The square is a choice

Origami is folded from a square. That is so universal a starting point that it is rarely stated as an assumption, and almost never examined as one — the tradition sells square paper, the traditional bases begin from a square, and every design tool defaults to it.

The mathematics does not require it. The seven axioms say nothing about the sheet’s outline. The flat-folding theorems are about interior vertices and are indifferent to the boundary. And the circle-packing account of design needs a region to pack into and does not care what shape it is.

So the sheet’s proportions are a free variable. What happens when it is varied is not what a smooth trade-off would look like.

What is held fixed, and why it has to be #

The comparison only means anything if the sheets cost the same, so every sheet in the sweep has area one. A sheet of proportions a to 1 measures √a by 1/√a.

That is the difference between asking “what shape of paper suits this design” and “how much paper does this design need”, and only the first is interesting. A large enough rectangle beats a square trivially and says nothing.

What is measured is the fraction of the sheet the flaps can claim: the discs’ total area over the sheet’s area, at the largest radius that fits. In design terms it is how much of the paper ends up in the model’s limbs rather than between them, and it is the same quantity the efficiency figures report for the square alone.

The number that keeps appearing #

78.5% is π/4, and π/4 is the density of a square grid of touching discs — each disc inscribed in its own square cell, with the corners wasted.

That is what the peaks are. A sheet of proportions p to q holds a p by q grid of discs perfectly, with every disc touching its neighbours and the sheet’s edges, and the efficiency is π/4 exactly whatever p and q are. Seven discs fit a 7-by-1 grid, so a 7-to-1 sheet is perfect for them.

So the shape of the curve follows from the arithmetic of the flap count. Every factorisation of n gives a proportion at which the sheet is exactly right, and every one of them reaches the same efficiency.

Twelve is the clearest case because it has three peaks and two near-misses between them, and the near-misses are proportions that look perfectly sensible. Two to one is a square cut in half; four to one is a familiar strip. Neither holds a grid of twelve discs, and both give away eight to ten points of efficiency to proportions that do.

Why the square has its reputation #

The square is not arbitrary and the sweeps say why.

A square is exactly right when the flap count is a perfect square: four discs in a two-by-two, nine in a three-by-three, sixteen in a four-by-four. Each reaches π/4 with a radius a reader can compute by hand — a quarter, a sixth, an eighth — and the search reproduces all three without being told them.

For every other count the square is somewhere on the slope. Six flaps on a square get 66.3% against the 78.5% available at 1.5 to 1. Seven get 66.8% against 78.5% at seven to one. Ten get 68.2% against 78.5% at two and a half to one.

The square’s real defence is different and is worth stating. It is never terrible. It is a peak at the square counts and within about twelve points of the peak everywhere else, and it never falls to the 57–60% that the wrong long rectangle produces. A designer who does not know the flap count in advance — which is a designer starting a design — is choosing the shape with the least bad worst case.

The prime case #

Seven flaps is the sharpest instance because seven is prime.

A prime count has exactly one factorisation, so there is exactly one proportion at which the sheet is right, and it is n to 1: a strip. Everything else is a compromise, and the compromises are all roughly as bad as each other.

The design reading is uncomfortable and is probably correct: a model with seven equal-length flaps and no other structure wants a long strip, not a square. That is not how anybody folds, and the reason is not that the geometry is wrong. It is that real designs do not have seven equal flaps.

What a designer would actually do with this #

The honest use of a result like this is narrow, and the narrow version is more useful than an oversold general one.

It is not an argument for abandoning square paper. The peaks are real and the losses between them are real, and both are a few per cent of flap length rather than a factor. A limb ten per cent shorter is a worse model and not an impossible one.

It is an argument for treating the outline as part of the design in the two cases where it pays. The first is a design at the limit, where the flaps are as long as the paper can possibly make them, and ten points of efficiency decides whether the model exists. The second is a design whose subject is strongly one-dimensional — a snake, a centipede, anything whose tree is a long chain — where the packing will be strung out in one direction whatever the sheet looks like, and giving it a sheet of that shape costs nothing at all.

