The sheet has two sides

A crease pattern is drawn as lines and is almost always read as lines. The theorems are about lines: how many meet at a point, what their angles do, which of them are mountains. The paper itself barely appears.

Turn the reading round. The lines cut the sheet into panels, the panels are what a folder actually handles, and the panels turn out to carry a condition of their own.

What the two colours are #

The rule for the painting is the shortest rule in the subject: cross a crease and change colour. Start anywhere, paint that panel, and work outwards.

For this to be a painting rather than a contradiction, every closed walk through the panels has to cross an even number of creases. Otherwise a walk returns to where it began demanding the other colour, and there is no consistent assignment at all.

That is not a hopeful condition. It is a strong one, and the surprise is that every flat-foldable crease pattern satisfies it.

The colours are also not arbitrary marks. A fold turns the paper over. A panel reached across one crease is showing the reverse of the panel it was reached from, a panel reached across two is showing the same face again, and the two colours are exactly the two sides of the sheet. Painting the panels is drawing the folded object’s outer and inner surfaces on the flat pattern, before anything has been folded.

Why every closed walk is even #

Any closed walk through the panels can be built from small loops, and the small loops are the ones that go once around a single interior vertex. So the question reduces to one vertex: how many creases does a loop around it cross?

Exactly as many as meet there. And Maekawa’s theorem says that at a flat-foldable interior vertex the mountains and the valleys differ by exactly two, which means their sum is even, which means the number of creases is even.

That is the whole argument. Maekawa is usually presented as a statement about the assignment — a constraint on which creases are mountains — and its parity consequence is left as a remark. Read from the panels it is the main event: the sheet has two sides, and Maekawa is the condition that lets it keep them straight.

A vertex on the sheet’s edge carries no such condition, and needs none: a loop around it is not a loop, because the walk runs off the paper. This is the same asymmetry that makes the edge of the sheet a place where the theorems stop, and it means the colouring is decided entirely by the interior.

The vertex that refuses #

The clean way to test a condition is to build the case it must reject.

Three creases at a point, evenly spaced. Nothing about the drawing looks impossible. Kawasaki’s alternating sum is not even defined for an odd count — the sectors cannot be split into two alternating groups — and Maekawa’s requirement that the two letters differ by two cannot be met by three, because three has the wrong parity for it.

And the panels do not colour. Two of the three sectors take the first colour by force and the third is asked for both.

This is one refusal doing two jobs. The pattern has no folded state, and the pattern has no colouring, and it is the same parity that settles both. A checker that only looked at panels would reject this pattern without ever computing an angle.

Five creases refuses the same way. Four creases does not, and it is worth having the control: an even star colours perfectly well, and whether it folds is then a separate question about angles and letters that the colouring has no view of.

The control matters more than it looks. A test that rejects everything is not a test, and a condition on crease patterns that happened to reject all stars would be measuring something other than parity — the number of creases, perhaps, or the fact that they all meet at one point. Three fails, four passes, five fails, and the alternation is the signature of a parity argument rather than of a size one.

It is also worth being precise about what the refusal establishes. The three-crease star has no colouring and no folded state, but the colouring did not prove the second. Both follow from the parity; neither follows from the other. A condition that refuses for the right reason and a condition that refuses for a reason that happens to coincide are hard to tell apart from a single example, which is why the parity is computed here separately from the colouring rather than inferred from it.

The condition is about the count and not about the angles, which is worth seeing on a vertex whose angles are nothing like the ones above.

What Maekawa’s proof is doing to the paper #

The reason Maekawa gives an even number rather than some other kind of constraint is worth pausing on, because it is the same reason the colouring works.

Walk around the vertex in the folded state instead of the flat one. The cross-section of the folded paper is a closed path, and each crease is a reversal in it — a mountain turns one way, a valley the other. Going once around the vertex has to come back to where it started, having turned through a full circle, and counting those turns gives the difference of two.

