The corner is worth four times the middle

A flap of length L uses up every point of the sheet within L of its tip. That is the observation the whole of modern origami design rests on, and it is usually drawn as a disc.

A disc, though, is only what the flap costs if all of it is on the paper. Stand the flap on the edge of the sheet and half the disc is off the sheet entirely — and paper that is not there cannot be consumed. Stand it in a corner and three quarters of the disc is missing.

So the same flap has three prices, and the cheapest place to stand is the place with the least paper around it.

Four is a fact about a square #

The factor in the title is not a property of corners. It is a property of right-angled corners, and writing the general version says which sheets are cheap and why.

A flap standing at a corner of interior angle $\theta$ has the wedge of its disc that lies on the paper, so it claims

$$\frac{\theta}{2\pi} \times \pi L^{2}$$

At a right angle that is a quarter, which is the square’s four-times. At a straight edge, $\theta = \pi$, it is a half. A sharper corner is cheaper, in exact proportion to its angle.

So a corner of 60° costs a sixth of the disc, and one of 45° an eighth. The corner of a square is not an especially good place to put a flap; it is the best place a square has.

The sheet decides the discount #

Read across shapes and the ranking is immediate. For a regular $n$-gon the interior angle is $\pi(n-2)/n$, so a corner costs

$$\frac{n-2}{2n}$$

of a full disc. A triangular sheet’s corners cost a sixth — its corners are worth six times its middle. A square’s cost a quarter, four times. A hexagonal sheet’s cost a third, only three times.

So a triangle is the convex sheet whose corners are worth most, and the advantage falls monotonically as the sheet rounds off, reaching a half — no advantage over an ordinary edge — in the limit of a disc.

That gives a design reading the disc argument does not usually carry. A base with a few very long flaps wants them at corners, and wants those corners sharp; a sheet chosen for such a base should be a triangle rather than a square, and the gain is fifty per cent on every corner flap.

What a triangle gives up is the number of corners: three against four, and a hexagon offers six at a third apiece. Multiplying, a triangle’s corners are worth $3 \times \frac56$ of a disc in total discount, a square’s $4 \times \frac34$, a hexagon’s $6 \times \frac23$ — 2.5, 3.0 and 4.0. More corners beats sharper corners in aggregate, and which matters depends on whether the design needs a few long flaps or many.

That is a genuine trade the square sits in the middle of, which is one more thing to add to the reasons a square is a choice rather than a given.

The disc is paper, and only paper that is there #

The conservation argument is worth restating in the form that makes the boundary case obvious. Folding is an isometry: it does not stretch the sheet, so a point of paper that ends up at the tip of a flap of length L started somewhere within L of wherever the flap’s base is. Every such point is spent on that flap and is unavailable to any other.

A flap costs a circle states the rule, and the argument as usually given quietly assumes the disc lies on the sheet. It never says so, because in the interior it is true and nobody looks at the boundary.

The correction is one sentence. The flap is charged for the paper it consumes; paper it would consume if the sheet extended further does not exist and is not charged. A disc centred exactly on an edge covers half of itself and costs half. A disc centred exactly in a corner covers a quarter and costs a quarter.

The boundary is worth more than the middle, by exactly two and exactly four. Nothing about that is approximate: the fractions are of a circle by a straight line and by two perpendicular straight lines, and they are one half and one quarter with no error term.

This site has met the same edge before, in a completely different field, with the same shape of consequence. Where the paper stops is about flat-folding, and its finding is that the conditions at a vertex — Kawasaki’s, Maekawa’s, the rest — are statements about a full turn of paper, so a vertex on the sheet’s edge is subject to none of them. The boundary removes conditions there and it removes cost here, and the reason is one reason: a theorem and a disc are both claims about all the paper around a point, and at the edge there is less of it. The two essays were written from opposite ends of the subject and are describing the same fact about a sheet having a rim.

Four flaps, and a factor of exactly two #

The consequence is best seen at four flaps, because four is the count where both answers are exact numbers a reader can check.

