A machine that can only crimp

The three machines of the last two rungs differ only in how many layers they hold. They share an atom: the simple fold, which turns everything on one side of a line through 180°. Holding a taxonomy fixed and varying one parameter is a good way to compare things, and it hides the possibility that the parameter is not the interesting one.

Here is a machine with a different atom, and it does not fit anywhere on that scale.

The move #

A crimp folds two adjacent creases in one motion. The segment between them turns back on itself and comes to rest between the two segments flanking it, the way a pleat does — and it is a pleat, which is why crimping is what a pleating machine does and what a person’s fingers do when a diagram says “crimp”.

The move has two conditions and both are geometric.

The creases must turn opposite ways. One mountain and one valley. Two creases of the same letter do not fold the paper back and forth; they fold it twice in the same direction, which is a different shape and not a crimp.

The middle segment may not be longer than either neighbour. If it is, it sticks out past the paper it is folding into, and the crimp is not a flat move. This is the big-little-big lemma arriving in one dimension and as a condition on a machine rather than on a vertex — the same inequality, doing the same job, in a different costume.

Both conditions look at three consecutive segments and nothing else. That is what makes the crimper analysable: its entire behaviour reduces to a rewriting on a list of numbers and letters, with no geometry to maintain and no pile to track.

The rewriting #

After a crimp the three segments become one. The middle has folded back, so it covers part of each neighbour, and what remains on the line runs from the start of the first to the end of the third: outer minus middle plus outer. The two creases are gone and the letters on either side are untouched.

That is the whole machine. A strip is a list of lengths and a string of letters, a crimp is a local rewrite, and the question “can this machine fold this strip” is the question “can this list be rewritten to a single number”.

The change of register is worth noticing. Every other machine on this ladder needs a simulator, because its state is a pile of paper with an order on it and the legality of a move depends on what is underneath what. The crimper needs no state beyond the list it started with. That is not because crimping is simpler to perform — it is a harder motion than a fold — but because its legality condition never reaches past three consecutive segments, so nothing has to be remembered.

There is no pile of layers in that description, and its absence is not an omission. A crimper’s moves never need to know the stacking, because a crimp always nests the middle segment inside the fold it makes, and a nested pleat cannot interleave with anything. The layer order comes out consistent by construction, which is why the crimper is the one machine here whose analysis is arithmetic rather than simulation.

Two creases at a time, and what that forbids #

Every crimp removes exactly two creases. A strip with an odd number of creases can therefore never be crimped away entirely: however the rewrites go, the count falls by two each time and an odd number never reaches zero.

This is not an argument about difficulty. It is a parity obstruction, the same shape of argument as the one that says a knight cannot reach a square of the wrong colour in an even number of moves, and it settles the question outright for half of all strips.

The sweep confirms it, and the confirmation is worth having, because a parity argument is exactly the kind of reasoning that quietly assumes the machine has no other move:

Six hundred and eighteen strips with an odd number of creases, and not one of them crimped. Meanwhile every even count has some that do, and the fraction that do is small — about a sixth at two creases and under a twentieth at six — because the length condition bites hard once there are several segments to satisfy it.

The odd rows are the ones worth pausing on, because every strip in them folds flat. A three-crease strip has eight assignments and on an even spacing all eight fold; the crimper reaches none of them. This is the widest gap between existence and reachability anywhere in the ladder, and it comes from counting rather than from geometry.

A machine whose primitive consumes two creases cannot fold an odd pattern, and no amount of strength, patience or lookahead changes it. That is a much stronger kind of negative result than the ones in the previous two rungs, which were about particular spacings. This one is about half of everything.

One ordinary fold removes the obstruction #

The parity result is the strongest negative in this ladder and it is also the most easily repaired, which is worth saying in the same breath because the repair says what kind of obstruction it is.

A crimp consumes two creases. A simple fold consumes one. So a machine allowed one simple fold and then any number of crimps can clear a strip of any crease count: fold once to make the count even, crimp the rest away.

That is not a loophole. It is a description of what a pleating line actually does — a lead-in fold that takes the web down to the right parity and register, and then a sequence of pleats. The parity obstruction is real, it is why the lead-in exists, and it is removed by the cheapest possible addition to the toolkit.

Which makes it a statement about the primitive’s arithmetic #

The general form is short and it covers machines nobody here has built.

A machine’s moves each consume some number of creases, and a sequence of moves clears a strip only if the crease count can be written as a sum of those numbers. So the crease counts a machine can clear are the sums generated by its moves’ consumptions — a machine whose only move consumes two reaches the even numbers and nothing else; a machine with moves consuming two and three reaches everything from two upward; a machine with a move consuming one reaches everything.

The crimper is the smallest interesting case: one move, consuming two, generating the evens. Half of all strips are outside it before any geometry is consulted, and the arithmetic that says so does not look at a single length or letter.

That also says where to look for other obstructions of this kind. A machine whose moves all consume an even number of creases has the same problem however many moves it has, and a machine with a single odd-consuming move never has it. The property that matters is not how powerful the moves are but whether the numbers they consume share a factor — which is a question about the toolkit’s arithmetic rather than about paper, and which a designer choosing between a pleater and a brake could answer before building either.

Neither machine is weaker #

Now put the crimper beside the all-layers machine, and the comparison refuses to come out as a chain.

Take a strip of 0.40, 0.10 and 0.50 with a mountain then a valley. That is the witness from the first rung, which no all-layers machine folds. The crimper folds it in one move: the letters differ, the middle is shortest, and the rewrite takes three segments to one.

Now take a strip of 0.25, 0.25 and 0.50 with two mountains. The all-layers machine folds it. The crimper cannot start, because the two creases turn the same way and a crimp needs them to disagree.

