The sheet decides which points exist

Two folds from a bare square put marks at a half, a third, a quarter, a fifth, a sixth, an eighth and a twelfth — and at no seventh, ninth or eleventh at all. That measurement, and every other measurement of the reference closure on this site, was made on a square.

Origami paper is sold square, so the choice is not arbitrary — and the rectangle that keeps its shape is what most of the world outside that tradition has instead. It is also not neutral, and the reason it is not is a fact about the square rather than about folding: a square has eight symmetries and a rectangle has four, so folds that would be distinct on a rectangle coincide on a square.

Hold the area fixed and change the proportion, and the reference closure changes by two orders of magnitude.

What is held fixed, and what is not #

Five proportions: the square, the A-series rectangle at one to root two, two squares side by side, the one-to-root-three rectangle, and the golden rectangle.

Each has area one. That is the comparison that makes the rows readable: a sheet with more marks on it is not simply a bigger sheet, and it is the same normalisation the packing efficiency measurement uses for the same reason.

Each starts with four corners and four edges, which is what a folder has before doing anything. The operations are the first four axioms — fold one point onto another, one line onto another, along the line through two points, and perpendicular through a point — applied to everything reachable, twice.

A mark is a crossing that a folder could tell apart. Two crossings a thousandth of a millimetre apart are one mark on paper, so the crossings are thinned: take one, discard everything within a third of a millimetre of it, repeat. Without that step the counting measures arithmetic rather than folding.

The counts #

One fold. A square admits twelve distinct folds and they cross at nine points. Every rectangle admits eighteen, and they cross at twenty-three or twenty-nine depending on the proportion. So before anything interesting has happened, a rectangle has three times as many marks on it as a square.

Two folds. The square reaches 92 distinct folds and 565 crossings. The A-series rectangle reaches 752 folds and 114,927 crossings, of which 45,705 are separable at a third of a millimetre.

Distinct folds after two rounds, and the marks a folder could tell apart:

The square is not slightly behind. It is behind by a factor of eighty.

Why the square loses #

The mechanism is symmetry, and it is a loss rather than a gain.

An axiom takes points and lines and returns a fold. On a square, many of those folds coincide: the fold that puts the bottom-left corner onto the bottom-right is the vertical midline, and so is the fold that puts the top-left onto the top-right, and so is the perpendicular bisector of the top and bottom edges’ midpoints. Three constructions, one crease.

On a rectangle the same three constructions give the same crease as well — but the square has four more symmetries than the rectangle does, and every one of them identifies a further pair of constructions that a rectangle keeps apart. The diagonals are the clearest case: a square’s diagonals are axes of symmetry and a rectangle’s are not, so a whole family of folds that a rectangle distinguishes are on a square the same fold reflected.

Fewer distinct folds means fewer crossings, and crossings are what a reference is.

The square is also the one proportion that needs no thinning at all. Its 565 crossings are 565 marks: no two of them are within a third of a millimetre of each other. Every rectangle’s crossings have to be thinned, some heavily — the A series’ 114,927 come down to 45,705. That is the same fact from the other side: the square’s own symmetry has already collapsed the folds that would have crossed near one another.

A folder’s check, in four folds #

The counts above are enormous and the mechanism behind them is small enough to verify by hand, which is worth doing because the claim is surprising.

Take a square. Fold it in half vertically, unfold, and fold it in half horizontally: two creases, crossing at the centre — one mark. Now fold the bottom-left corner onto the top-right: the diagonal. On a square that diagonal is a symmetry axis, and folding the bottom-right corner onto the top-left gives the other diagonal, which is also one. Four folds, and the crossings are the centre and the two diagonal-with-midline meetings.

Take a rectangle and do the same four folds. The two midlines are the same. The two diagonals are not symmetry axes, so each of them meets each midline somewhere that is not the centre, and the four crossings are four distinct points rather than a smaller set. Same four folds, more marks — and every subsequent round compounds it, because each new fold is constructed from the points already there.

