Closure is not the identity

There are two ways to check whether a crease pattern has a flat folded state, and only one of them is normally taught.

The taught one is local. Go to each interior vertex in turn and ask four questions about the creases meeting there: do the sectors close to a full turn, do the alternating sums agree, do the letters differ by two, and does the smallest sector have creases of different letters bounding it. Every one of those is a statement about a point, and a pattern passing all of them at every vertex is still not guaranteed to fold.

The other is global and geometric, and it is the one this essay is about. It runs the fold rather than checking it.

The walk #

A flat folded state assigns each panel of paper a rigid motion of the plane — where that panel ends up. Two panels sharing a crease are related by the reflection in that crease, because folding along a line is exactly reflecting one side of it onto the other. So the motion of any panel can be computed by starting somewhere, walking across creases, and composing a reflection at each one.

That works as long as there is only one way to get anywhere. Usually there is not: the panel graph has loops, a loop offers two routes to the same panel, and the two routes must agree. Composing the reflections all the way round a loop and requiring the answer to be the identity is exactly the statement that the two routes agree.

On a disc of paper that is the whole story, and it is a good story. The composition is a product of two-by-two matrices with a translation each, it is cheap to compute, and it returns a distance rather than a bit: a pattern that does not fold misses by an amount that can be printed and drawn. Nothing about the local conditions gives a magnitude.

The step the disc hides #

Now do the same walk on a glued sheet.

Start on a panel, cross every crease going once round the band, and arrive back at — not the panel the walk started on. The walk ends at the copy of it on the other side of the seam, which is the same panel of the sheet and a different piece of the drawing.

The paper knows they are the same. The drawing has them a band’s length apart.

So the condition is not the composition is the identity. It is the composition is the map that identifies the two ends. On a cylinder that map is a slide along the band. On a Möbius band it is a slide with a flip.

And the disc’s rule is the case where that map is the identity — which it is, on a disc, because on a disc a closed walk really does come back to the same paper and the identification is trivial. Nobody writes the general condition down because on a disc there is nothing to write: the right-hand side of the equation is what is left when there is nothing to put there.

What the general form immediately gives #

Two results fall out in a line each, and both were reached differently elsewhere.

The parity. A composition of $k$ reflections turns the paper over exactly when $k$ is odd; that is elementary and needs no origami. A cylinder’s gluing map is a slide and does not turn the paper over, so a cylinder needs $k$ even. A Möbius band’s is a slide with a flip and does, so a Möbius band needs $k$ odd. The colouring argument reaches the same place by multiplying signs round a loop, and the two share no arithmetic.

A distance. Where the colouring answers yes or no, the composition answers with a number: the largest disagreement between the two motions, in widths of the strip. A band that folds reads nought to rounding; the rest do not all read the same, and the differences turn out to matter, because a band that misses by a translation is a solvable problem and a band that misses in its linear part is not.

Why the equation is linear, and what that buys #

The composition splits into two halves that behave completely differently, and separating them is what makes the condition usable rather than merely correct.

The linear part — the two-by-two matrix — depends only on the crease angles. Reflecting in a line at angle $\phi$ has a linear part that depends on $\phi$ and not on where the line sits, so composing $k$ of them gives a matrix determined by the alternating sum $\phi_1 - \phi_2 + \phi_3 - \cdots$ and nothing else. That half of the condition is decided before a single crease is placed, which is a constraint on the angles alone.

The translation part depends on where the creases are, and it depends on them affinely: move a crease by a certain amount and the composition’s translation moves by an amount proportional to it. So once the angles are fixed, finding the positions is solving a system of two linear equations, not searching.

That is a considerable practical difference from the local conditions. The local conditions are a constraint satisfaction problem over letters, and finding a satisfying assignment is a search. The closure condition, on a sheet with no interior vertices, is a linear solve.