So each machine reaches something the other does not. There is no sense in which one is a restriction of the other, and the word “weaker” — which the previous rung used freely — turns out to have been available only because those three models genuinely were nested by their layer rule.

That matters beyond the bookkeeping. When a manufacturing process is described as “restricted”, the useful question is not how restricted but restricted in which direction, because two processes can each do things the other cannot and the pattern that suits one may be unmakeable by the other. A pleating line and a brake are not a fast machine and a slow machine. They are different machines.

How much a crimper reaches #

The incomparability says the two machines are different. It does not say how much either one gets, and the counts are worth having because they are not close.

On an evenly creased four-crease strip, sixteen assignments fold flat, all sixteen are reachable by an all-layers machine, and six are crimpable. The six are the ones whose letters alternate somewhere: a crimp needs a mountain beside a valley, and an assignment with long runs of one letter offers nowhere to start.

Move to the spacing with a short segment in the middle and the two bars swap character. The all-layers machine drops to nothing while the crimper keeps four, because a short middle segment is precisely what a crimp wants and precisely what buries the next crease for a machine that must hold everything.

That inversion is the incomparability made quantitative. The two machines are not merely different on a couple of contrived witnesses; they prefer opposite spacings, and the preference runs the whole way through the census.

Where the length condition comes from #

It is worth being precise about the middle-segment rule, because it is easy to state as a fact about crimps and it is really a fact about paper.

When the middle segment folds back, its far end travels to a point inside the neighbouring segment. If the middle is longer than that neighbour, the far end travels past the neighbour’s other end — off the paper it was supposed to nest inside — and the fold is no longer flat, because the overhanging piece has nothing beneath it to lie against and the strip is left with a step.

The condition is therefore middle ≤ both neighbours, with equality allowed. Equality is the interesting boundary: at exactly equal lengths the folded segment lands flush with the end of its neighbour, which is the case a folder recognises as a clean pleat and which is why evenly spaced strips crimp so readily.

What the machine cannot see #

Two limits, and the second is the one that keeps this rung honest.

The crimper has no memory. It looks at three consecutive segments, and a rewriting system that only ever looks at three consecutive anything cannot express a constraint between the first crease and the last. In one dimension that turns out not to cost it anything, because the flat-folding conditions on a strip are themselves local after the folding is taken into account. In two dimensions the corresponding statement is false, and it is false in exactly the way that makes flat-foldability NP-hard: the difficulty is a constraint between distant parts of the sheet.

The order of crimps is not free. The figure with the uneven spacing shows it: merging three segments into one changes the lengths, so a crimp that was legal can stop being legal and one that was blocked can open up. The implementation here searches over the orders with backtracking, which is the definition rather than an algorithm anybody would run; whether a greedy rule suffices — always crimp the shortest available middle, say — is a real question that this repository has not settled, and the code carries all three strategies precisely so that the difference can be measured rather than assumed.

That second limit is a genuine gap in what is being claimed. The counts in this essay come from the backtracking search, so they are counts of what a crimper can do given a perfect choice of order. A crimper with a fixed rule would reach fewer, and how many fewer is unmeasured.

The same strip, three atoms #

It helps to see one pattern folded three ways, because the sequences look nothing alike even where all three succeed.

The simple-fold machines take four moves and the crimper takes two, on the same strip, arriving at the same pile. That is not an efficiency comparison — a crimp is a harder motion to make than a fold, and a machine that performs one is a more complicated machine — but it does show what “different atom” means concretely. The crimper’s moves are coarser, so it finishes sooner where it finishes at all, and it fails outright where the coarseness does not fit.

The reason an accordion is the pattern everyone folds first is that it is reachable by every machine in this ladder except the crimper on odd counts. It is the intersection of what everything can do, which is a good definition of “the obvious pattern”.

The idealisation, named #

Everything here treats a pleat as taking no thickness. The middle segment folds back and lies flat against its neighbours, three layers occupying the space of one.

Real pleating machines are built around exactly the fact that this is false. The sheet has a thickness, and a pleat’s inner fold has to travel further than its outer one, which is why pleated fabric and pleated foil are made with a taper that the geometry says is unnecessary. The length condition of this essay — middle no longer than either neighbour — becomes, in a real machine, middle shorter by an allowance that depends on how many layers have accumulated.

That does not change any negative result. A machine with a stricter length condition reaches fewer patterns, so the parity obstruction and the incomparability both survive. It does change the positive ones: some of the strips counted as crimp-foldable here are crimp-foldable only for paper of no thickness.

Who found this, and when #

Crimping as a named primitive is old in the craft and recent in the mathematics. The pleat is in every tradition that folds anything; the observation that a machine restricted to it defines its own class of foldable patterns belongs to the same body of work as the simple-fold models, and the length condition is the one-dimensional shadow of the big-little-big lemma, which is usually credited to the systematic study of single-vertex flat-foldability rather than to any one paper.

What is new here is small and specific: the parity obstruction stated as a fact about the machine rather than about the patterns, and measured across 618 odd-crease strips; and the pair of two-crease witnesses that make the crimper and the all-layers folder incomparable. Both are the kind of thing that is obvious once written and that nobody writes, because the interesting machines are the two-dimensional ones.

Where the ladder goes next #

Three machines that fold one line at a time, one that folds two, and none of them reaching everything that folds flat. That invites the question the whole ladder has been circling: is there a machine that loses nothing?

There is, and it is one of the three already met. A machine allowed to choose which layers it takes reached every flat folding of every strip in the sweep — 117 spacings and not one exception — which would put all of the loss in this ladder down to being forced rather than to folding one line at a time. That is a strong claim, it is measured rather than proved, and the next rung is about what it would mean and how far it can be trusted.