A stair does not use all its faces

The anamorph that crosses a corner establishes the piecewise case. A design cast onto a floor and a wall is two homologies rather than one: the map is continuous across the join, because a point of the join is in both planes and the ray through it lands there either way, and its scale is not, because the two planes make different angles with the ray.

The obvious next case is a flight of steps, and it has been the obvious next case for some time. A flight of nine is eighteen planes rather than two.

The count is the least interesting thing about it.

What it has instead is a distinction that a corner cannot show, because a corner has too few faces for the distinction to have anywhere to live.

Two questions that are the same question on a corner #

On a corner, “which planes are there” and “which planes can be painted” have the same answer. There are two, the design eye sees both, and every ray of the design lands on one or the other.

On a flight they come apart completely, and which way they come apart depends on which end of the flight the reader is standing at.

From the top, no riser is reachable #

Stand at the top of a flight going down. Every riser faces away — down the stairs, toward the bottom — so no ray from the eye can reach one.

Measured across eye heights from 1.4 metres to 5.0, on a nine-step flight: the number of reachable risers is zero at every one of them, and the design lands entirely on the treads. That is not a conditioning statement or a matter of grazing angles; it is that half the flight’s surface has its outward normal pointing away from the eye.

So a descending flight offers a nine-plane picture surface out of an eighteen-plane object, and no choice of where to stand changes it. From a standing height eight of the nine treads take paint, which is the most the object has to offer that reader.

That is worth separating from a nearby and much weaker statement. It is not that the risers are hard to paint from the top, or that the design lands on them badly. There is no ray from the eye that reaches one, so there is no design on them at all — the question of how the design would look there does not arise.

A surface that faces away from the centre of projection is not a picture surface. That is obvious for a camera and it is easy to forget for an anamorph, because the design feels like something applied to the object rather than something projected onto it. It is projected, from one point, and everything a projection cannot reach is not part of the picture.

From the bottom, both kinds — and the picture concentrates #

Stand at the bottom of a flight going up and the geometry inverts, but not symmetrically.

The risers now face the reader, so all nine are reachable. And the near treads are still visible, because a standing eye is above them — four of them, at an eye height of 1.65 metres and 2.2 metres back.

So seventeen of the eighteen planes are used, which is the best any single eye does on this object — nine risers and eight treads, with only the topmost tread out of reach.

What is more interesting is where the picture goes. Fifty-eight per cent of the design lands on the risers, and the risers are only 36 per cent of the flight’s surface — 17 centimetres of riser against 30 of tread.

The disproportion is the number worth carrying. Fifty-eight per cent of the picture on 36 per cent of the surface means the design is sixty per cent denser on the risers than an even distribution would put it, and that density is what a painter feels as the risers being the part that carries the image.

Why the risers get the picture #

The concentration is not a coincidence of the numbers chosen and it has a one-line cause.

A tread seen from a standing eye at the bottom of a flight is nearly edge-on. The ray meets it at a shallow angle, so a small step across the design corresponds to a large distance along the tread — the design’s local stretch across a tread reaches 34.9 at the worst place in this arrangement, which is a centimetre of picture becoming a third of a metre of stair.

A riser seen from the same eye is nearly face-on. The ray meets it near normal, the stretch is close to one, and the paint goes where the picture wants it.

So the picture concentrates on the risers because the risers are where the picture is cheap, and the treads are where a centimetre of design becomes half a metre of floor.

The relation between the two is one line and it is the whole cause. A face whose normal makes angle $\iota$ with the arriving ray stretches the design by $1/\cos\iota$. A riser’s normal is horizontal and a tread’s is vertical, so for a ray at depression $\delta$ below the horizon the two stretches are $1/\cos\delta$ and $1/\sin\delta$, and their ratio is

$$\frac{\text{riser stretch}}{\text{tread stretch}} \;=\; \tan\delta.$$

Risers and treads cost the same only for a ray at 45°, and every ray shallower than that favours the riser. A standing eye 1.65 m up and 2.2 m back looks down at the nearest step at 36.9° — already inside the crossover — and every ray further up the flight is shallower still, so the riser is the cheaper surface at every point of the design and increasingly so with height.

The two ends of the range make the point on their own. The worst tread stretch measured here is 34.9, which is $1/\sin\delta$ at $\delta = 1.64°$ — a ray running almost level, near the top of the flight and near the horizon. The riser at that same place is stretched by $1/\cos(1.64°) = 1.0004$. One surface is at the arithmetic identity and the other has run away, at the same point, from the same ray.

It also explains why the concentration does not depend on the flight’s length. The ratio carries only $\delta$, and lengthening the flight adds steps at ever shallower depressions — where the riser’s advantage is largest and the tread contributes almost nothing. A longer flight is more riser-dominated in the same proportion, which is what the twelve-step figure shows.

That is why every stair mural ever painted is painted on the risers. It is not a convention and it is not an aesthetic choice; it is the arrangement’s own arithmetic, and the alternative is to paint a design stretched thirty-fold across surfaces the viewer sees nearly edge-on.

Where the design has to be measured rather than chosen #

There is a second thing a flight forces that a corner does not, and it is a practical one.

On a single plane, the design’s extent is set by the eye’s own height — a design point level with the eye casts a ray parallel to the floor and lands nowhere, so the picture stops below the horizon and the floor needed runs away as the top approaches it.

On a flight, the limit arrives earlier and for a different reason: the flight is a finite object, and rays aimed above it go over the top step. Where that happens is a fact about the flight’s own slope against the sightline, and it has to be measured on the object rather than derived from the eye.

Both limits are real and they bind at different places. A designer who applied only the first would write a design that fits comfortably below the horizon and half of which misses the stairs entirely.

