A flight that has ends

One flight, two pictures found that a flight of stairs carries two anamorphs that never collide: the treads from the top and the risers from the bottom, on disjoint sets of faces. It proposed a rule from that — an object supports as many pictures as it has families of faces — and recorded in its own shortfall that the flight it measured had no step ends, so the rule’s prediction of three for a real staircase could not be checked.

Here is a flight with ends. The rule is nearly right and it counts the wrong thing.

The object #

A flight of stairs in this collection is a ribbon: treads and risers, swept across a width, with nothing on the sides. That is the right model for the question a stair does not use all its faces asked, and it is not a staircase.

A staircase is a solid. Its side is a staircase-shaped polygon, and the least that makes it one is a rectangle per step: the exposed side under each tread, running down to the flight’s own foot. Two sides, nine steps, eighteen further faces on a flight that had eighteen — so the object doubles its face count and grows its area by about a third.

The design plane has to be free #

The first thing the ends demonstrate is not about the ends. It is about where a picture may stand.

The anamorph field pins the design plane: the intended picture stands upright at the origin and the eye is in front of it, which is right for a picture on a wall and a person in a room. Under that convention the step ends are nearly unreachable — the rays run broadly along the flight, the ends are edge-on to them, and a design that lands on the ends lands on almost nothing else.

Free the design plane — put it perpendicular to whatever direction is being cast along, which is what a picture with no eye does — and the ends are reachable at once, by a direction pointing across the flight.

So the shortfall’s question could not have been answered inside the convention that raised it. A flight’s step ends carry a picture that no pinned design plane can reach, and that is a fact about the pinning rather than about the ends.

The count #

With the plane free, the question is well posed: how many disjoint pictures does the object support?

Two pictures are disjoint when the sets of faces their designs land on do not meet, so the question is a covering question over the sphere of directions. The unit is a group of faces pointing one way — because a design along one direction reaches only faces whose outward normal opposes it, so two faces with opposite normals can never be in the same picture whatever they are called.

The flight without ends has two such groups: treads pointing up, risers pointing along the flight. Two families of faces, two groups, two pictures. The rule agrees with itself.

The flight with ends has three families and four groups: treads, risers, the left ends and the right ends. And it supports four pictures, each drawn along the direction opposite its own group’s normal, each with every one of its design’s marks landing on that group and on nothing else.

What the rule was counting #

The rule said families and the answer is orientations, and the difference is exactly one on this object.

“End” is a word. It names two sets of faces that point in opposite directions, and no direction reaches both of them, because a direction that opposes one normal agrees with the other. So the two sides are two pictures however they are named, and a rule that counts words undercounts by however many of its words cover more than one orientation.

That is a small correction and it generalises cleanly. A picture is fixed by a direction, so what it can be drawn on is a set of faces pointing one way, and a word is not a direction. A cluster of blocks has five kinds of face and five orientations and the rule is exactly right on it; a flight with ends has three kinds and four orientations and the rule is out by one.

Why the purity is not exactly one #

A measurement detail that turns out to carry the argument.

A group “has a picture of its own” when a direction’s design lands on it and on essentially nothing else. The direction that makes that most likely is straight along the group’s own normal, which is where the search starts; a small cone around it is swept too, because a face can be shadowed from straight ahead and reachable from a few degrees off.

The threshold is a share of the marks, not of the faces, and the distinction matters. A direction that grazes one riser with two rays out of six hundred is drawing a picture of the treads, and a test demanding not one stray ray would report that no object anywhere has a picture of anything.

On this flight the purities come out at exactly one for all four groups, which is a cleaner result than the threshold was written for — looking straight along a group’s normal, nothing else is edge-on enough to be hit. On a cluster of blocks it is also one; on objects with more oblique faces it is not, and the threshold does work.

The corrected rule, and the ceiling it implies #

Stated in the unit the correction supplies, the rule is short: an object supports as many disjoint pictures as it has distinct exposed face normals. A design cast along a direction reaches exactly the faces whose normals oppose it, so grouping the faces by normal partitions them automatically and the groups can never overlap. Counting normals is counting the object’s Gauss map, and it is the only count that can be right.

That has a consequence for architecture which the seven objects here all obey without it being noticed. A building made of axis-aligned surfaces has at most six normals, and one of them points at the ground. So a room, a stair, a corridor, a cluster of blocks — anything rectilinear — supports at most five pictures, however many faces it has and however elaborately they are arranged. The cluster of blocks reaches that ceiling with twenty-one faces; the flight with ends reaches four of it with thirty-six. Adding faces does not add pictures once the normals are used up, which is why the two columns of the table come apart so sharply.

Exceeding five needs faces that are not axis-aligned: a chamfer, a pitched roof, a splayed reveal. And a curved object breaks the count entirely — a cylinder has a continuum of normals, so it supports a continuum of pictures, each landing on an infinitesimal strip and each therefore carrying no picture at all. At that point the count stops being the right question and the right one is how much solid angle of directions a usable picture occupies, which is a measurement this object does not need and a smooth one would.

The control that makes it a finding #

The flight without ends is run through the identical machinery and returns two pictures on two families. Without that, four on three would be a number rather than a comparison — the machinery might simply be generous.

It also settles a question the earlier essay left open. One flight, two pictures found its two pictures with a pinned design plane and two eyes at the flight’s two ends; this finds two with a free plane and two directions. The counts agree, which is not obvious — a free plane could easily have found more — and the agreement is what licenses using the free-plane count on the ended flight.

What a picture on the ends would look like #

Worth asking, because a count of pictures is abstract and the ends are a real surface.