The two computations, kept apart #

The closure condition and the colouring answer the same yes-or-no question about the same sheets, and they are computed here without sharing any code, deliberately.

The colouring walks the panels flipping a bit and multiplying signs. It never touches a coordinate. The closure multiplies six numbers at a time and never mentions a colour. Neither of them can be made to agree with the other by an accident of implementation, because there is no implementation in common to be accidental in.

The agreement is not a theorem being tested. Both statements follow from the same underlying fact — that a folded state is a map from the sheet to the plane, well defined on the sheet rather than on a drawing of it — and the point of computing them separately is that a mistake in either one shows up as a disagreement rather than as a wrong answer both of them give.

What the condition is not #

It is not a decision procedure for flat-foldability in general, and the difference is worth stating flatly because the condition looks stronger than it is.

On a sheet with interior vertices, the closure condition holds round every loop of the panel graph, and there are many such loops. Requiring all of them is equivalent to requiring the local conditions at every vertex plus a global consistency, and it says nothing at all about the layer ordering — two panels can be mapped to overlapping places in the plane by motions that satisfy every closure condition, and the paper still has to decide which of them is on top without passing through the other. Deciding that is the part the theorems do not reach, and it is where the hardness lives.

What is special about the sheets in this essay is that they have no interior vertices at all. Their creases run from one edge of the strip to the other and meet nothing, so the panel graph is a single cycle, and there is exactly one loop to close. That is why the condition is a linear solve here and a search everywhere else.

A worked instance, in six numbers #

It helps to see the two sides of the equation written out on a case small enough to hold in the head.

Take a cylinder with two creases, both square across the strip, at positions $a_1$ and $a_2$ measured along a band of length $L$. Reflecting in a vertical line at $x = a$ sends $x$ to $2a - x$ and leaves $y$ alone. Compose the two: the first sends $x$ to $2a_2 - x$ and the second sends that to $2a_1 - (2a_2 - x) = x + 2(a_1 - a_2)$. The linear part is the identity — two reflections in parallel lines always compose to a translation — and the translation is $2(a_1 - a_2)$ along the band.

The gluing map runs the band back on itself by $-L$. So the condition is $2(a_1 - a_2) = -L$, which says the two creases are half a band apart. Put them at a quarter and three quarters of the way along and it holds exactly; that is the flattening of a tube everybody already knows, arrived at from the algebra.

Four creases give $2(a_1 - a_2 + a_3 - a_4) = -L$, which is one equation in four unknowns and therefore has a three-parameter family of solutions. That family is why a tube can be flattened in many ways and why the creases need not be evenly spaced, and the equation is the whole of the constraint.

Odd counts have no solutions at all, and the reason is visible in the algebra rather than argued for: an odd number of reflections in parallel vertical lines composes to a reflection, a reflection is not a translation, and no choice of positions changes that. The linear part fails first and the positions never get a chance to matter.

Where the magnitude earns its keep #

A yes-or-no answer is enough to decide whether a sheet folds. It is not enough to do anything else, and three things become possible once the condition returns a distance.

A near miss can be recognised. A drawing that misses by a thousandth of a width is a drawing somebody meant to be right, and it is almost always a rounding error or a strip cut slightly wrong. A drawing that misses by two is wrong in kind. A bit reports both as no.

A search can be steered. Anything that hunts for a foldable arrangement needs to know whether it is getting warmer, and a bit never says so. The distance is a function of the crease positions with a gradient, which is what makes the linear solve possible in the first place.

A refusal can be attributed. The distance separates into the linear part and the translation, and the two failures have different causes and different cures. A generator drawing one of these figures asserts on the mismatch and reports which half of it failed, so a wrong drawing produces a sentence naming the reason rather than a picture with something subtly off in it.

That last one is the reason the condition is computed at all rather than reasoned about. A figure claiming a band folds is drawing an object whose closure has been checked to within a rounding error, and a figure claiming one does not is drawing an object whose mismatch has been measured.

