The corner premium, with no corners

A corner is worth four times the middle is one of the more useful facts in design, and it is worth restating before it is taken apart.

A flap costs a circle: a flap of length $L$ consumes every point of the sheet within $L$ of its attachment. Put the attachment in the middle of the sheet and the whole disc has to be there. Put it at a corner of a square and only a quarter of the disc is on the paper, because the other three quarters are off the edge — and paper that is not there costs nothing.

So a corner flap of the same length costs a quarter as much paper. That is the premium, and it decides a great deal about where a designer puts things.

What the premium actually depends on #

Not on there being a corner. On the interior angle of the boundary at the attachment point.

A flap attached at a point where the paper subtends an angle $\theta$ consumes a sector of $\theta$ rather than a full turn, so it costs $\theta / 2\pi$ of a disc. At a square’s corner $\theta$ is a right angle and the fraction is a quarter. Along a straight edge $\theta$ is a straight angle and the fraction is a half. In the interior $\theta$ is a full turn and the fraction is one.

Written that way the premium is a statement about angles and the word corner is a description of where small angles happen to occur on the shapes people use.

The arithmetic of the premium #

The rule deserves to be written out once, because it is short and it is usually quoted only in its special case.

A flap of length $L$ attached at a point where the paper subtends an interior angle $\theta$ consumes a sector of radius $L$ and angle $\theta$, whose area is $\tfrac12 \theta L^2$.

In the interior, $\theta = 2\pi$ and the area is $\pi L^2$ — a whole disc.

On a straight edge, $\theta = \pi$ and the area is $\tfrac12 \pi L^2$ — half.

At a right-angled corner, $\theta = \tfrac{\pi}{2}$ and the area is $\tfrac14 \pi L^2$ — a quarter, which is the familiar four times.

At a hexagon’s corner, $\theta = \tfrac{2\pi}{3}$ and the area is a third of a disc — three times rather than four.

At a triangle’s corner, $\theta = \tfrac{\pi}{3}$ and the area is a sixth — six times.

So the premium is $2\pi / \theta$, it is continuous in the angle, and corner names the places where the angle is small rather than a category of its own. A sheet’s value to a designer, on this measure, is decided by the distribution of interior angles round its boundary.

Why a sharper corner is not simply better #

The arithmetic suggests cutting sheets with very sharp corners, and it is worth saying why nobody does.

A flap attached at a sharp corner is cheap in area and it is also thin. The sector of paper feeding it subtends a small angle, so the flap it produces is a spike rather than a limb, and a base needs limbs with some width to them.

More practically, a sharp corner has very little paper near its tip, so a flap attached there can only be short before it starts consuming paper from further in, at which point the discount is diluted by the parts of the disc that are not in the corner.

So the premium is real and it is bounded by geometry that the area calculation does not see. A triangle is a good sheet for three flaps and a bad one for eight, and the corner count is only half the story.

That is a general shape in this subject: an accounting that prices one resource correctly and ignores a constraint that binds first.

What the two ends of a cylinder are worth #

A cylinder’s boundary is two circles, and the same arithmetic applies to them, so it is worth completing.

Every point of either circle has an interior angle of a straight angle, so every flap attached there costs half a disc. Uniformly — there is no variation round the circle at all, since a circle has no corners and no distinguished points.

That uniformity is the interesting part. On a square a designer chooses where on the boundary to attach, and the choice matters by a factor of two between an edge and a corner. On a cylinder there is no such choice: every boundary point is the same, and the only decision is which of the two ends.

So a cylinder is not merely poorer in cheap paper. It is homogeneous along what boundary it has, which removes a whole dimension of design decision.

That is a fair description of what a tube is: an object with no distinguished places, which is why its design questions are about lengths and counts rather than about placement.

The hole, priced properly #

Since holes and gluings are the two operations that change a sheet’s boundary, it is worth putting the pricing side by side.

