Dividing without measuring

Halving a square by folding is obvious: bring two edges together. Quartering it is halving twice. Dividing it into eighths, sixteenths, sixty-fourths — all easy, because every one is a power of two and folding in half is the natural operation.

Thirds are not a power of two, and there is no obvious fold that produces them.

The usual answer is to estimate, roll the paper until it looks even, and crease. It works, it is what most folders do, and it is the source of a surprising amount of the frustration in complex folding.

Haga’s theorem #

Kazuo Haga, a biologist rather than a mathematician, noticed the following in the 1970s.

Take a unit square. Fold the bottom-right corner up so that it lands exactly on the midpoint of the top edge. That single fold produces, elsewhere on the sheet, a point that divides an edge in the ratio $1:2$ — an exact third, arrived at by a fold that was aiming at a half.

The arithmetic is a right triangle and the Pythagorean theorem. If the corner lands at distance $k$ along the top edge, the crease produces a division at $\frac{k^2}{1+k^2}$ and another at $\frac{2k}{1+k^2}$ — both rational whenever $k$ is, and neither of them the number that was aimed at.

So folding a corner to a half gives thirds. Folding it to a third gives fifths and sevenths. Each construction produces new rational points, and the set of reachable divisions grows quickly.

Why any rational division is reachable #

The general statement is stronger than the specific constructions, and it follows from what a fold can do.

Axioms 1 to 5 are enough to construct any rational point on an edge. Given a length $p/q$, the construction is elementary: repeated bisection reaches any dyadic rational, and Haga-type folds reach the rest.

There is also a beautifully simple method for $1/n$ that most folders learn instead, because it needs no theorem. Fold a diagonal from one corner, and another line from a corner to the midpoint of the opposite side. Where they cross is at $1/3$ of the width. Repeat the construction using that point and the crossing moves to $1/5$, then $1/7$. Each iteration is one fold and the sequence is exact throughout.

Error, and why it compounds #

The practical argument for exactness is about what happens downstream, and it is worth being quantitative.

Suppose a division is out by one per cent. Fold on that reference and the next crease inherits the error. Fold again and it inherits it plus its own. Errors from independent estimates grow like the square root of the number of steps; errors from reused references grow linearly, and folding reuses references constantly.

A model on a 32-grid has creases aligned to creases aligned to creases, five or six deep. A one per cent error at the first division is a five per cent error where it matters, which on a 20-centimetre square is a centimetre — enough that the flaps do not meet and the model does not close.

The exact construction contributes nothing to that chain. The folder’s hands still contribute, but hand error is small, random and does not systematically accumulate in one direction.

That is the whole practical case, and it is why serious designers specify exact reference folds rather than trusting the folder to estimate — particularly for a design laid out on a grid.

The grid, and why it is worth its cost #

Once exact division is available, a design can be laid out on a grid — and a great deal of modern complex origami is.

Box pleating gives up the efficiency of a freely-packed arrangement — the flaps are not as long as they could be for the paper used — and buys something worth more. Every crease meets every other crease where the grid says it should, every reference is exact, and a mistake is visible immediately because a line fails to land on a grid intersection.

The grid is, in effect, an error-correcting code for folding. A free-angle design gives no signal when a crease is slightly wrong; a grid design gives one at every intersection.

What the divisions are actually for #

It is worth being concrete about why a folder needs sevenths.

Flap lengths. A circle-packed design puts flap centres at positions determined by the packing, and those positions are rarely at halves. A design calling for a flap centred at $3/7$ of the width needs sevenths, exactly, or the flap comes out the wrong length.

Grid conversion. A design on a 24-grid needs the square divided into 24, which is $8 \times 3$ — the eighths are free and the thirds are not.

Proportion. Some models want a rectangle of a specific ratio cut from a square, and cutting is not available, so the proportion has to be folded.

In every case the requirement is the same: a reference at an exact position, produced by folding, from which later creases can be measured.

The same problem without a square #

Everything above assumes a square, which is a convention rather than a necessity, and dropping it changes the constructions.

A rectangle of unknown proportion cannot be divided by the standard methods, because those methods use the square’s symmetry. The first step with an arbitrary rectangle is usually to make a square from it — fold a diagonal, and the part that overhangs is removed or folded away.

Traditional Japanese folding used square kami almost exclusively; European folding used rectangles more freely, because European paper came in rectangles. The A-series is $1:\sqrt2$, chosen precisely so that halving preserves the proportion, which makes it excellent for repeated bisection and awkward for everything else.

The choice of starting shape is therefore not neutral. It determines which divisions are cheap.

The theorem nobody was looking for #

There is something worth pausing on in Haga’s result, and it is not the mathematics.

Haga was a biologist. He was folding paper with schoolchildren, noticed that a fold aimed at a midpoint produced a third somewhere else, and worked out why. The theorem is elementary — it is a right triangle and the Pythagorean theorem, well within reach of anybody who did school geometry — and nobody had written it down.

The subject is full of this. The mathematics of folding is not deep in the way that, say, algebraic topology is deep; most of its central results are accessible to a determined sixth-former — the flat-folding conditions included. What made them hard to find was that nobody was looking, because folding was a craft and crafts are not where theorems are expected to be — the same reason the design algorithm waited until the 1990s.

Exactness is not the same as accuracy #

A distinction that matters, and that the figures on this page cannot show.

Exact means the construction produces the right number, with no approximation in the method. Accurate means the crease ends up in the right place on the actual paper.

Haga’s construction is exact and a folder’s execution of it is accurate to perhaps a quarter of a millimetre — the width of a crease, roughly, and better than an estimate by a factor of ten or so, but not zero.

