The patient machine is the weak one

The all-layers machine of the previous rung fails because it is forced: it must take every layer crossing the fold line, and a layer with no crease there stops it dead. The obvious repair is to let it take fewer. Let it peel off one layer, fold that, and come back for the rest — the patient machine, the one that never grabs more than it can manage.

The patient machine is worse. Not slower: weaker. It reaches strictly fewer patterns, and on every strip tested here it reaches exactly two.

Paper is joined #

The reason is not subtle once it is said, which is why it is worth saying carefully: a sheet of paper is connected, and folding part of it moves the rest.

Consider a strip whose last segment has already been folded back onto its neighbour. Two layers now sit at that end of the pile. To fold the next crease along, a machine must move the segment underneath — and that segment is attached, at the crease already folded, to the segment lying on top of it. Moving one without the other does not fold the paper; it tears it.

So the machine has no choice about how many layers to take. It must take the one it wants and everything joined to it beyond the fold line. On a strip, the only piece that carries a free end is an end piece, so a one-layer machine can only ever fold inward from one of the two ends.

That constraint is not a modelling choice. It is enforced in the simulator by a connectedness test, which after every move checks that each already-folded crease still joins two pieces meeting at the same point on the line. Without it the simulator cheerfully rolls a strip by pulling the fourth segment out from under the fifth, and reports the roll as one-layer foldable. It is not, and no machine or pair of hands has ever made one that way.

Exactly two, every time #

Folding inward from the ends is a strong restriction, and its consequence is arithmetic rather than geometric.

Each fold takes the outermost remaining segment and lays it over the pile — above it or below it, according to which end the machine is working from and which way the crease turns. A sequence of such folds produces a stack in which each segment sits immediately outside the one before, which is the layer order of an accordion and of nothing else. The assignment that produces that order alternates: mountain, valley, mountain, valley, or its mirror image.

Over 117 spacings at two, three, four and five creases — generated from a seeded stream so the sweep repeats exactly — the one-layer machine reached the two alternating assignments on every one of them and nothing else. Not approximately, not usually: the set was exactly MVMV… and VMVM… on every spacing, with no exceptions.

That flat line is the whole result. On an evenly creased strip every assignment folds flat — which is itself a small surprise and comes from the layers always being nestable — so the number that fold rises as 2ⁿ. The number a one-layer machine can build does not rise at all.

Why “patient” was the wrong word #

The intuition that fails here is a good one and it fails for an interesting reason, so it is worth locating precisely.

The intuition is that a machine allowed to fold fewer layers is a machine with more options, because folding one layer is one of the things a machine that folds all layers could choose to do. That is true of the choice and false of the constraint. The one-layer machine is not permitted to fold all the layers; it is required to fold exactly one. The models are not nested by permission — they are three different requirements.

And the requirement to take exactly one is severe, because paper is joined. The set of moves available to a one-layer machine is not a subset of the all-layers machine’s moves plus some extras. It is a different, much smaller set, most of whose members are illegal for a reason that has nothing to do with layers and everything to do with connectivity.

The model that does have more options is the one in the middle.

Some-layers — take any contiguous block from the top or the bottom of the pile — is the model with real freedom, and it is the only one of the three that can roll a strip: rolling means folding two, then three, then four layers together, each time taking the whole outer coil. That is a block at the top, so it is legal; it is not one layer, so the patient machine cannot do it; and it is not every layer, so the all-layers machine cannot do it either.

The roll, which nobody thinks of as difficult #

The clearest case is the one everybody has done without noticing. Take a strip, and roll it up.

A roll has every crease turning the same way — four mountains, no valleys. It folds flat, and the stacking search finds it immediately. The one-layer machine cannot make it, because after the first end fold the next crease needs two layers moved at once. The all-layers machine cannot make it either, on an unevenly creased strip, because it must take the flat paper still lying underneath the coil.

Only the middle model rolls, and rolling is the single most natural thing anybody does to a strip of paper.

