The lengths are free

A crease pattern is a drawing, and in a drawing the lines have ends. Where each one stops looks like part of what has been specified — as much a part of it as the direction the line points in.

Every condition this site checks disagrees. Kawasaki adds alternate sectors and compares the two sums. Maekawa counts mountains against valleys. The big-little-big lemma looks at a sector and both of its neighbours. All three read angles, or letters, and none of them contains a length.

So the lengths are free, and a single vertex is not one pattern but an entire family of them.

What each condition actually reads #

It is worth being pedantic about the inputs, because the claim is entirely a claim about inputs.

Developability, the prerequisite, asks whether the sector angles around the vertex sum to a full turn. Kawasaki asks whether the alternating sums are equal. Maekawa asks whether the number of mountains and the number of valleys differ by two. Each takes the angles, or the letters, and returns a verdict.

The third condition is the one that most looks as though it might sneak a length in, and it does not.

The big-little-big lemma says that a sector strictly smaller than both its neighbours must be bounded by creases of opposite kinds — one mountain and one valley. “Smaller” is a comparison of angles. A crease drawn twice as far out does not make its sector any bigger, because a sector is a wedge and a wedge has no end.

There is a fourth thing a crease pattern records, and it is worth putting beside the other three so that the omission is complete rather than partial. The mountain-and-valley letters are read by Maekawa and by the big-little-big lemma; they are not read by Kawasaki or by developability at all. So of the four pieces of data in a drawing — the vertex positions, the crease directions, the crease lengths, the letters — exactly one is consulted by nothing.

That is the whole argument, and it takes three sentences. What is worth spending an essay on is what follows from it.

One vertex is a family of patterns #

Fix the sector angles of a flat-foldable degree-four vertex. Two of the four angles are then free — the other two are their supplements — and that two-angle freedom is the one the subject talks about constantly.

Underneath it sits a second freedom nobody quotes. Four creases have four lengths, and every one of them can be set to anything positive without any condition noticing. The vertex in the hero figure is drawn at four sets of lengths, from all four equal to a longest-over-shortest ratio of four, and the count of assignments that fold is four in every one of them.

Two members of that family, folded, make the point better than the count does.

The two folded states are not variations on a theme. They are different shapes with different outlines, different areas and different silhouettes, and the pattern that produced each is, to the theorems, the same pattern.

This is the connection that seems worth pausing on. In a drawing of a crease pattern, the lengths are most of what is on the paper — they are the ink. They carry none of the information about whether the thing folds, and all of the information about what it folds into.

What the lengths decide instead #

Nothing in that section says the lengths do not matter. It says they do not matter to the conditions, which is a much narrower statement, and the complement of it is where the lengths live.

Where a piece of paper lands under a flat fold is settled by the crease positions alone. Fold along a line and everything on one side reflects; do that repeatedly and there is a map from the flat sheet to the plane which never consults a mountain or a valley. That map is built out of the crease lines, and where a crease stops is where the panel it bounds stops.

The flat line in that figure is the essay. Thirteen patterns, each a different drawing, each answering the foldability question identically — and the quantity that does move is the one a reader of the folded model would actually notice.

So the division is clean, and it is worth stating in the form a designer would use. The angles decide whether a folded state exists. The lengths decide which folded state it is. Neither half of that sentence is the whole pattern, and the reason is the third thing.

The third thing, which is neither #

A folded state has to say two things: where every piece of paper ends up, and wherever two pieces land on the same spot, which one is nearer the reader. The first is the lengths and positions. The second is the layer ordering, and it is not read off the angles or off the lengths.

That is why the two-part division above stops where it does. Move a crease’s endpoint and the footprint changes; move it far enough and the set of legal stackings can change too, because two panels that used to overlap may stop overlapping and a constraint disappears with them. The lengths reach into the ordering, but they do not determine it, and the ordering is a separate combinatorial object with a count of its own.

The honest summary is a three-way split rather than a two-way one. Angles settle existence, lengths settle the footprint, and the stacking is a third question that both of the others constrain and neither answers.

It is a split worth carrying, because each part behaves differently under change. The angles are brittle: disturb them and foldability is gone immediately. The lengths are inert: disturb them by any amount and foldability is untouched while the shape moves continuously. The stacking is neither continuous nor brittle — it is a count, it changes in jumps or not at all, and there is no small perturbation of it because there is nothing there to perturb.

At one vertex the split is two-way, not three #

The three-way division — angles for existence, lengths for the footprint, stacking for neither — is the right account of a pattern. At a single vertex it collapses, and the collapse has a reason worth having, because it explains a number the figures report and do not account for.

The stacking count is four in every member of the family. Not four on average and not four usually: four at every set of lengths tried, alongside the four assignments that fold. So at one vertex the lengths do not reach into the ordering at all.

The reason is that every panel of a single vertex touches the vertex. Folding flat maps each sector onto a wedge with its apex at the folded image of that point, so two panels whose wedges share any direction share the paper immediately around the apex — however short either of them is. A crease drawn to a tenth of the radius still produces a sector that overlaps everything its angular image meets.

So the overlap structure at one vertex is a function of the angles alone, and the constraints between panels are read off the overlaps, and the count of legal stackings follows. Lengths cannot remove an overlap that starts at the apex.

Which is why two vertices are different #

The essay’s remark that lengths can change the set of legal stackings is correct and it needs a second vertex to bite.

With two vertices, a panel need not touch both of them. Two panels can be far apart on the sheet, land on the same region of the folded plane or not depending on how far their creases run, and a length that shortens one of them can remove an overlap outright — taking its constraint with it and changing the count.