The side of a flight is a staircase polygon: a sequence of rectangles of increasing height, each a tread’s depth wide. A design cast across the flight lands on them at one scale — the map is affine, by one face, one scale — and the picture is therefore a picture in the ordinary sense, cut into vertical strips by the steps and reassembled without distortion.

That is a real decorative arrangement, and it is one that has been used — the anamorph that crosses a corner is the same problem where the surfaces meet rather than alternate: a device painted on the side of a stair so that it reads from across a hall. What the count adds is that such a device is compatible with a different device on the treads and a third on the risers, none of them interfering, because the three land on disjoint faces.

Four pictures on one staircase, none of them visible from where any of the others is read. That is the finding in the form a reader can picture.

Why a normal and not a name #

The unit the count is taken over deserves a paragraph of its own, because choosing it is the whole of the correction.

A face has an outward normal, and a design carried along a direction reaches it only if that normal opposes the direction. That is not a modelling choice; it is what “the design lands on the front of the face” means, and a design landing on the back of a face is a design landing on nothing.

So faces sharing a normal are exactly the faces that can share a picture, and faces with opposite normals are exactly the faces that cannot. Grouping by normal is therefore not a convenience — it is the finest grouping the geometry permits and the coarsest that is safe, which is what a unit should be.

Grouping by name is neither. It is coarser than the geometry on this object, because “end” spans two normals; and on a different object it could be finer, because two differently-named faces can share a normal — a flight’s treads and the landing at the top of it point the same way and would be two names and one picture.

A count taken over names is a count over the vocabulary of whoever built the model, which is a fact about the modeller. A count over normals is a fact about the object.

Getting the normals right, which was not automatic #

A short note on a mistake this measurement made and caught, because it is invisible in every picture.

The two sides of the flight are built from the same two in-plane axes, so their computed plane normals are identical and one of them has to be flipped to point outward. The first version flipped the wrong one, and nothing in any figure changed: both sides still drew, both still received designs, and the only thing that differed was which direction the object claimed to face.

What it broke was the count. With both sides’ normals pointing inward, the grouping put them in one group with the wrong direction opposite it, no design reached either, and the object reported two pictures instead of four.

That is the shape of a defect this collection has met before and it is worth naming again: the symptom is a number rather than a picture, and every gate that asks whether a figure draws correctly passes. The half of the object’s geometry that no drawing shows is the half a drawing cannot check.

What a fifth picture would need #

The count is four and it is worth asking what would raise it, because the answer says what kind of object supports many pictures.

A fifth picture needs a fifth orientation with enough area to be a picture, and a flight has no more: treads up, risers along, two sides across, and nothing else. Adding a landing at the top adds area to the treads’ group rather than a new one; adding a handrail adds many small faces at many angles, none of them with area enough to matter.

What does add orientations is faceting. An object cut into more planes at more angles supports more pictures, up to the point where each group is too small to carry one — and that is a real trade, because a finely faceted object supports many small pictures and a coarsely faceted one supports few large ones.

A cluster of blocks is the worked example in the other direction: five orientations, five pictures, and each one large. A flight is four and uneven — the treads carry nearly half the object’s area and each side an eighth.

The limit of the counting #

Two things the count does not claim.

It is a lower bound. The search is greedy over the sphere and over the groups, so it finds a set of disjoint pictures rather than proving that no larger set exists. On these objects the answer equals the number of normal groups, which is an upper bound too — no picture can span two groups — so the bound is tight here. On an object with curved faces or with many nearly-parallel ones it would not be.

And “disjoint” is about faces, not about visibility. Two pictures on disjoint faces do not interfere in the sense that no mark of one lands where a mark of the other did. They may still both be visible from some third direction, which would show two pictures at once and neither correctly. That is not a collision and it is not nothing, and the earlier essay’s phrase “never collide” is the right one for what is being claimed.

What the ends change about the earlier reading #

One number in the earlier work moves, and it is worth recording rather than leaving to be rediscovered.

A stair does not use all its faces reports that no eye reaches more than fifteen of a flight’s eighteen planes. On the ended flight the object has thirty-six planes and the same eye reaches the same fifteen, because the ends are edge-on to a pinned design and receive nothing.

So the fraction falls from fifteen in eighteen to fifteen in thirty-six, and the earlier essay’s headline — that a great deal of an object is unpaintable — is strengthened rather than weakened by modelling the object more honestly. That is the direction a correction should go and it is not always the direction one does.

What the ends do to the eye’s own reading #

The count above is taken over directions, and the anamorph field’s questions are taken over eyes. It is worth running one of them on the ended flight, because the answer moves in an interesting direction.

The eye that reaches the most finds an interior optimum: the highest eye is not the one that reaches the most of a flight, because a very high eye sees the treads at a glancing angle and its design’s rays cross the flight rather than landing on it. On the ended flight that optimum is unchanged, because the ends contribute nothing to a pinned design at any height.

So the ended flight has the same best eye, the same reached set, and a third more surface. Every quantity reported as a share of the object drops by a quarter, and every quantity reported as a set of faces is identical.

That is a useful way to state what the ends are. They are surface the eye’s convention cannot see at all, so they change every denominator and no numerator — which is why modelling them makes the earlier essay’s headline stronger rather than putting it in doubt.

The short version #

A real flight of stairs has three kinds of face and supports four disjoint pictures, because its two sides are one kind pointing two ways and no direction reaches both. The rule the earlier round proposed counted kinds; the quantity is orientations.

And the picture on the ends is only reachable if the design plane is free to stand perpendicular to the direction being cast along, which the anamorph field’s own convention does not allow — so the shortfall could not have been discharged inside the convention that recorded it.