The second case is one the tradition already agrees with. Strip folding and modular work start from rectangles because the technique wants them, and nobody regards that as a departure from anything.

There is a third use and it is the cheapest. The peaks are at ratios of small whole numbers, which means they are at proportions a folder can produce by folding: halving a square gives two to one, dividing it in thirds gives three to one and three to two, and dividing a square into any whole number of parts is a fold rather than a measurement. Every proportion this essay recommends is reachable from a square with no ruler, which makes the recommendation actionable in a way it would not be if the optima had landed at irrational ratios.

They did not land there for a reason. The peaks are at p to q with pq = n, and p and q are whole numbers because they count discs. An optimum whose position is dictated by a factorisation is always going to be at a rational proportion, and rational proportions are exactly what folding constructs.

The square’s virtue is its variance #

“Never terrible” can be made exact, and the sweeps’ own numbers do it.

Off its peaks the square barely moves. Six flaps give 66.3%, seven give 66.8%, ten give 68.2% — a spread of under two points across counts that have nothing in common, one of them prime. A strip of proportions $n$ to 1 reaches 78.5% for its own $n$ and falls to the high fifties for the others: a spread of about twenty.

The reason is that a peak is a match to a factorisation, and a square matches $p = q$. A count with no square factorisation misses it, but misses it by the same amount whatever the count is, because the square is equally far from every rectangular grid. A long rectangle is exactly right for one factorisation and badly wrong for the rest.

So the choice is a familiar one under uncertainty: the square is the minimax option and the matched rectangle is the maximum one, and a designer who knows the flap count before choosing the paper should not be taking the minimax.

What the points are worth in limbs #

Efficiency is not the quantity a designer feels, and converting is one line. Total disc area is $n\pi r^2$, so at a fixed count the flap length goes as the square root of the efficiency.

Giving up the peak therefore costs

$$\sqrt{66.3 / 78.5} = 0.919,$$

or 8% off every flap on the model. Ten points of efficiency is not ten per cent of a leg; it is eight, and the square root is what keeps a large-sounding loss inside the range where a design still works.

It also sets the scale for the near-misses. Twelve flaps at two to one reach 68.6% against 78.5%, which is 6.5% shorter limbs — enough to lose a segment on an antenna and not enough to lose the model.

Which theorem was checked, and how #

The peaks are the check. π/4 is not in the code, and neither are the proportions at which it appears; both fall out of a search that anneals disc positions and knows nothing about grids.

Two exact values anchor everything. Four discs in a unit square have optimal radius exactly a quarter, and nine have exactly a sixth; both give π/4, and both are reproduced to five figures. Those numbers are the site’s standing yardstick for the packing search and it may never beat one — beating a published optimum would be either a discovery or a bug, and a figure is the wrong place to find out which.

Fairness is the third check, and it is the one a comparison like this most easily fails. Every sheet in a sweep is packed by the same function with the same restarts and the same steps, including the square, which is not handed to the site’s older square-only search. A comparison in which one shape got a better-tuned algorithm would be measuring the tuning.

Every value quoted here comes from a search, so the size of a search’s own error belongs beside them.

Where π/4 stops being the ceiling #

The grid is not the best arrangement of discs in general and this essay must not imply that it is.

The densest packing of equal circles in the whole plane is hexagonal, at about 90.7%, and any bounded region large enough to hold a decent piece of a hexagonal arrangement will beat π/4. Nothing in these sweeps does — the best value found anywhere, at every count tried up to sixteen, is π/4 — but that is a statement about small counts in a bounded sheet, not a theorem.

The reason small counts do not reach the hexagonal density is the boundary. A hexagonal arrangement’s advantage is in its interior, and a packing of a dozen discs is almost all boundary; the rows have to end somewhere and the ends waste more than the staggering saves. Where the crossover happens is a question this essay does not answer and the sweeps do not reach.

The sheet’s shape is chosen before anything else in a design, and everything after it inherits the choice.

What the model leaves out #

Three things, and each of them matters in a real design.