That is the entire mechanism, and the factor of eighty at two folds is what it comes to when the compounding is followed through rather than reasoned about.

The advantage is in the growth rate, not the head start #

A factor of eighty at two folds and a factor of three at one invites the obvious reading — the rectangle starts ahead and stays ahead — and the two rows say something stronger than that.

Divide each proportion’s second-round count by its first. The square goes from nine marks to five hundred and sixty-five, which is a factor of sixty-three. The A-series rectangle goes from twenty-nine to forty-five thousand seven hundred and five, which is a factor of fifteen hundred and seventy-six. So the rectangle is not merely three times ahead at the first fold and carried along by it: its closure grows twenty-five times faster per round.

Those two numbers account for the whole gap exactly. Three and a fifth times ahead after one fold, twenty-five times the growth rate through the second, and three and a fifth by twenty-five is eighty — which is the ratio the table reports.

That matters because it says the symmetry is not a one-off deduction. Each round builds folds out of the marks already present, so a coincidence that collapsed two constructions at the first round removes not one mark but every mark that pair would have generated afterwards, and every mark those would have generated. Symmetry compounds, and the compounding is what turns a factor of two in the symmetry group into a factor of eighty in the closure.

It also makes a prediction the essay’s own numbers support and do not state. If the growth rates hold for another round, the third-round gap should be around eighty times twenty-five — two thousand — and it is the reason the square’s third round is computable while the rectangles’ are not. The square’s tractability at depth three is the same fact as its poverty at depth two.

And how much of it is the tolerance #

The second thing the two rows settle is how much of the eighty is a statement about paper rather than about geometry.

At a third of a millimetre the square holds 565 marks and the A series 45,705. Coarsen the tolerance to a millimetre and the square drops to 541 — a loss of four per cent — while the A series drops to 10,683, losing three quarters of everything it had.

So the square’s marks are genuinely separated and the rectangle’s are crowded, and the advantage is largely a supply of references that are close together. Threefold coarser tolerance costs the rectangle a factor of four in its lead: eighty times becomes twenty.

Continue that at the same rate and the advantage would reach one somewhere near a centimetre of tolerance, which is a crude but not absurd figure for a beginner working by eye on a sheet the size of a page. That extrapolation runs from two measured points and should be read as an order of magnitude rather than a number — but the direction is not in doubt, and it says what the eighty is worth. A folder working to a third of a millimetre has eighty times the references. A folder working to a centimetre has none of the advantage at all.

What this does not mean #

It does not mean a square is a bad sheet to fold on, and three things are worth saying before the conclusion runs away.

Marks are not what a construction wants. Folding a strip into thirds needs one crease in one place, not a supply of them. A folder looking for a particular fraction wants that fraction, not a large supply of arbitrary crossings, and the systematic route to any nth reaches it in n folds on any sheet. Having eighty times as many marks is having eighty times as many places to be nearly right.

Crowding is the other half. More marks in the same area means marks closer together, and a reference closer than a crease is wide is not a reference. The thinning above is exactly that correction applied once; applied at a millimetre rather than a third of one, the A series drops to 10,683 and the square only to 541.

And the square’s symmetry is why anybody uses it. A pattern folded on a square can be rotated a quarter turn and still fit, which is what makes the traditional repertoire work at all. The count above prices that symmetry, and the price is paid in references rather than in models.

What it would change for a folder #

The measurement has one practical consequence and it is small, which is worth admitting before drawing anything larger from it.

A folder who needs a reference that a square does not offer has three moves available. They can fold the systematic ladder, which reaches any nth in n folds on any sheet and is what a careful folder does anyway. They can use one of the square’s own approximations, which is what most published sequences do and is why exactness and accuracy come apart. Or they can cut the sheet to a different proportion before starting, which costs one cut and multiplies the supply of accidental marks by eighty.