So the freedom is exactly as local as the conditions are. At one vertex the lengths are free in the strongest sense available: they change the footprint and nothing else, not even the combinatorics. The moment a pattern has two vertices, a length is simultaneously somebody else’s position and a possible overlap, and both of the things it was free of come back at once.

That is a cleaner statement of where the result stops than the freedom does not survive contact with a neighbour. It survives being drawn at any size; it stops surviving when there is a second place for the paper to land.

Which theorem was checked, and how #

The claim being made is an invariance, which is an awkward thing to check, because a measurement that does not move is indistinguishable from a measurement that was never taken.

So the check is built the other way round. The fold checker is run on every member of the family and asked for the count of assignments that satisfy every local condition; that count is required to be identical across the family, and it is four in the hero figure and four at all thirteen sweep positions. At the same time Kawasaki’s two alternating sums are computed at each of those positions and their disagreement recorded. The worst disagreement anywhere in the sweep is 1.8 × 10⁻¹⁵ radians — floating-point noise, arrived at from the drawn geometry rather than from the formula that guarantees it.

That would still prove nothing on its own, because a figure in which nothing moved would pass it. So the generator also requires the folded footprint to spread, by a factor of at least 1.05, and refuses to draw a family in which it does not. The refusal is not decorative: it fires on a set of equal lengths, and it fires on two sets that are rotations of one another, which are exactly the two ways of writing down a family that has no variety in it and looks as though it does.

The pairing is the point. A disturbance to the angles destroys foldability at once. A disturbance to the lengths, of any size, destroys nothing.

The idealisation the whole family rests on should be said plainly. The paper here has no thickness, the creases are perfect rays from a point, and the vertex is a point rather than the small crumpled region a real crease crossing makes. A stack of four layers costs nothing in this model. On real paper the four-layer region in the second folded state is thicker where the arms are long, and a crease that runs a long way is harder to place accurately than a short one — so the lengths are free in the geometry and not free on a table.

Where the freedom runs out #

The picture cannot show the sheet, and that is the limit worth naming.

A vertex drawn on its own has creases that stop wherever the figure stops. On an actual sheet a crease has to go somewhere: to the edge of the paper, or to another vertex. The moment there is a second vertex, one crease’s length is the other vertex’s position, and the position of a vertex changes the sector angles at both ends of every crease that reaches it. The freedom does not survive contact with a neighbour.

That is the precise sense in which the result is local. It says that at one vertex, in isolation, the lengths carry no condition. It does not say that a pattern’s crease lengths are free, because in a pattern most of them are not lengths at all — they are the distances between vertices, and each is doing angular work at its far end.

Two further places the reasoning must stop. The sheet’s edge removes conditions rather than adding them, and a crease that reaches the boundary is subject to nothing at all there, which is a separate argument with its own consequences. And a pattern all of whose vertices pass every local test can still fail globally, which is the standing gap between a local test and a decision and is untouched by anything here.

The same fact, seen from a design office #

There is a version of this that practitioners use without stating it, and it is what makes the result feel less like a technicality.

A designer working from a circle packing sets the positions of things first and reads the creases off afterwards. A designer working from a grid does the opposite, fixing a lattice of allowed directions and then choosing where each line begins and ends. Both are exploiting the same split: one is choosing the shape and letting the conditions be satisfied automatically, the other is choosing a set of angles known to satisfy them and then treating the lengths as the design.

The second is why box pleating is a workable way to invent things by hand. Every vertex on the grid has 45° and 90° sectors, which satisfy the conditions permanently, so the whole design activity is the placement of endpoints — a search through a space in which no move can break foldability. That is a strange and very comfortable place to work, and it exists because of a fact about what the theorems read.

Who noticed it, and when #

Nobody announced this, because it is a property of statements rather than a discovery about paper, and it has been visible in the statements since they were written down.

The angle condition carries Toshikazu Kawasaki’s name from a 1989 paper, sits in Koji Husimi’s book a decade earlier, and is in Jacques Justin’s 1986 statement of the local conditions as well; the site’s own account is in the two conditions at a point, and the gap between a result and the name it goes by is a subject of its own here. The parity condition is Maekawa’s by eponym, Husimi’s and Justin’s by date, and the winding argument behind it is worth reading for how little it needs. The big-little-big lemma is Justin’s from 1986 and was named in the algorithmic literature of the 1990s. Every one of the three is stated over angles, and none of the three has ever had a length in it.

What is genuinely modern is the habit of treating a vertex as a family rather than as a drawing. That comes from the tessellation and computational side of the subject, where patterns are generated rather than sketched, and where it becomes obvious very quickly that two drawings which differ only in where the lines stop are the same input. Anyone who has written a fold checker has met the fact from the inside: the data structure that holds a vertex holds a list of angles, and the coordinates are only there so the thing can be drawn.

Traditional practice knows it in a third way. A great many classical models are variations of the same base at different proportions — the same vertex, the same conditions, different paper covered — and a folder who scales one arm of a bird base to change a beak is using this result without a name for it.

Where the ladder goes next #

The immediate continuation is what happens when the same vertex is repeated. Tile the plane with one flat-foldable vertex and the lengths stop being free in the way described here, because every crease now runs to a neighbour — and the sheet becomes a material whose properties come from the geometry the tiling forced.

In the other direction, the small size of the angle-family is its own subject. There are not many flat-foldable degree-four vertices, and the same one keeps being found independently in buckled cylinders, in crumpled sheets and in designed patterns. The lengths being free is part of why: two people who arrive at the same vertex with different arm lengths have arrived at the same vertex, and the drawings do not look alike.

And the third question stays open. A pattern with its angles fixed and its lengths fixed still has not named a folded object, because the stacking is a separate count — which is the next rung of this ladder.