The condition on a sheet with two loops #

Everything above concerns sheets with exactly one closed path that cannot be shrunk. A torus has two, and the pattern generalises in the way one would hope, with one wrinkle worth naming.

Gluing both pairs of a rectangle’s edges gives two independent identifications: a slide across and a slide along. A walk that goes once round in each direction has to satisfy the closure condition for each, and a walk that goes round one way and then the other has to satisfy the condition for the composition of the two maps — which is automatic, since the two slides commute.

The wrinkle is that on a torus the two conditions are not the only ones. Panels can also be identified with panels a cell away in both directions at once, and the collection’s own consistency test for such sheets had to be rewritten for exactly that reason: a loop of the drawing that ends one cell over is not a loop of the sheet. The closure condition sees it correctly, because a walk ending one cell over composes to the map that identifies that cell, and that map is not the identity.

So the general statement survives the extra loop without amendment. What does not survive is any shortcut that assumes a walk returning to the same drawn panel has returned to the same paper.

The general statement, written out #

For the sake of having it in one place:

A sheet is a piece of paper together with a rule saying which of its boundary points are the same point. A flat folded state is a map from the sheet to the plane, isometric on each panel. Choosing a panel and a motion for it determines the motion of every other panel by composing reflections along a path, and the state exists exactly when every closed path in the sheet imposes a consistent condition — which is that the composed reflections equal the isometry the identification demands along that path.

For a disc, every closed path bounds and every identification is trivial, so every condition reads the composition is the identity, and the general statement collapses to the familiar one.

For a cylinder there is one path that does not bound, and its condition reads the composition is a translation by the band’s length.

For a Möbius band there is one such path, and its condition reads the composition is that translation composed with a flip.

Three sentences, one of which is quoted everywhere and two of which are not, and the reason the two are not is that until a sheet existed whose identification was not trivial, they said nothing that the first one did not.

Why the local conditions cannot be patched into it #

A reasonable thought at this point is that the vertex conditions could simply be extended to cover glued sheets, and that the whole business of composing motions is an unnecessary detour. It is worth saying why that does not work.

The vertex conditions are conditions at a point, and every point of a glued sheet is an ordinary point of paper. The identification does not create a vertex, destroy one, or change the angles at any of them: the rectangle’s edges are placed to miss every vertex precisely so that this is true, which is what makes the four sheets a controlled comparison rather than four different drawings.

So whatever the vertex conditions say about a cut rectangle, they say the same thing about every gluing of it. They cannot distinguish sheets they cannot see the difference between, and the difference between the sheets is not at any point.

That is the general shape of the situation and it is not special to gluing. A loop of paper with three creases has no interior vertex at all and no flat folded state, and every vertex condition passes on it vacuously. A checker built out of point conditions is complete for exactly the questions that are about points, and whether a sheet’s own shape lets a walk close is not one of them.

What was actually changed #

For the record, since the correction is small and its scope matters.

The comparison used to be between the composed motion and the identity. It is now between the composed motion and the map the sheet’s identification demands along the path in question. On a disc that map is the identity, so every number this collection has ever published from that computation is unchanged, and the drawings that were checked by it are checked by it still.

What is new is that the comparison takes an argument. The argument is supplied by the sheet rather than by the drawing, which is the point the whole essay is about: the condition was always about a sheet and a drawing together, and one of the two was being supplied by a default nobody had noticed making.

What it costs to have said it #

Very little, and that is worth noting because the general form is sometimes treated as an abstraction bought at the price of usability.

Every existing computation continues unchanged. A disc’s identification is the identity, so the right-hand side is the identity, so the code that compared against the identity was computing the general condition all along and did not know it. Nothing had to be rewritten, no result moved, and the only thing that changed is that the comparison now takes an argument that was previously a constant.

What is bought is the two glued sheets, and with them the first case in the collection where the parity condition of the whole subject reads the other way round.