faces, and the protection each gets from the other’s picture is precisely the margin by which the allocation was decided. Where the margin is small the two pictures compete visibly; where it is large, one of them disappears into texture. A flight is a good object for this because its margin is infinite on half the faces and 1.58 on the rest, and nothing about that required either reader to give anything up.

One flight, two models of it #

A note on how the measurement is made, because a reader following the geometry will notice something odd about it.

The two readers are modelled as two objects: an “ascending” flight and a “descending” one, each with the eye 1.65 metres up and 2.2 metres in front of the intended picture. That is not two staircases. It is the field’s own convention for one flight seen from its two ends, and it is why the stair takes a flag rather than a rotation: the design plane stands in front of the reader in both cases, at the landing in one and at the foot in the other, which is where a real design would be.

The face correspondence between the two models is exact and worth stating: tread $k$ of the descending flight and tread $k$ of the ascending one are the same physical slab, and likewise for the risers. So an allocation made in the two models is an allocation of real faces, and a painter could carry it out with a single flight and two stencils.

What the models do not share is a coordinate frame, which is why the essay quotes shares and stretches rather than positions. A statement like “the ascending picture is 68 per cent riser” is frame-free; a statement about where a particular mark lands would need the two models reconciled, and nothing here needs one.

Why two eyes at the same end do not work #

The obvious variation fails, and it fails for a reason worth having.

Take two eyes at the same end of the flight — one standing and one on a child’s shoulders, say, or two seats in the same room at different heights. Their reachable face sets overlap almost completely: measured on an ascending flight from 1.65 metres and 2.9 metres, the two share fifteen faces.

So two readers at the same end are competing for the same surface, and one picture must be painted over the other. There is no allocation that gives both an undamaged picture, because there is no face that one of them can use and the other cannot.

The disjointness comes from the fold’s orientation, not from the eyes’ separation. Two eyes a metre apart at the same end share everything; two eyes at opposite ends share nothing that matters, because the flight’s risers turn away from one of them by construction. It is the object that separates the pictures, and only an object with faces pointing in opposite directions can do it.

What other objects could do it #

Once the mechanism is named — faces pointing in opposite directions, each reachable from one eye only — the question of which objects support two pictures answers itself.

A cluster of blocks does, and in more than two ways — though occlusion decides which of its faces any given eye actually reaches: a block’s four sides face four directions, so four eyes around it could each have their own faces. What limits it is size rather than geometry — each picture gets a quarter of the surface, and how high each eye stands decides how much of that quarter is reachable.

A corner does not. Its floor and wall both face into the room, so any two eyes that can see the design at all reach both faces, and the two eyes’ sets are identical.

A bare floor certainly does not: one face, one picture, and any second eye is simply a wrong eye.

A corridor does, weakly — and its doorway adds a face in another room that only a reader at one end can use: the two side walls face each other, so a reader at each end reaches both but at very different angles, and the design could be allocated by which wall each reader sees more squarely.

So the rule is that an object supports as many pictures as it has families of faces pointing in different directions, and each picture gets the family that faces its own reader. A staircase has two families and is the cleanest case; a smooth surface has one family per point and none of them is reserved to anybody.

A third reader, which this model cannot answer #

The obvious extension is a reader standing beside the flight, and the model here has nothing to say about it — which is worth admitting rather than extrapolating.

A real flight of stairs has ends. The steps stop at a stringer or a wall, and from the side a reader sees a sawtooth of step ends: a third family of faces, pointing sideways, reachable from neither the top nor the foot.

The flight in this collection has no such faces. The stair the field built is a set of treads and risers extending to a half-width, with nothing closing them off, because every question asked of it so far has been asked from in front. So the sideways reader’s picture cannot be measured here, and the rule above — an object supports as many pictures as it has families of faces pointing in different directions — predicts that a real flight supports three rather than two.

That prediction is testable and is not tested, and the honest form of it is a sentence with a condition attached: if the step ends are large enough to carry a design and the sideways reader’s rays reach them without being blocked by the stringer, a staircase is a three-picture object. Both of those are architecture rather than geometry, which is why the measurement was not attempted rather than attempted and reported vaguely.

What it does not solve #

The allocation gives each reader a picture on faces the other cannot use. It does not give either of them a picture from more than one point.

Each of the two designs is still an anamorph and still correct from exactly one eye, with the same few-centimetre tolerance and the same total loss to a second eye in the same head. Two pictures on one object is a statement about the object’s capacity, not about either picture’s audience.

Nor does it help with the descending reader’s own problem, which is that 68 per cent of their design lands on the first tread. A design cast from the top of a flight is crowded onto the nearest face and spread very thinly over the rest, so the “picture” the treads carry is mostly one tread’s worth — and a designer would do better to treat that as a single-face anamorph on the top tread than as a picture across nine. The stretch that decides a design’s band is the measurement that would set its size.

Two pictures, or one picture twice #

There is a variation that sounds like the same idea and is a completely different object, and separating them is worth a paragraph because the difference is the field’s oldest distinction.

The arrangement above gives two different pictures to two eyes. The variation gives one picture to two eyes — a design painted so that it reads correctly from either end of the flight.

The second is impossible, and the impossibility is not a matter of degree. A face carries one set of marks; the marks that make a given picture from one eye are a different set from the marks that make it from another; and no allocation of faces helps, because the trouble is on each face individually rather than in the division between them.

That is an anamorph has one eye restated for a faceted object, and it is worth noticing that the two-picture result does not weaken it at all. Each of the two pictures here is still correct from exactly one point, still fails at a step of a few centimetres, and still gives itself away to the reader’s own second eye. What the flight provides is not tolerance; it is capacity, and capacity and tolerance are separate quantities that a reader is entitled to confuse until somebody separates them.

The same distinction settles a question about curved displays in the neighbouring field: a curved screen serves one seat, and the reason it cannot serve three is that it is being asked to show one picture to three eyes. If it could show three different pictures — which a display cannot, since it has one panel — the audience would be a solved problem there too.

What a painter would actually do with it #

The allocation is geometric and the job it implies is not obvious, so it is worth walking through once.

The risers are painted as a strip picture. Nine risers of 0.17 metres each, seen from the foot at a median stretch of 1.4, is a picture in nine horizontal bands with 0.30 metres of tread between them — and from the design eye those gaps close, because the treads are edge-on. So the ascending reader’s picture is composed on a canvas that is 68 per cent present and whose missing 32 per cent is invisible from where they stand.

The treads are painted as a fan. The descending reader’s design is on the treads, and 68 per cent of it is on the first one, so the job is one large panel and eight thin slivers.

And the two are painted at different scales. The riser marks are nearly the design’s own size; the tread marks for the descending reader are within a factor of 3.3.

The practical consequence is that the two pictures are entirely different kinds of work. One is a nine-band mural on vertical strips; the other is a single large image on one horizontal slab with a tail. A designer who thought of “two pictures on one staircase” as a symmetric arrangement would be surprised by both halves.

The short version #

A flight of stairs has two families of faces pointing in opposite directions. The reader at the top can reach only the treads; the reader at the foot can reach both, and prefers the risers, which take 68 per cent of their design at a median stretch of 1.4.

Allocate treads to one and risers to the other and one staircase carries two pictures with no face doing double duty. The reader at the top sees only their own, because the risers face away from them entirely; the reader at the foot sees both, with the other picture compressed by a median factor of three into slivers between the riser strips.

Two eyes at the same end share fifteen of eighteen faces and cannot be given separate pictures at all, which is what makes the fold rather than the separation the thing that does the work.