The third is almost never done, and the reason is not ignorance of the arithmetic. A square is what the model was designed for: cutting it to a rectangle changes every angle in the pattern, so the extra references are references on a sheet the design no longer fits. What the measurement prices is therefore an option that is available only when the design is being made at the same time as the choice of paper — which is the designer’s situation and not the folder’s.

Which theorem was checked, and how #

Every proportion is compared at the same area, checked rather than assumed: each sheet’s width times its height is one to the last bit a double holds.

A proportion of one must reproduce the square exactly. The general construction contains the case it generalises, and if it did not the comparison would be between two different pieces of machinery.

A sheet with no width is refused rather than folded, which is the ordinary defence against a caller who has computed a proportion from something that came out zero.

The thinning is done on a grid rather than pair by pair. A hundred thousand crossings is ten billion comparisons and a pass over cells one gap across is a hundred thousand; the two give the same answer and only one of them finishes.

The square must need no thinning, which is asserted rather than observed — it is the sharpest form of the symmetry claim, and a run in which the square lost marks to thinning would mean the coincidence story was wrong.

Where the model stops #

Two folds, and one of the counts is enormous. The closure at three folds on a rectangle specifies more folds than can be held at once, so the comparison stops at two — and the square’s advantage in tractability is real: it is the only one of the five whose third round is computable with the full operation set.

Axioms five and six are left out. They are the ones that need a circle and a cubic, and including them multiplies every row without changing the ordering; the comparison is over the four that always return a fold.

A mark is not a reference until a folder can name it. The count is of distinguishable crossings, and a crossing whose coordinates are an unnameable irrational is a mark a folder can use and cannot ask for. What fraction of each row is rational is not measured here.

And the sheet’s proportion is not usually free. A folder has the paper they have. The measurement is about what a proportion offers rather than about a choice anybody makes often, and its practical form is a comment about which sheet to reach for when the choice does exist.

What the picture cannot show #

A bar chart of marks cannot show where they are. A sheet with forty thousand references scattered unevenly is a different tool from one with forty thousand spread out, and the density map that would show the difference is a different figure that this rung does not draw.

Nor can any of these figures show a fold that failed to be distinct. The square’s missing folds are the whole explanation and they are invisible: what is drawn is what exists, and the argument is about coincidence — which appears as an absence.

The idealisation, named #

A fold is exact and a crease is a line with no width. The thinning above is the one place where the width of a crease enters, and it enters as a threshold in millimetres — a third of one on a sheet of a hundred and fifty — rather than as a model of a crease.

That threshold is doing a great deal of work. At a millimetre the A series’ advantage falls from eighty times to twenty, and at three millimetres it would fall further; the ordering survives all of them, and the magnitude is a function of what a reader counts as two marks.

The generalisation #

A closure is a property of what it starts from, not only of the operations that build it. The axioms make no mention of the sheet, and every measurement of what they reach must; that is easy to miss precisely because the operations are the interesting part and the sheet is the boring part.

The specific form is worth carrying because it is counter-intuitive twice over. Symmetry in the starting configuration reduces what a closure reaches, because symmetric configurations make distinct constructions produce the same result. And the reduction is not small: eight symmetries against four costs a factor of eighty at two folds.

That is a general fact about generated sets, and it is the reverse of the way symmetry is usually priced here — requiring a packing to be symmetric halves the coordinates a search must find, which is a saving; requiring a sheet to be symmetric costs references, which is not. A symmetric seed generates a smaller closure than an asymmetric one under the same rules, and anybody measuring the reach of a construction system on a deliberately tidy example is measuring the tidiness as much as the system.

Where the ladder goes next #

The measurement suggests a question it cannot answer: which fractions each proportion reaches, rather than how many marks. A rectangle’s extra marks are worth something only if they are at nameable places, and the count of distinguishable crossings says nothing about whether a folder could ask for one.

The other direction is the practical one. A folder cutting a sheet to a chosen proportion before starting is making a decision that costs one cut and changes the reference supply by two orders of magnitude — which is either the cheapest improvement available in the subject or an artefact of counting the wrong thing, and separating those two is what the fractions question would settle.