The discs are equal. A real design’s flaps are not: a beetle’s legs, antennae and abdomen want different lengths, and unequal discs do not form grids. The whole spiky structure above is a consequence of grids being available, so it should be expected to soften considerably — the peaks blur, the arithmetic of n stops mattering, and what is left is a broad preference rather than a set of exact proportions.

There are no rivers. Flaps hanging from different points of the tree need a strip of paper between them, and rivers are the part of a packing that responds most to the sheet’s shape, because a river is a long thin thing.

The classical bases assume a square. A preliminary base, a bird base, a waterbomb base — all are built from a square’s diagonals and midlines, and none means anything on a rectangle. A designer working from a base is working from a square whether or not it suits the flap count; the freedom measured here is available only to a designer working from a packing.

One feature of real packings is especially sensitive to the outline and none of the sweeps contains it.

The other thing the outline decides #

There is a second consequence of the sheet’s shape that no efficiency number captures, and it points the opposite way.

The outline is where the conditions stop. Every vertex on the sheet’s edge is exempt from Kawasaki, from Maekawa and from the big-little-big lemma, so a sheet with more perimeter for its area has proportionally more unconstrained vertices and more freedom in its crease pattern.

A long thin sheet has a great deal of perimeter. That is a real advantage and it is invisible to a circle packing, which knows nothing about creases. It is part of why strip folding is a tractable subject with a decidable flat-folding problem while the general two-dimensional case is intractable: a strip is almost entirely boundary.

So the proportions affect a design twice, once through how well the flaps pack and once through how constrained the crease pattern is. The two do not agree. The efficiency argument makes a square right for four flaps and nine; the constraint argument makes a square the hardest shape of a given area to fold anything on, because it has the least perimeter of any rectangle.

The unequal case, and why it should soften #

The whole spiky structure above depends on grids being available, and grids need equal discs. It is worth saying what should happen when the flaps are not equal, since that is every real design.

Two discs of different radii do not tile a rectangle in rows. The best arrangement for a mixed set is generally irregular, and the neat correspondence between factorisations of the count and proportions of the sheet has nothing to attach to. So the peaks should blur into a broad preference, the arithmetic of the flap count should stop mattering, and the curve should become the smooth trade-off it was expected to be in the first place.

What should survive is the crude part: a design whose flaps are mostly long and few wants a sheet that is longer than it is wide, and a design with many similar flaps wants something nearer a square. That is a weaker statement than the peaks, and it is the one a designer can actually use, because no real tree has seven equal branches.

The measurement is not made here and the reason is worth stating rather than hiding. Packing unequal discs is a harder search than packing equal ones, the results are noisier, and a sweep over sheet proportions would multiply that noise by the number of proportions. A figure whose peaks were within the search’s own error would be a figure that could not fail, which is the one thing a figure on this site may not be.

There is one design style for which the square is not a convention at all but a premise.

Why paper is square anyway #

The reason is commercial and cultural rather than geometric, and it is worth stating so that the convention is not mistaken for a result.

Square sheets are what the tradition standardised on, and a standard shape makes diagrams reproducible: a folding sequence that says “fold the corner to the centre” needs everyone to start from the same outline. Once the diagram became the way models were transmitted, the square became a requirement of the format rather than of the paper.

The tradition is also less uniform than it looks. Strip folding, the division of a strip into thirds and much modular work start from long rectangles, and paper for those is sold in the shape the technique needs. Where the shape is part of the technique, the shape varies.

What is new is the ability to ask the question quantitatively for a given design, and that arrived with the circle-packing formulation in the 1990s. Before there was a computation from tree to base, “what shape of paper” had no formulation; after it, the answer is one more parameter in a search that was already being run.

Where the ladder goes next #

The neighbouring question, and the more consequential one, is what a symmetry imposed on the packing costs. The sheet’s shape is a constraint nobody notices; a symmetry is one everybody imposes deliberately. Both are measured the same way, and both have counts where they are free and counts where they are expensive.

Further out is the accounting the whole subject runs on. Every visible feature is bought from the same sheet, and the shape of that sheet decides how much there is to spend.