Even is not enough

There is a sentence that gets left out of the proof, and leaving it out costs nothing at all until somebody cuts a hole in the paper.

The sentence is the sheet is a disc.

What the colouring is a statement about #

Reading a crease pattern as a set of panels rather than a set of lines produces a condition no vertex theorem states. The panels of a flat folded pattern take two colours so that no crease has the same colour on both sides, and the colours are not a convention: they are which side of the paper each panel ends up showing. A fold turns the sheet over, so crossing a crease must change the colour, and the colouring exists exactly when that demand can be met everywhere at once.

On a square of paper it is a parity and nothing more. Walk a small loop round one interior vertex, crossing every crease that meets it. The colour flips once per crossing, so the loop closes on the colour it started with exactly when the vertex has an even number of creases — and Maekawa forces that, since mountains and valleys differ by two and their sum is therefore even. A vertex where three creases meet has neither a colouring nor a folded state, and the two facts are the same fact.

So the colouring is Maekawa’s parity read off the faces instead of the creases. That is a satisfying thing to be able to say, and it is true, and it is true of a disc.

The loop of paper #

Take a square, cut a square hole out of the middle, and run three creases from the hole to the rim.

Count the interior vertices. There are none. Every crease begins on the hole’s edge and ends on the sheet’s edge, and no crease meets another anywhere. Developability, Kawasaki, Maekawa and the big-little-big lemma are all conditions on the creases meeting at an interior vertex, so all four are satisfied — vacuously, in the way that a statement about every member of an empty collection is satisfied.

Now count the panels. There are three, and each is adjacent to the other two across a crease. The colour has to change across each of those three creases, and a cycle of three faces cannot alternate. There is no two-colouring. There is therefore no flat folded state, and the ring cannot be folded flat by any means whatever.

The parity did not go anywhere. What went away is the vertex that was carrying it.

Two computations, one answer #

A refusal deserves a second witness, and this one has an entirely different kind of witness available.

Folding a pattern flat is composing reflections. Put one panel down where it is; its neighbour across a crease is that panel reflected in the crease line; the neighbour after that is another reflection, and so on. Going all the way round a cycle of panels and back to the one that was put down first, the composition of those reflections has to be the identity, or the sheet does not close.

Round the three-crease ring it is not. It is a reflection, and the panel it puts down is 1.87 sheet-widths from where the other route puts it — not a small residual, not a tolerance question, a gross geometric contradiction. At five creases the gap is 1.52 and at seven it is 1.84; at four, six and eight it is a few parts in ten thousand million million, which is what zero looks like in floating point.

That agreement is worth pausing on. One computation walks a graph of panels and flips a bit; the other multiplies two-by-two matrices and measures a distance. They know nothing about each other, and they agree about every ring and every disc at every crease count. The colouring’s refusal is a fact about the paper rather than an artefact of the way it was counted.

The count that does not depend on the count #

There is a third way to see it, and it is the one that makes the obstruction feel like a property of the sheet rather than of the drawing.

Draw any closed path in the paper that goes round the hole and count the creases it crosses. Not which creases, not where — just how many. On the three-crease ring the answer is three, for a square path, for a hexagon, for a twelve-sided polygon, at a fifth of the way out and at four fifths. Twelve different paths, twelve different orders of crossing, one number.

That number’s parity is the whole obstruction. The colouring assigns a colour to each panel and demands that it flip at every crossing; a closed path that flips an odd number of times cannot come back to the colour it started with, and so no assignment of colours exists. A path that does not go round the hole crosses each crease an even number of times or not at all, which is why the condition never arises on a square of paper: on a disc every closed path can be shrunk down to nothing, and a path shrunk to nothing crosses nothing.

Why the theorem is about a disc and never said so #

The proof that even degrees give a two-colouring goes like this. Pick a panel and call it white. Walk to any other panel along some route, flipping colour at every crease crossed, and paint it accordingly. The colouring is well defined provided any two routes to the same panel agree — and two routes differ by a closed loop, and a closed loop can be pushed across the sheet until it is a collection of little loops round vertices, each of which crosses an even number of creases.

Every step of that is correct, and the middle step is where the sheet’s shape enters. A closed loop can be pushed across the sheet until it is a collection of little loops round vertices. On a disc, yes. On a loop of paper, a path that goes round the hole cannot be pushed anywhere useful: it cannot be shrunk past the hole, because there is no paper there to shrink it across.

Reading that obstruction as a loop that cannot be shrunk is a piece of topology, and a good one; it belongs to the mathematics of loops up to deformation and not to this subject, and nothing here derives it. What belongs here is smaller and entirely sufficient: the panels form a graph, the graph has a cycle of odd length, and a graph with an odd cycle has no two-colouring. That is all the argument needs, and it is decided by counting rather than by continuity.

What every checker on this site said about it #

The most useful thing about the three-crease ring is what happened when it was handed to the machinery that checks every pattern here before it is drawn.

The gate reads the interior vertices, tests four theorems at each, and passes the pattern if all of them hold. On the ring it found no interior vertices, ran no tests, and returned success. Nothing was wrong with any of the four theorems and nothing was wrong with the code that applies them; the pattern was certified by a check that had performed no checks, and a pass with an empty record is indistinguishable from four theorems holding.

That has been repaired in the only way that is honest: a pattern with no interior vertices is now refused rather than certified, with a message saying that there was nothing to test. A figure whose whole subject is a single fold can still say so explicitly, which is the difference between a blank answer and a deliberate one.