Cutting a hole adds a boundary circle. Every point of it is worth half a disc, so the hole creates cheap places — and it does so in the middle of the sheet, which is where the paper was dearest. That is why a hole is cheap paper: it removes the least valuable material and turns its edge into the second most valuable.

Gluing a pair of edges removes a boundary. The places that were worth half a disc become worth a whole one, and there is no compensation anywhere.

So the two operations are opposite in the design accounting as well as in the topological one, and only one of them is a bargain. A designer who needs a closed sheet is paying for the closure in every currency at once: fewer cheap places, less freedom in the pattern, and a parity condition.

Which is a fair summary of why closed sheets are products rather than design surfaces.

A sheet with a smooth boundary #

The first consequence is about a shape nobody uses much, and it isolates the point.

A disc of paper has a boundary with no corners at all. Every point of its edge has an interior angle of a straight angle, so every edge placement costs half a disc, and there is nowhere that costs less.

So a circular sheet has an edge discount and no corner premium, and the two are usually run together because on a square they coincide at four points.

That already says the premium is not about corners as such. A hexagon’s corners subtend two thirds of a straight angle rather than half of one, so a hexagon’s corners are worth a third rather than a quarter — less of a bargain, and there are six of them.

And a sheet with none #

A cylinder has two circles of boundary and no corners, so the same discount applies at its two ends and nowhere else.

A torus has no boundary at all. Every point has a full turn of paper round it, every flap costs a whole disc, and there is no discount anywhere.

That is the strongest form of the result and it is not a curiosity: it means the whole apparatus of placing flaps at corners, ranking shapes by their corner count, and choosing a square because it has four cheap places, is an apparatus about sheets with boundary, and on a closed sheet it is empty.

Why a hole is not a corner #

A related result is worth setting beside this one, because the two point in opposite directions.

A hole is cheap paper: cutting a hole in the middle of a sheet costs very little in what the sheet can produce, because the middle is the part a uniaxial base uses least. The hole’s boundary is new boundary, and a flap attached at it gets a discount.

So cutting a hole adds discounted places. It also adds a parity condition the sheet did not have, which is the price, and the two are measured in different currencies.

Gluing does the reverse: it removes boundary, so it removes discounted places, and it adds a parity condition as well. Both operations add a condition and only one of them adds cheap paper.

What the premium is not about #

Three misreadings, each of which the angle formulation rules out.

It is not about the boundary being straight. A curved boundary works the same way; what matters is the interior angle at the point, and a smooth curve has a straight angle there.

It is not about the corner being convex. A reflex corner — one that bites into the sheet — has an interior angle larger than a straight angle, so a flap attached there costs more than an edge flap. Notches produce these, and a notch’s two ends are cheap while its inner corner is dear.

It is not about the sheet’s outline being polygonal. A disc of paper has an edge discount everywhere and no premium anywhere, and it is a perfectly good sheet for designs that want their flaps distributed evenly.

That last one is worth a moment. A circular sheet is rare in this subject and it is the natural sheet for anything with rotational symmetry — a flower, a wheel, a deployable dish. Its uniform half-disc discount is exactly the property a cylinder’s ends have, which is a small piece of tidiness: the objects with smooth boundary behave alike whether they are flat or closed.

Reference points, which follow the same map #

The corner premium is about material, and there is a second thing corners are good for that follows a similar pattern.

A fold needs something to align, and the only marked things on a fresh sheet are its own boundary features. A corner is the intersection of two edges, so it is a point, and a point is worth much more than a line to a construction: bringing a point onto a line is one of the axioms, and bringing a point onto a point is another.

A square has four such points before anything is folded. A disc has none: its boundary is a smooth curve with no distinguished point on it anywhere.

So a circular sheet is generous in material and destitute in references, and a square is the reverse of a disc in exactly one respect and the same in the other. That is part of why the square won.

On a cylinder the situation is worse again: two smooth circles, no points, and a seam that is not a mark. Constructions on such a sheet have almost nothing to start from.

The ranking, redone #

Putting the two measures together gives a ranking of sheets that is more honest than the corner count.