Requiring whole discs inside the sheet gives four flaps of length 0.2500 apiece, placed at the centres of the four quarters. Allowing the discs onto the boundary gives four flaps of 0.5000, placed one in each corner. The flaps are exactly twice as long from exactly the same sheet.

The second number is not an approximation the search happened to land near. Four quarter-discs of radius ½, one in each corner of a unit square, exactly tile the square — total area 4 × (π/4)(¼) against a sheet of 1, so the discs claim π/4 of it and touch each other along the edges. The search is never told this. It is started from the interior grid, so every point it puts in a corner it has to arrive at by pushing against the clamp, and it comes out at 0.5000 to nine decimal places.

That is what makes the number worth asserting. A search seeded with the answer would demonstrate nothing; a search that has to find the corners establishes that the corners are where the paper is.

What the boundary is worth, flap by flap #

Four is the cleanest case, not a special one. Sweeping the flap count gives a gap that is never negative and never zero.

Across nine flap counts the boundary rule is never worse, and at best it is 2.41 times better — 141% more flap, at two flaps, where the interior rule wastes almost the whole sheet on margin. At the narrowest the advantage is still 41%.

The reason the gap can never be negative is worth having in one sentence, because it is what makes the sweep a check rather than a comparison of two searches. Every arrangement that satisfies the interior rule also satisfies the boundary rule — a disc wholly inside the sheet is a disc that is allowed to be on the boundary and simply is not. So the boundary rule’s best is at least the interior rule’s best, always, and a sweep in which it came out lower would be reporting a search failure rather than a fact about paper. The construction refuses to draw in that case.

The gap narrows as the flap count rises, and the reason is geometric rather than mysterious. The boundary of a square is a one-dimensional thing and the interior is two-dimensional, so the number of flaps that can be got onto the edge grows more slowly than the number a design might want. At two flaps the whole design is boundary; at fifty it is mostly interior; the advantage decays accordingly.

Which claim was checked, and how #

Three separate assertions carry this, and each is put against something that does not know the answer.

The first is the halves and quarters themselves. It would be trivial and worthless to compute a half-disc’s area by halving a disc’s. Instead the paper each of the three flaps covers is integrated over the sheet — a quadrature over the columns of the disc, exact in the vertical direction at every column, clipped to the sheet — and the three results are compared with πL², πL²/2 and πL²/4, which the quadrature never consults. They agree to 2.1 × 10⁻⁸, which is the quadrature’s own resolution and not a tolerance chosen to pass.

The second is the search. Every interior packing quoted here is checked against the published optimum where one exists — the search never consults those values, and the figure refuses to draw if it ever exceeds one, which would mean either the search or the literature was wrong.

How much paper is wasted is where those numbers come from, and the honest state of the problem is that for most counts nobody knows the optimum. That matters here in a specific way: the gap between the two rules is a difference of two searches, so a poor search on either side would understate or overstate it. The relation that cannot be faked is the one asserted — never worse — and it is asserted at every count rather than in summary.

The third is the exact half at four flaps, described above. What would have made all three fail is easy to name: a quadrature that agreed only to three digits would mean the clipping was wrong; a boundary result below the interior one at any count would mean the search was; and a four-flap answer of 0.4998 would mean the corners were being approached rather than reached.

Necessary, and not sufficient #

Everything above is about a condition that a design must satisfy. It is not a condition that makes a design.

The discs are a necessary condition and not a sufficient one. A packing that satisfies every non-overlap requirement is not a proof that a base exists. The regions left between the discs still have to be resolved into creases, and the construction that fills them needs those regions to be convex; a packing can be perfectly legal and leave a leftover the construction refuses. The count of nine in the figure is a count of discs that fit, not of flaps somebody has folded.

The second half of the caveat is sharper and is the one a designer feels. A corner flap’s paper is not interchangeable with an interior flap’s. The quarter-disc in a corner is stuck in that corner: it is the only place on the sheet with two boundary lines meeting, and there are four of them. The three quarters of the disc that the corner flap did not pay for were never available to anybody, so nothing has been saved that another flap could borrow. The corner is cheap and the corner is scarce, and a design that wants six long flaps cannot put six of them in four corners.