Each face is exactly a projective map #

The piecewise structure is worth verifying rather than assuming, because “a plane, so a homography” is the kind of statement that is true and is occasionally false for a reason.

Fit a homography from four design points to their marks on one face, and ask the remaining marks on that face where they landed. The worst error across every face with enough marks to test is $10^{-15}$ metres.

So each face carries a genuine projective map, exactly as the floor anamorph’s single homology does — and the eye comes back out of it the same way, from any one face with enough marks on it, and the flight is eighteen of them stitched at seventeen edges.

The stitching has the same character the corner’s does. The design is continuous across each nose and each internal corner, because a design point aimed at an edge lands on the edge from either side; and its scale is not continuous, because the two faces meet the ray at different angles. Continuous and not smooth, seventeen times.

The band the design can occupy #

A practical point that took a measurement rather than an assumption, and it is the one that made the two directions comparable.

The first version of this cast the same design — a band of the picture plane from 15 centimetres to 2.6 metres above the ground line — onto both flights. From the bottom of an ascending flight that is right. From the top of a descending one, four fifths of the rays went straight over the top step and out into the room.

The reason is that a descending flight seen from its own top occupies a narrow band low in the view. Nine steps at 30 centimetres of going each is 2.7 metres of floor, dropping away, so the whole object subtends a few degrees below the horizontal — and a design written across the same band as the ascending case is mostly aimed at nothing.

So the band is measured rather than chosen. Bisect on “does this ray hit the flight” and take the range that does, minus a small inset. Each direction then gets the picture it can actually carry, which is what makes the comparison a comparison.

What this adds to the anamorph field #

The field has three results about what an anamorph’s surface costs and this is the fourth.

The design that outruns the floor measures how much floor a design needs and finds it runs away as the design’s top approaches the eye’s own height. That is about extent.

The ceiling that is not a plane casts onto a vault and finds the collineation stops where the curve starts. That is about smoothness.

The anamorph that crosses a corner finds a piecewise map, continuous and not smooth. That is about joins.

And this one is about visibility — that a surface can be present and unreachable, and that half of a very ordinary object is.

What a second eye does not fix #

The obvious repair is a second design eye — paint the risers for a viewer at the bottom and the treads for one at the top — and it is worth saying exactly why that is not one design but two.

An anamorph is a picture with one eye, and the field measures how completely: even the two eyes of one head see different pictures, by a disparity that the collection quantifies. Two design eyes at opposite ends of a flight are not a refinement of that; they are two separate pictures sharing an object, each unreadable from the other’s position.

That is a perfectly good thing to make and it is not a solution to the problem this essay names. The unreachable faces are unreachable from the design eye, and adding a second eye adds a second set of reachable faces along with a second set of unreachable ones. Nothing about the flight has become fully paintable; the object now carries two partial pictures rather than one.

The eye height, and what it buys #

Sweeping the eye height at the bottom of an ascending flight moves two things in opposite directions.

Higher brings the treads closer to square-on, so the worst stretch falls sharply — 34.9 at a standing height of 1.65 metres, 9.8 at 2.4 — and the picture spreads more evenly over the flight.

Higher also moves the picture off the risers, which were the cheap surfaces: the riser share falls from 58.5 per cent to 53.6 across the same change. And a child’s eye at 1.2 metres reaches only eleven of the eighteen planes at all, with three quarters of the picture on the risers.

There is therefore an optimum somewhere and it is not at either end, which is the ordinary situation for a design decision with two competing terms. Where it sits depends on what a designer is optimising — the fraction of the flight used, the worst stretch, or the fraction of the picture on surfaces the viewer sees squarely — and those three do not have the same answer.

What the measurement supplies is the ability to ask. Before it, the choice of eye height for a stair mural is a matter of taste; after it, it is three curves and a decision about which one matters.

The stitching, counted #

Seventeen edges, and it is worth saying which kinds they are because they are not all the same.

Each nose — the far edge of a tread, where the riser begins — is a convex edge, and the design crosses it from a nearly edge-on surface to a nearly face-on one. That is where the scale jump is largest.

Each internal corner — the bottom of a riser, where the next tread begins — is a concave edge, and the design crosses it the other way.

So the scale alternates: compressed on a tread, expanded on a riser, compressed again, seventeen times up the flight. A straight line in the intended picture becomes a polyline of eighteen segments whose lengths alternate by a factor that the two angles determine.

That alternation is visible in any photograph of a stair mural and it is usually read as the paint following the steps. It is the other way round: the paint is a single projective picture and the steps are what a single projective picture looks like when the surface it lands on folds seventeen times.

What was carried and what it turned out to be #

This case has been the obvious next one for four rounds of work, described each time as “a dozen homologies where the corner anamorph is two”. That description is accurate and it is about the wrong quantity.

The count is arithmetic — two per step, so nine steps is eighteen — and nothing follows from it that does not follow from the corner. Every one of the eighteen is a homology for the same reason the corner’s two are, and they stitch the same way.

What does not follow from the corner is that a flight cannot be fully painted. A corner uses both of its faces from any eye that can see the picture at all. A flight uses at most seventeen of eighteen from the best available eye, eleven from a child’s, and nine of eighteen from the other end, and the unreachable ones are unreachable by construction rather than by a bad choice.

That is the sentence the four deferrals were carrying without knowing it, and it took casting the design and counting what the rays actually hit rather than counting the planes.

It also generalises past stairs, which is the reason to record it rather than to file it as a fact about steps. Any object with a fold has faces that face away, and an anamorph on such an object is a picture on a subset of it decided by where the eye is. A corner is the special case where the subset is everything, and it is the case the field happened to do first — so the field’s intuition was formed on the one example that does not show the phenomenon.