The seam carries a sign

Take a strip of paper, bring its two ends together and glue them. Crease the loop straight across in a few places and press it flat. Whether the pressing works — whether the loop has a flat folded state at all — depends on how many creases there are and on nothing else about them: an even number and it flattens, an odd number and it does not. That is the argument a ring of paper makes, and it is one of the cleanest results in the subject, because it needs no vertex, no angle and no theorem about a point.

Now glue the strip with a half twist, and everything inverts. Three creases flatten and four do not.

The drawing did not change. The creases are in the same places, at the same angles, in the same numbers. What changed is a single sentence about the paper, and the whole of this essay is about where in the argument that sentence enters.

The colouring, and what it is a product of #

The two-colouring is the oldest global statement in the subject and the only one that is not about a point. Fold a sheet flat and look down at it: every panel of paper shows the reader either its front or its back, and crossing a crease swaps which. So the panels take two colours, and a pattern whose panels cannot be two-coloured has no flat folded state — it is the one condition that survives having no vertices at all.

On a disc of paper the condition is not much of a condition. Every closed path can be shrunk to a point, so every loop in the panel graph is a boundary of something, and the parity works out automatically wherever the vertex conditions hold. That is why the colouring reads as a consequence rather than as a constraint, and why it is usually mentioned once and then left alone.

A loop of paper is the first sheet where it bites. A path running once round the loop cannot be shrunk to a point — the hole is in the way — so the parity of the creases it crosses is not forced by anything local, and it becomes a genuine condition on the whole sheet.

Written as arithmetic rather than as a walk, the condition is a product. Give every step round the loop a sign — minus one if crossing it swaps front for back, plus one if it does not — and the colouring exists exactly when the signs multiply to plus one. On a loop of paper with four creases that is $(-1)^4 = +1$, and the loop flattens. With three creases it is $-1$, and it does not.

The step nobody was counting #

There is a step in that loop which is not a crease.

A walker going once round a glued band crosses every crease, and then crosses the seam to get back where it started. On a cylinder the seam is not a crease, it is not drawn, and nothing happens at it: the paper continues, the same side stays up, and the step contributes $+1$. Because it contributes $+1$, it contributes nothing, and because it contributes nothing, it is easy to leave out of the account entirely — which is exactly what the even rule does.

On a Möbius band the seam contributes $-1$.

That is the whole of the difference. The half twist is an instruction about which points of the two ends are the same point, and the instruction says that the front at one end meets the back at the other. A walker crossing the seam arrives with the paper the other way up, having crossed no crease. The step is still there, it still carries a sign, and now the sign is not one.

So the condition on a cylinder is $(-1)^k \times (+1) = +1$, which asks for $k$ even, and the condition on a Möbius band is $(-1)^k \times (-1) = +1$, which asks for $k$ odd. Both are the same statement — the signs multiply to one round the loop — and the two sheets disagree about the answer because they disagree about one factor.

What the sign is a fact about #

It is worth being careful about what kind of statement the seam’s sign is, because it is easy to hear it as a convention.

It is not a choice of drawing. Whether the front of one end meets the front or the back of the other is a fact about which sheet of paper was built, and the two sheets are genuinely different objects — one can be painted in two colours and the other cannot be painted at all, in the sense that a painter starting anywhere and never lifting the brush comes back to the start on the other side. The sign records that fact, and it is the only place in the whole account where it enters.

It is also not a fact about the creases. Nothing in the crease pattern knows it. Every count the drawing supports — how many creases, how many panels, what angles they meet the edges at, which conditions hold at which vertices — is identical on the two sheets, and every one of them is silent about the seam. A file holding the drawing holds none of it.

And it is not a statement that needs the folded state. The colouring is computed on the flat pattern, before anything is folded, by walking the panels and flipping a bit. What the gluing supplies is one extra edge in that walk, with a sign on it.

Both sheets, at every count #

Setting the two side by side across a range of crease counts makes the inversion easy to see and easy to check, and the checking matters because a claim of the form this rule is exactly reversed on that object invites the suspicion that a sign has been dropped somewhere.

Two computations produce that table and they share no code. One walks the panel cycle multiplying signs, which is combinatorics and knows nothing about geometry. The other composes a reflection at every crease and asks what the composition is, which is geometry and knows nothing about colours. They agree about every band in the table, and the second returns a distance where the first returns a bit — a matter taken up in the essay about what the composition has to equal, because the geometric version turns out to say something the counting version cannot.

Why no vertex is involved #

A reader who has spent any time with this subject will have noticed what is missing. Developability, Kawasaki, Maekawa and the big-little-big lemma are the four conditions this subject checks, and not one of them appears above.

They cannot appear, because there is nowhere for them to hold. Each reads the creases meeting at one interior point of the paper and says something about the angles or the letters there. A band’s creases run from one edge of the strip to the other and meet nothing on the way, so the sheet has no interior vertex at all, at any crease count, on either gluing.

This is not a defect in the conditions and it is not a trick. It is the exact statement of what a local check is: a claim that nothing goes wrong at any point, which on a sheet with no points where anything could go wrong is true and empty. A checker that reads vertices reports a three-crease cylinder as satisfying every theorem it knows, and a three-crease cylinder cannot be pressed flat by anybody.

The colouring is the instrument that sees it, and the colouring is a statement about the sheet.

The ring, and why the seam was invisible #

The result on the loop of paper is not new here; it has been the standing example of a global obstruction since the collection first had one. What is new is noticing that it was stated with a hypothesis and the hypothesis was not written down.