That is why the boundary advantage is a fact about arrangement rather than a discount. Moving a flap outward makes it cheaper; it does not make the sheet larger, and it commits that flap to a place.

There is a third caveat, and it is the idealisation the whole picture rests on. The disc model charges a flap for a full disc of radius L because the base it belongs to is uniaxial — every flap along one line, each as long as its own edge of the skeleton — and the argument is exactly as good as that restriction. A flap that is not on the axis is not priced by this rule at all. Nor does the rule say anything about the strips of paper that join the flaps together, which have their own widths and their own claim on the sheet, and which the corner rule does not discount. What is measured above is the price of a flap’s tip and nothing else.

Symmetry arrives at the corners on its own #

There is a second route to the same arrangement, and it does not mention the boundary at all.

Requiring a packing to be symmetric usually costs something — when symmetry costs measures how much, and across nine disc counts the mirror-symmetric search falls as much as 14.6% short of the free one. At four discs it costs nothing at all, because the best arrangement is already symmetric.

The connection worth carrying is that a designer imposing symmetry for reasons of subject — a creature with two sides, four legs, a body on an axis — is pushed towards arrangements that use the corners, because the corners are the symmetric points of a square. The traditional repertoire got there first and got there by hand. The bird base and its relatives take the square’s four corners to a point and produce four flaps from them, and every folder who ever made one was exploiting the quarter-disc rule several centuries before anybody wrote it down.

A different question about the same sheet #

The square is a choice asks a question that sounds like this one and is not. It holds the flap count fixed and varies the proportion of the sheet, at constant area, and asks which shape lets the flaps claim the most.

That is a question about the outline of the sheet. This one is about where in the sheet a flap sits, and the outline is held fixed throughout. Nothing in the earlier essay asks the second question, and nothing here asks the first: the sheet is a unit square in every figure above, and the only thing that varies is whether a disc may hang over its edge.

The two interact, and the interaction is where a design decision actually gets made. A longer sheet has more edge per unit area, so it has more of the cheap paper — which is a reason to prefer a long sheet that has nothing to do with efficiency at packing, and which the sweep above cannot see because the sweep counts area claimed rather than where it was claimed from. Two different rung-1 questions about a sheet, and neither one answers the other.

Who noticed it, and when #

The half-disc and quarter-disc rule is part of the tree theory as it was set out in the 1990s, and is generally credited to Robert Lang and Toshiyuki Meguro, who developed the method independently. Lang’s treatment names the three cases outright — a flap in the middle of the sheet, a flap on an edge, a flap in a corner — and prices them as a full circle, a half and a quarter. It is presented as a piece of bookkeeping rather than as a discovery, which is fair and is also why it is easy to read past.

The practice is very much older than the bookkeeping, and this is the ordinary situation in this subject rather than an exception. Every classical base with points in it puts them in the corners, and the reason given in the traditional literature is that the corners are where the points are — which is a description of the fold rather than an explanation. What the disc argument added was not the arrangement but the reason, and a reason is what lets somebody design a base nobody has folded before.

It is worth resisting the tidy version in which the folders knew this all along. They knew where to put the points. They did not have a way to say how long a point in a corner could be, or to decide in advance whether a wanted set of flaps would fit — and that difference is the whole of what the method is for.

Where the ladder goes next #

The immediate continuation is what the boundary does to the rest of the pattern. Discs pushed against the edge leave a different leftover region from discs in the interior, and the construction from a packing to a crease pattern has to fill it — so a boundary-heavy packing is cheaper in paper and can be harder to complete.

The other direction is growth. If a corner flap is cheap and a corner is scarce, then adding a flap to a finished design usually means adding an interior one, at full price — unless paper is inserted for it, which is what a graft costs and is priced by a strip’s width rather than by a disc. Those two prices are the two ways a design gets bigger, and choosing between them is the design decision this rung leads to.