A triangle. Three corners at a sixth of a disc each, three reference points, very little paper. Excellent for three flaps.

A square. Four corners at a quarter, four reference points, and the whole apparatus of binary division built on its edges. The default, and it is the default for reasons that survive scrutiny.

A hexagon. Six corners at a third, six reference points, and more paper near the boundary. Good for six-flap subjects and worse per corner.

A disc. No corners, a uniform half-disc discount, no reference points. Generous and hard to construct on.

A cylinder. Two smooth circles, uniform half-disc discount at them, no reference points, no corner premium, and a parity condition. The product rather than the design surface.

A torus. Nothing at all: whole discs everywhere, no references, two parity conditions.

Read down that list and what falls is not a single quantity. Cheap material and reference points fall together on smooth boundaries and separately on polygonal ones, which is why a single ranking has never been quite right.

The ranking that has nothing to rank #

Design lore ranks sheet shapes partly by their corners: a square has four right angles and is a good sheet; a rectangle has four but two of them are further from the middle; a hexagon has six shallower ones; a triangle has three sharp ones and is excellent for three-flap subjects.

On a cylinder the ranking has two entries — the two end circles, both smooth, both worth half a disc — and no way to distinguish them. On a torus it has none.

That is not a defect in the ranking. It is the ranking correctly reporting that a closed sheet has no cheap paper, which is a real property of a closed sheet and part of why nobody designs bases on one.

What replaces it #

For a closed sheet the useful questions are different, and it is worth saying what they are even though nothing here answers them.

How much surface fits in a volume. A closed sheet is good at enclosing, which is a question this collection has asked in another field.

How far it packs along its own axis. A tube’s value is that it collapses to a fraction of its length, and the fraction is the design quantity.

How it deploys. A closed sheet has a motion from packed to extended, and the force, the reliability and the path are what the applications care about.

None of those is a paper-efficiency and none of them has a corner premium in it. The design vocabulary for closed sheets is a different vocabulary, and this collection has been using the flat one throughout.

A note on the number four #

The familiar statement is that a corner is worth four times the middle, and the four is worth a sentence because it is the least general part of the rule.

Four is $2\pi$ divided by a right angle. It is the premium of a square’s corner and of no other shape’s, and quoting it without the shape is the same kind of omission the rest of this phase has been about.

A rectangle’s corners are also right angles, so four is right there too. A hexagon’s are a third rather than a quarter, giving three. An equilateral triangle’s are a sixth, giving six. A shape with a very sharp point has a very large premium and a very thin flap.

So four is a fact about squares, the interior angle over a full turn is the rule, and the two have been used interchangeably because the square is the sheet.

That is the smallest instance in this phase of the thing the whole phase is about, and it is a good one to end on: a number that is a property of the object everybody uses, quoted as though it were a property of the subject.

What is measured and what is argued #

The distinction matters here more than usual, because the essay is mostly geometry rather than computation.

Measured: the free letters, panels and search costs of the four sheets, which are what the claims about a closed sheet’s poverty in freedom rest on. Those are counts on built objects.

Arithmetic: the sector area $\tfrac12 \theta L^2$ and the premium $2\pi/\theta$. Elementary, exact, and not a computation this collection runs.

Argued: everything about references, rankings and what a designer would do. Those follow from the arithmetic and from what the boundary is for, and none of them is a measurement.

So a reader wanting a number for how much worse a cylinder is as a design surface will not find one, because no design has been attempted on one. What is established is that the quantity the ranking is built on is nought there, which is a statement about the accounting rather than about any design.

What survives #

The premium survives on every sheet with boundary, in its corrected form: a flap’s cost is the interior angle at its attachment, divided by a full turn.

Corners are where that angle is smallest, on the shapes people use, which is why the rule of thumb works.

And on a sheet with no boundary the cost is a whole disc everywhere, the premium is nought, and a design method built on placing flaps cheaply has nothing to place them on — which is a clean statement of why the uniaxial method is a method for flat sheets rather than a method for folding.