There is a lesson in that which generalises beyond folding. A machine model that cannot do the obvious thing is usually the wrong model of the machine, and the way to find out is to try the obvious thing rather than the interesting one.

The roll is also the pattern that shows the two failures are genuinely different rather than two views of one. The one-layer machine fails at the second fold, having made a legal first one; the all-layers machine, on a strip whose creases do not line up, fails at the second fold too but for the opposite reason — too much paper rather than too little. A repair for one is not a repair for the other, and a machine built to avoid both is the middle model, which is neither a relaxation of the first nor a tightening of the second.

Something similar is true of the tessellations. A material made of creases is manufactured by pressing a whole sheet at once, which is an all-layers move applied to a pattern whose creases were designed to line up — and the alignment is usually described as a consequence of the pattern’s periodicity rather than as the manufacturing requirement it also is.

What each model can see #

The three machines differ in strength, which is the essay’s subject, and they differ in something else that the table above makes visible and nobody names: what information about the strip each of them is sensitive to at all.

The one-layer machine’s answer is two, on every one of the hundred and seventeen spacings. Move the creases anywhere and it is still two. That is not a robust result; it is a machine that cannot see the crease positions. Its argument — fold inward from a free end, build an accordion — never mentions a length, so lengths cannot reach it.

The all-layers machine is the opposite. On an evenly spaced strip it reaches all sixteen assignments; on a strip with one short segment it reaches none. The letters did not change and the count went from everything to nothing, so this machine is decided entirely by the geometry and barely at all by the labelling.

The some-layers machine matches flat-foldability, which depends on both.

Three models, three sensitivities #

Set out that way the taxonomy has a second axis, and it explains the shapes of the three results rather than merely recording them.

One-layer: purely combinatorial. Reaches a constant, independent of every length. Its failures are about connectivity, which is a fact about the strip’s topology.

All-layers: purely geometric. Reaches anything or nothing, decided by whether the creases line up when the paper is folded. Its failures are about a layer with no crease at the fold line, which is a fact about positions.

Some-layers: both, and therefore neither restricts it. It is the only model whose answer requires knowing the letters and the lengths, and it is the only one that reaches everything.

That is why the middle model is the strong one, restated in a form that does not depend on the sweep. The two restricted machines are each blind to half the problem, and a machine blind to half the problem can only get the answer right when that half happens not to matter. The some-layers machine sees both halves because its move — a contiguous block from the top or the bottom — is chosen against the paper in front of it rather than fixed in advance.

It also predicts the shape of the one-layer result before the sweep runs. A machine that cannot see lengths must return the same count on every spacing, so the only question was which constant, and “exactly two on every spacing” stops looking like a suspicious number the moment the model is read this way.

A folding diagram is a list of machine moves #

The result has a consequence for something much older than any of these models, and it explains a feature of origami instruction that otherwise looks like convention.

A folding diagram is a sequence of steps, each of which says: fold here, in this direction, taking this much paper. That is a machine model with the machine left implicit, and the steps are almost always simple folds. What varies between diagrams is precisely the parameter this essay is about — how many layers a step takes — and diagrams are careful about it. “Fold the near layer only” and “fold all layers” are distinct instructions with distinct symbols, and a diagram that omits the distinction is a diagram somebody will get wrong.

So the traditional bases are, in effect, patterns selected over centuries for being reachable by a particular machine: hands, which can hold any contiguous block of layers and which are therefore a some-layers machine with a strength limit. That is the strongest of the three models, which is why so much is foldable and why the restriction only becomes visible when somebody builds a machine that is not a hand.

It also explains why the instruction “roll it” almost never appears in a diagram. A roll is a some-layers move whose block grows at every step, and a diagram that spelled it out honestly would need a different picture for each turn. The tradition writes “roll” and leaves it to the reader, which is a tacit admission that the move is not atomic.