The reason it was not is that on the sheets available at the time, the hypothesis was always true. An annulus — a square with a square hole cut out of it — has two sides. So does a disc, so does a disc with any number of holes, so does a torus. Every sheet the collection could build was orientable, so the seam’s factor was always $+1$, so the rule was always even, and a factor that is always one is a factor nobody writes.

The moment a sheet with one side is available, the omission becomes visible and slightly awkward, because the missing sentence is not a technicality. A closed path on a flat-foldable sheet crosses an even number of creases is false as stated. The true statement is that the signs multiply to one, and on a two-sided sheet that reduces to the even count because every step but the creases contributes nothing.

Doing it with a strip of paper #

None of this needs a computer, and the whole of it fits in about four minutes with a sheet of A4 and a pair of scissors.

Cut two strips a couple of centimetres wide and thirty long. Join the ends of the first into a plain loop with a piece of tape; join the ends of the second after giving one end a half turn, so that the tape holds the front of one end against the back of the other. The second loop is the awkward one: run a finger along it and the finger comes back to where it started having travelled both faces of the paper without crossing an edge.

Now crease. On the plain loop, put four creases across it at roughly equal spacings and flatten the loop onto the table. It goes flat, and it goes flat in a particular way — the four panels stack into two pairs, and the two creases that end up at the two ends of the flattened shape are the ones the loop chose. Add a fifth crease and try again; the paper resists, and what it is resisting is a contradiction rather than a shortage of dexterity.

On the twisted loop, put three creases across it and flatten. It goes flat, into a triangle. Add a fourth and it will not.

The reason the demonstration is worth doing is that it makes the sign tangible. On the plain loop the eye can follow a single face all the way round; on the twisted one it cannot, and the moment where the followed face becomes the other face is exactly the moment the walker crosses the seam. That instant is the factor of minus one, and it is a physical event rather than a bookkeeping choice.

One caution about the second loop, which catches people out. A half twist is a half twist and not a full one. A strip joined with a full turn is an ordinary cylinder with a twist in it — two sides, an even rule — and it will not flatten at three creases. The distinction is not about how the paper looks; it is about whether the two ends were joined front to front or front to back, and only the second changes the sign.

The same argument, said with reflections #

The counting argument above is complete, and it has a geometric twin worth stating because the twin says more.

Folding a sheet flat sends every panel to the plane by a rigid motion, and the motions of two panels sharing a crease differ by the reflection in that crease. So the motion of any panel is a product of reflections, one for each crease on some path from the panel the walk started at, and a walk that comes back to where it started imposes a condition: the product of the reflections round the loop has to be a motion the sheet can actually have.

On a disc that condition reads the product is the identity, because the walk came back to the same paper and the paper has to be in the same place. On a glued sheet the walk does not come back to the same paper — it comes back to the image of where it started, one band-length along — so what the product has to equal is the map that does the gluing.

And a product of reflections turns the paper over exactly when there is an odd number of them. A cylinder’s gluing map is a slide, which does not turn the paper over; a Möbius band’s is a slide with a flip, which does. The parity falls out of that in one line, and it falls out with a magnitude attached rather than as a yes or a no, because two motions that are not equal differ by a measurable amount.

That is the same result by a route that shares no arithmetic with the first, which is the reason both are computed here rather than one.

What a hole would do instead #

There is a family resemblance between a glued band and a sheet with a hole in it, and it is worth saying exactly how far it goes.

An annulus — a square of paper with a square hole in the middle — is the same shape as a cylinder: bend the annulus and the inner boundary becomes one end of a tube and the outer becomes the other. Creases running from the hole to the rim become the creases running across the band. Every count matches, the parity condition matches, and the loop that cannot be shrunk is the loop that goes round the hole.

What does not match is that an annulus is made by removing paper and a cylinder is made by joining it. That is a difference in how the sheet was obtained rather than in what it is, and neither the colouring nor the reflections can tell the two apart, which is the right answer: they are the same sheet.

The Möbius band has no such twin. There is no way to make one by cutting a hole out of a flat sheet, because everything obtained that way has two sides. A sheet with one side has to be built by an identification, and that is why it took a gluing to get one, and why the collection had no example of the sign being anything but plus one until it had a way to say which points of a boundary were the same point.

The sentence that was missing #

The rule as it is usually stated — a closed path on a flat-foldable sheet crosses an even number of creases — is not wrong so much as under-quantified. It is true of every sheet that has two sides, which is every sheet anyone normally folds, and it is false of the sheets that do not.

The corrected statement costs one clause. Assign each step of a closed path a sign, minus one where the step exchanges the two faces of the paper and plus one where it does not; the product round the path must be plus one. Crossing a crease always exchanges them. Crossing an ordinary point of the paper never does. Crossing a seam does exactly when the seam was glued with a flip.

Stated that way it is a condition about the sheet and the drawing together, which is what it always was. Both of the usual statements — the panels take two colours and every closed path crosses an even number of creases — are the special case where the sheet contributes nothing, and both of them are quoted in this collection and elsewhere without the sheet being named at all.

What this does and does not settle #

It settles the parity, and the parity is a necessary condition rather than a sufficient one. A band with the right count still has to have its creases in positions and at angles that let the paper close up, and that turns out to be a much sharper constraint than the counting — a Möbius band creased square across the strip has the right parity at every odd count and folds at none of them. The sign is the first thing to check and it is nowhere near the last.

It also does not touch the layer ordering. A sheet that has a two-colouring has, at most, the possibility of a flat folded state; which panel lies over which is a separate question with its own machinery, and on a sheet with no edge it loses a feature it has always had, since an order needs somewhere to start.

What it does settle is where in the argument the sheet enters. The colouring is a product round a loop. Creases contribute to that product and so does the paper, and until there was a sheet whose contribution was not one, the second half of that sentence had never been said out loud.