A full circle is the same object as a closed walk of panels. In the folded state the walk is turns of the paper; on the flat pattern it is crossings of creases. Maekawa’s difference of two and the panels’ two colours are one statement seen from two sides, and that is why neither can be true without the other.

Which theorem was checked, and how #

The site’s habit is that a claim gets a test it could fail. This one is tested three ways, by three computations that share no code.

The first counts crossings. Starting from one panel, it walks the panel adjacency and flips colour at every mountain and every valley, and reports either a colouring or the loop that refused one. It never looks at a vertex.

The second counts creases at vertices. It takes the edges, throws away everything that is not a crease, adds up how many meet at each interior point, and reports which counts are odd. It never looks at a panel.

The third is the folded state. Every panel’s position is computed by composing reflections — one reflection per crease crossed on the way to it — and a composition of an odd number of reflections turns the sheet over while an even number does not. So each panel arrives with an orientation, and the orientation is a third opinion about its colour.

The assertion in the pattern library refuses three separate disagreements: a pattern that will not colour although every vertex is even, a pattern that colours although some vertex is odd, and a pattern that colours in a way the folded orientations contradict. None of the three has ever fired on a pattern this site publishes, which is the point of writing them.

The third check is the one worth dwelling on, because it is the only one that involves the fold. The other two are statements about a drawing: a walk over panels, a count over vertices. Composing reflections actually folds the thing, in the sense that it produces a position for every panel, and the orientation each panel arrives with is a by-product nobody asked for. That the by-product agrees with a colouring computed by counting crease crossings is not a restatement — it is two different pieces of geometry landing on the same partition of the panels, and either one could have been wrong.

The clearest way to see what the colouring is blind to is to look at a condition it cannot express.

Where the colouring stops being useful #

It is a necessary condition and it is a cheap one, which is a combination this subject offers rarely. It costs one pass over the panels, it needs no angles, and it kills an entire class of drawings before any theorem is consulted.

It also decides almost nothing.

A pattern can two-colour perfectly and fail to fold for every other reason there is. The angles can be wrong, in which case Kawasaki refuses it. The assignment can be wrong, in which case Maekawa does. The smallest sector can be flanked by the wrong pair, in which case the big-little-big lemma does. And even with all four conditions satisfied at every vertex, the sheet may still not fold, because deciding that is NP-hard and no local test reaches it.

There is a sharper limit still. The colouring says which face of the paper each panel shows. It says nothing at all about which panel lies above which, and that is a different object with its own rules and all of the difficulty. Two folded states can have identical colourings and look nothing alike.

The idealisation the whole argument rests on should be named, because it is the site’s standing one: the paper has no thickness. A real sheet’s two sides are separated by something, and a real fold puts a small curved region between them rather than a line. Neither changes the parity, and both are why a physical model of a deeply folded pattern comes out thicker and shorter than the drawing.

And the limit has a worked example on this site already.

How much the colouring rejects, counted #

The claim that the condition “kills an entire class of drawings before any theorem is consulted” has a size, and the size is larger than the phrase suggests.

A pattern two-colours exactly when every interior vertex has an even number of creases. For a drawing that was not built to fold — lines placed on a sheet with no theorem in mind — each vertex’s degree is as likely to be odd as even, and the vertices are independent. So a drawing with $V$ interior vertices two-colours with probability about

$$2^{-V}$$

At four interior vertices that is one in sixteen. At twenty — a Miura of modest size — it is one in a million. At the hundred and twenty-six of a tessellation patch it is a number with no name.

So the colouring is not a weak filter that happens to be cheap. It is the strongest filter in the subject per unit of effort: one pass over the panels, no angles, no letters, and it refuses all but a vanishing fraction of drawings that were not constructed to fold.

What makes it invisible as a filter is that every pattern anybody draws deliberately passes it, because Maekawa hands it over for free. A condition that rejects almost everything and is satisfied by everything one meets is easy to mistake for a triviality.

Which is the first factor of two, and only the first #

The star family shows exactly where the colouring’s work ends and the letters’ begins, and the division is clean.