Two ends, and what they buy #

There is one refinement worth making, because it explains why the answer is two rather than one.

A one-layer machine folding only from the right-hand end produces a stack in which each new segment lands on the same side of the growing pile every time — which is a roll, and a roll is not what it produces, because the segment it lands on top of is attached the wrong way. Working from a single end, the letters it can realise are forced, and the count would be one.

The machine has two ends, and it may alternate. That does not double the reachable set, because both ends are producing the same accordion from opposite directions; what it doubles is the labelling, since an accordion built from the left has its letters exchanged relative to one built from the right. The two reachable assignments are one folded object described twice, which is why the count is exactly two and not three or four.

That is a small point and it is the kind of small point worth checking, because “exactly two on every spacing” is the sort of number that usually means a bug. It does not here; it means a symmetry.

What is actually being counted #

Two cautions about the numbers above, because both are easy to over-read.

The counts are over assignments, with the crease positions fixed. That is a natural thing to count and it is not the same as counting patterns: a spacing with sixteen assignments is one crease pattern asked sixteen questions. Where the essay says the one-layer machine reaches two, it means two of the sixteen letter strings, on that spacing.

And the sweep is a sweep. One hundred and seventeen spacings with no exception is strong evidence and it is not a proof. The argument in the second section — that a one-layer machine can only fold from a free end, and folding from free ends produces an accordion — is a proof sketch, and the sweep is what checks the sketch against a machine that has no opinions. Where this site can prove something it says so; here it has an argument and a measurement that agree, which is a weaker thing and worth labelling as one.

The honest limit of the simulator #

The simulator tracks pieces of a line and their order in a pile. That is the whole state, and it is enough in one dimension because a folded strip is a pile of intervals.

In two dimensions it is not enough. A folded sheet is a set of polygons in the plane with a partial order on the pairs that overlap, and the order is only defined where they overlap — which is where the difficulty of the whole subject lives. A two-dimensional simple-fold simulator would need that partial order maintained under every move, and deciding whether a consistent one exists is the hard problem rather than a subroutine of it.

So nothing here is evidence about two-dimensional simple folding. The taxonomy is the same, the connectivity argument is the same, and the counts are not transferable. What the literature has for two dimensions is a different shape of result: for maps ruled into a grid, some models are polynomial and others are hard, and the boundary is not where the one-dimensional case would suggest.

The other limit is the strength assumption. Every machine here can close on any number of layers at once, which is exactly what real material cannot do. A some-layers machine rolling a strip is holding an ever-thicker coil, and a real one runs out of throat depth. That does not weaken the negative results — a machine with a layer limit is more restricted, not less — but it does mean the some-layers model’s strength is an idealisation, and the strongest model here is the one leaning hardest on it.

Who found this, and when #

The three-model taxonomy is from Arkin, Bender, Demaine, Demaine, Mitchell, Sethia and Skiena, in the map-folding paper that gave the previous rung its definitions. Their interest was the two-dimensional grid case and the complexity classes it falls into; the one-dimensional comparison here is the case a reader checks first and then usually stops thinking about.

What is contributed here is the connectivity condition as a piece of running code, and the observation that leaving it out is a mistake with a specific and recognisable symptom: a simulator without that test reports that a roll can be made one layer at a time, which is exactly the answer somebody hoping the patient machine is the strong one would like to see. The check was added because the simulator gave that answer, and the answer looked too convenient to be true.

Where the ladder goes next #

The three models are all simple folds, which is one particular atom. Change the atom and the picture changes again: a machine that can only crimp folds two adjacent creases as a single motion, reaches strips no simple-fold machine reaches, and is defeated by an odd number of creases for reasons that are pure arithmetic.

And the some-layers row of the first figure has been left unexplained on purpose. It matches the flat-folding row exactly, on every spacing tried, which would mean that allowing a machine to choose costs nothing at all. Whether that survives a wider sweep, and what it would mean if it did, is the next question.