So exactness removes the systematic component of the error and leaves the random one. That is the useful thing: random errors partly cancel and systematic ones do not, and a folding sequence with no systematic error can be long.

Finding the shortest sequence #

Knowing that an exact construction exists is not the same as knowing how to make it in four folds rather than eleven, and for a designer the difference matters a great deal.

Robert Lang’s ReferenceFinder attacks exactly this. Given a target point or line on the square, it searches through millions of folding sequences built from the seven axioms and returns the shortest one that lands within a stated tolerance. A folder asks for a point at $3/7$ of the width and gets back a five-fold sequence, with an error of a thousandth of the sheet.

Two things about that are worth noticing.

It is a search, not a construction. The mathematics guarantees a sequence exists; finding a short one is a separate computational problem with no closed-form answer.

And it returns approximations by choice. A sequence that is exactly right may take eleven folds where one that is right to a tenth of a millimetre takes four, and for a sheet cut with scissors the second is better. Exactness is a means here, not an end.

The same idea, at every scale #

Exact reference-making is not a peculiarity of paper. It is what any process does when it has to position something without an external measure.

A machinist without a scale uses a dividing head — a mechanism that produces exact angular divisions by counting gear teeth rather than by measuring. A draughtsman without a protractor constructs angles with a compass. A carpenter marks equal spacings with dividers, stepping them along rather than reading a rule.

In every case the same trade is made: give up direct measurement, gain exactness, and accept a construction that takes longer. Folding is unusual only in that the medium and the instrument are the same object.

That also explains why the constructions feel satisfying out of proportion to their difficulty. They produce certainty from nothing but the sheet, and there is no step at which anybody has to trust a ruler.

Reference points are the hidden half of a diagram #

Anybody who has followed a complex folding sequence has met a step that appears to accomplish nothing: fold, crease firmly, unfold. Sometimes several in a row.

Those are constructions. Each one is producing a reference — a point or a line for a later step to align to — and the model does not change because nothing has been shaped yet. A diagram that labelled them “make a reference at three-sevenths” would be far clearer than one that says “fold and unfold”, and almost none do.

That opacity is a real cost of the notation. The Yoshizawa–Randlett system is excellent at saying what to do and has no vocabulary at all for why, so the structure of a folding sequence — which steps are construction, which are shaping, which are the collapse — is invisible on the page and has to be inferred.

A crease pattern has the opposite problem. It shows the whole structure and gives no order, so it says everything about where the creases go and nothing about how to get there.

Where the model stops #

Zero-width creases. Every construction here treats a crease as a line. A real crease has a width, and on thick paper the width is a real fraction of a small division — dividing a 15 cm square into sixteenths gives cells under a centimetre, and the creases bounding them are a tenth of that.

Paper does not slip. Aligning a corner to a point assumes the sheet stays where it is put. It does not, quite, and the error is larger for stiff paper.

One sheet, no cuts. Everything here divides a square without removing material. Cutting makes all of it trivial and is not available.

The figures show the result, not the fold. The left-hand panel draws the divisions and the crease that produced them, and it cannot show the sequence — which fold came first, and what it was aligned to. That sequence is most of the practical difficulty.

“Estimated by eye” is drawn generously. The right-hand panel’s errors are a per cent or two, which is roughly what a careful folder achieves. A hurried one does worse, and the figure would be less flattering if it showed that.

What a fold cannot divide #

The constructions reach every rational proportion, and it is worth asking what they miss.

Irrational divisions are reachable too, within the degree limit: $1/\sqrt2$ is a diagonal fold, and anything of degree $2^a 3^b$ is available from the axioms. What is not reachable is a division at a transcendental proportion — a point at $1/\pi$ of the width does not exist in any folding sequence, however long.

That is a curiosity rather than a limitation, because nobody wants one. Designs call for rational proportions, occasionally for a square root, and never for anything transcendental. The reachable set comfortably contains everything the craft asks for, which is a slightly unusual position for a constructibility result to be in.

The genuine limitation is elsewhere and is practical: the number of folds. Every reference costs a crease, every crease is a mark on the finished model, and a design needing a dozen construction folds before the shaping starts has a dozen unwanted lines in it. Designers therefore trade exactness for tidiness routinely, and the mathematics has nothing to say about that trade.

Why exactness is not obvious #

It is worth pausing on how surprising Haga’s result should be.

Folding a corner to a midpoint is an operation with an obvious purpose: it puts the corner at the midpoint. That it also produces an exact third somewhere else on the sheet is not something anybody would predict, and it is not a special property of that particular fold — the same construction with the corner going to a third gives fifths, and so on.

What is going on is that a fold is a reflection, reflections preserve lengths, and the resulting configuration is a right triangle whose sides are constrained by Pythagoras. Rational inputs give rational outputs, and the outputs are not the inputs.

So the sheet is a small algebraic machine: put in a rational point, get out several more. Nobody designed it that way, and it is available to anybody who folds a corner onto a mark and looks at where the crease crosses the edges — which is a very large number of people over a very long time.

The ladder from here #

Later rungs: Haga’s theorem derived. The general rational construction. The $1/n$ iteration and its proof. Error propagation, measured. Grid systems and their trade-offs. Reference finding as an optimisation problem, which is what Lang’s ReferenceFinder does. Non-square starting shapes. The A-series and its silver ratio. And the question of how accurate a fold actually is, which has been measured and is better than most folders assume.

Lang’s ReferenceFinder searches millions of folding sequences to find the shortest one producing a given point to within a stated tolerance. It exists because knowing that an exact construction is possible does not tell a folder how to make it in four steps rather than eleven.