Of all stars — $k$ creases from a point at equal angles — the colouring rejects precisely the odd ones and passes the even ones, whatever their angles. That is a factor of two, taken at no cost.

Of the even stars that survive, the letters then decide. At degree $2k$ with equal sectors, Maekawa admits $\binom{2k}{k+1} + \binom{2k}{k-1}$ of the $2^{2k}$ letterings — a half at degree four, seven sixteenths at degree eight, and a share falling as one over the square root of $k$.

So the colouring removes one factor of two and the letters remove a share that keeps shrinking. The first is free and bounded; the second is expensive and unbounded.

That is the honest ranking of the two conditions, and it explains why the colouring is a lemma rather than a headline. It does a great deal against arbitrary drawings and almost nothing against the drawings a folder produces, and the second is the population anybody working in the subject actually sees.

What it is good for #

Two things, and the first is a design technique rather than a theorem.

A colour change is a fold that brings the reverse of the paper to the outside, and every model with a black beak on a white bird is one. The two-colouring is the map of what is available: the panels of one colour are the only ones that can show the reverse, and no amount of cleverness moves a panel from one class to the other without changing the creases. What a colour change costs is a separate and surprisingly steep question.

The second use is diagnostic. The colouring is computed from the pattern alone, and so is the folded state’s orientation. A drawing that comes from somewhere else — traced, scanned, reconstructed from a photograph — can be put through both and asked whether they agree. They will not agree if a crease was missed, because a missing crease merges two panels that ought to differ.

That failure mode is the common one and it is nearly invisible by eye. A crease pattern reproduced from a printed diagram loses the faintest lines first, and a lost line does not leave a gap: it leaves a larger panel that looks entirely plausible. The parity notices immediately, because removing one crease from a vertex turns an even count odd, and the colouring then refuses somewhere that may be several panels away from the damage. It is a poor locator and an excellent alarm.

What the areas do, and what that is not #

Every tessellation on this site puts almost exactly half its area on each side. The Miura is fifty-fifty to the last decimal, the preliminary base likewise, the waterbomb likewise, the corrugated leaf likewise. The square twist is 50.7% and 49.3%, and the difference is one small panel in the middle with nothing to pair against.

This is not a theorem and the essay will not pretend otherwise. It is what happens when a pattern is built by repeating a unit: the repetition pairs the panels off, and the balance follows from the repetition rather than from the folding. A designer who wants one side to dominate has to arrange it deliberately, and the arrangement is the whole content of colour-change design.

A last pattern, chosen because most of its vertices are not interior ones.

Who noticed it, and when #

The two-colourability of a flat-foldable crease pattern is old enough that its first statement is hard to pin down, which is a familiar problem on this site — a name is not a date, and folding’s results have a habit of being known to folders long before they are written down as mathematics.

What can be said is that it appears as a lemma rather than a headline. It is the kind of fact that gets used in a proof about something else, stated in a line, and left. Its natural home is the modern computational treatment of flat-foldability, where the pattern is a plane graph and its faces are objects the algorithm has to name anyway — and once the faces are named, their two-colouring is the first thing anyone notices about them.

The practical tradition knew it first and by a different route. Anyone who has folded a colour change knows perfectly well which regions can be made to show the reverse, because the paper tells them. The mathematics did not discover the fact; it explained why the paper was right.

Where the ladder goes next #

Three directions run out of this one, and each is a different kind of question.

The first stays with the panels and asks what else they know. Where the paper goes when the pattern is folded is a panel question too: the panels land somewhere, they overlap, and how deep the overlap gets is the same quantity as how small the folded object is.

The second asks how much of a coincidence any of this is. Every pattern here satisfies the parity because it was built to; almost no pattern does, and the fraction that does can be measured.

The third is the design question, and the one a folder cares about: given that only one class of panels can show the reverse of the sheet, what does it cost to put it where a design wants it? The answer is twice the area, every time, and it is the reason a colour change is used sparingly by people who could use it everywhere.