It is worth being clear about what kind of mistake this was. It is not the mistake a checker makes when it is asked a question it cannot answer, which this subject has known about since its first essays and which is honest as long as it is said out loud. It is the mistake of an assertion that has nothing to assert about, and the only defence against it is to notice that the record is empty.

Where the parity actually lives #

Both sheets carry the same parity and only one of them has somewhere to put it. That is the whole difference between them, and it can be stated exactly.

On the disc, the three creases meet at a point, and that point is an interior vertex of degree three. Odd degree, so the sheet fails at that vertex, and every checker in the subject reports it in the same breath.

On the ring, the hole has removed the paper the vertex would have occupied, along with the vertex. The three creases are still there and they still make an odd cycle of panels, so the sheet still fails — but there is now no point at which anything is wrong. The failure has stopped being local without becoming any less definite.

There is one more way of putting it that a folder might find more convincing than any of the above. Number the panels round the ring: one, two, three, back to one. Each step turns the paper over. Three turns leave the paper the wrong way up, and the paper it has to join is the paper it started from.

The vertex went and its theorem stayed #

The reflection test was introduced above as a second witness, agreeing with the colouring and computed from nothing the colouring uses. It is worth asking what condition it is actually enforcing, because the answer is that a theorem this essay says has nothing to bite on is still biting.

The creases on the ring are rays from the middle of the sheet, so they all lie on lines through one point — the point the hole removed. Composing reflections in lines through a common point gives a rotation about that point, through twice the alternating sum of the angles between successive lines. The composition is the identity exactly when that alternating sum is a whole number of straight angles.

That is Kawasaki’s condition, written out. The sheet has no vertex at the centre and no paper there either, and the condition the centre would have carried is still the condition the loop has to satisfy. Cutting the hole removed the vertex and left its theorem behind, restated as a demand on a closed path rather than on a point.

Which is why the even rings close and close exactly. The creases in the census are equally spaced, so the sectors are equal, and an alternating sum of an even number of equal terms is zero — Kawasaki, satisfied for the same reason it is satisfied by any equally-spaced vertex. The odd rings fail the parity before the angles get a hearing at all, and the sheet-and-a-half of separation the reflections report is the sheet failing to be even, not the sheet failing to be Kawasaki.

And on a ring it is two conditions, not one #

That account depends on the creases being radial, and nothing requires them to be. Move one crease so that its line misses the centre, keeping both of its ends where they were, and the reflections no longer share a point.

An even composition of reflections in lines that are not concurrent is not a rotation about anything; it is a rotation about some point, or — when the lines’ angles cancel — a translation. So closure now asks for two things instead of one: the alternating sum of the crease directions must vanish, which is Kawasaki’s half, and the leftover translation must vanish too, which is a demand Kawasaki has never had to make.

On a disc that second demand does not exist, and the reason is worth stating plainly. Every crease at an interior vertex passes through the vertex by definition, so concurrency is not an assumption there; it is what being a vertex means. The ring is the first sheet in this collection where a cycle of creases can fail to meet at a point, and the extra condition is the price of that freedom.

So the ring is not merely a sheet where the parity has nowhere to sit. It is a sheet whose closure condition is strictly stronger than the disc’s: one parity, one angle sum, and one displacement, where a vertex has the first two and gets the third for free. An even ring with equally spaced radial creases satisfies all three, which is why the census finds zeros there; an even ring drawn carelessly would satisfy the first two and fail the third, and no theorem in this subject would have anything to say about it.

Where the model stops #

The hole is square and the sheet is square, and neither matters. The construction uses straight creases along rays from the centre because they are easy to draw and easy to check; the argument uses only the cycle of panels, and any hole of any shape with any odd number of creases reaching it gives the same refusal.

Nothing here decides that an even ring folds. Four creases on a ring give an even cycle and a two-colouring, and a two-colouring is necessary and not sufficient — the same footing every local condition on this site stands on. What the even case establishes is that the refusal at three is about the parity and not about the hole.

The reflections are a flat-folding test and not a rigid one. The composition round the cycle asks whether a folded position exists, not whether the sheet can be moved into it without tearing. The distinction between existence and reachability is a separate matter and this argument does not touch it.

One hole, one loop. A sheet with two holes has more independent ways round, and the parities add; the arithmetic is not hard and none of it is drawn here.

Why nobody folds a ring #

Origami is defined by a rule about the sheet — one square, uncut — and the rule is so nearly universal that its consequences for the shape of the paper have never needed stating. A folder who cuts a hole has left the subject by most people’s reckoning, and the small industry of cut-and-fold work that does exist is about what a cut buys at a point rather than about what it does to the sheet as a whole.

So the case has been left alone, and the theorem has been stated in the form that is true of the sheets people use. There is nothing wrong with that. What is worth noticing is that the missing hypothesis is invisible from inside the ordinary case: every pattern on a square of paper satisfies it, so no amount of experience with squares of paper will ever suggest that it is there.

Where the ladder goes next #

The obvious continuation is the one this essay has deliberately not taken: what happens with two holes, or with a sheet joined into a band, where there is more than one independent way round and the parities have to be tracked together rather than singly. That is a small piece of bookkeeping and it would be an essay about bookkeeping.

The better one is about the size of the thing that was removed. Cutting a hole of any positive area gives the refusal above, and cutting nothing at all gives no refusal, so somewhere between them a quantity changes discontinuously — which is not how cuts normally behave in this subject, where taking a wedge out of a sheet changes the angle at a point by exactly the angle of the wedge. A cut whose effect does not depend on its size at all is a different animal, and it is the one the next rung is about.