Fourteen states are one pile

Where the machine catches up found the weakest machine in this subject complete where nobody expected it. A machine that must fold every layer a fold line crosses — a guillotine brake, and the machine that is stopped by a strip with two creases in it — reaches every folded state of a strip of equal stamps, up to six stamps. At seven it misses twenty-eight of 924 states, and deciding is not making observed that the twenty-eight are fourteen states each with its turned-over twin, one in each of fourteen markings, and that what the fourteen have in common was not established.

A shallow machine pays in states, not folds then ruled out one explanation: the all-layers machine is exactly as fast as any machine on every state it does reach, so the misses are not about sequences running long. What they are about is the piles.

They are one pile. Seen from seven different stamps.

The heavy line in each pile is stamp 0, and following it from panel to panel shows the pile being turned: the stamp at the bottom of one panel is at the top of the next, and every panel is a legal folding of a different marking of the same strip. Before the grouping is argued for, it is worth recalling how the shortfall first appeared, because nothing in its first appearance suggested it was one thing.

The measurement it starts from is the one drawn above. Up to six stamps the all-layers machine produces every folded state there is, and at seven the bar falls short by twenty-eight. Nothing about the shortfall’s size suggests a pattern: it is three per cent of the states, spread across fourteen of the sixty-four markings, and fourteen markings is neither a symmetry class of markings nor a family anybody would name.

A pile as a list #

A folded strip of equal stamps is a pile, and a pile is a list: which stamp is at the bottom, which next, up to the top. Seven stamps numbered 0 to 6 along the strip give a list like 0 6 1 2 3 4 5, read from the bottom. Stamps next to each other along the strip are joined by a crease at one end of the pile — the crease between 0 and 1 at the right, between 1 and 2 at the left, and so on alternately — and a list is a folding exactly when the creases at each end can be drawn as arcs that do not cross.

That is the picture in the first figure, and it is the picture of a meander: points on a line, arcs above for one end and below for the other, no two arcs crossing. The number of such lists for seven stamps is 462, which is the classical stamp-folding count, and every missed state is one of them. The count counts labels is the account of what that count distinguishes and what it does not.

Turning a pile is a symmetry of the foldings #

Take the stamp at the bottom of a pile and put it on top. The list 0 6 1 2 3 4 5 becomes 6 1 2 3 4 5 0. If the first list is a folding, so is the second, and the reason takes three sentences.

The bottom stamp has at most one crease at each end of the pile, and each is an arc from the bottom to some height $h$. Every other arc at that end lies wholly below $h$ or wholly above it, because it cannot cross the arc from the bottom. Moving the bottom stamp to the top turns its arc into one from $h$ to the top, and the arcs that were below $h$ are now outside it and the arcs above $h$ are now inside it — still not crossing anything.

So turning is a map from foldings to foldings, and doing it seven times returns the pile. Reading a pile downward instead of upward is another — it is the same pile turned over. Together they group the 462 foldings of seven stamps into classes.

Every class has exactly twice as many piles as the strip has stamps: 16 piles of four stamps in 2 classes of 8, 50 of five in 5 of 10, 144 of six in 12 of 12, 462 of seven in 33 of 14. No pile is left unchanged by any turn, so no class is smaller. That is why each stamp-folding count is the stamp count times another count — 4, 10, 24, 66 — which is the count of semimeanders, and the classes here are half of those because reading a pile downward pairs them. The identity is classical; the oldest open problem is where these counts enter the subject, and turning a pile is the symmetry that the identity is a shadow of.

The machine respects the symmetry #

The classes are a fact about foldings. What was not obvious is that they are a fact about the machine as well.

Every pile the all-layers machine reaches, turned, is a pile it also reaches, and every pile it misses, turned, is a pile it also misses. Measured on every strip from four to eight stamps, the machine’s reach is a union of whole classes. So the question of which piles it misses is a question about classes, and at seven stamps the answer is one.

The missed class is the pile 0 6 1 2 3 4 5 and its thirteen companions. Read as a folding, stamps 1, 2, 3, 4 and 5 lie one on another in order — an accordion of five stamps. Stamp 0 is joined to stamp 1 by a crease at the right-hand end and lies under the accordion. Stamp 6 is joined to stamp 5, at the top, by a crease at the left-hand end; it wraps down round the whole left side of the accordion and comes to rest between stamp 0 and stamp 1 — inside the fold of the crease that joins them. The last stamp wraps round the accordion from one end and is tucked into the fold that holds the first stamp at the other.

The other thirteen are that arrangement seen from different starting stamps, and each is a folding of a different marking — the markings MVMVVV, VMVMVV, MMVMVV, MVVVMV, VVMVMV, MMMVMV, MVMMMV and their mirror images. Fourteen markings, fourteen states, one pile.

Why a machine that takes everything cannot tuck #

A fold that takes every layer the line crosses cannot choose to fold one flap and leave the flap beside it. To put stamp 6 between stamp 0 and stamp 1, the pile has to open a gap between two stamps that are already stacked and slide a third into it — which is what a hand does when it tucks, and what the machine that may choose does by taking a block that stops partway down the pile.

The all-layers machine can make an accordion, and it can wrap one end of a strip round an accordion. What the missed pile needs is a wrap that ends inside another fold — stamp 6 brought round one end of the pile and pushed into the crease between stamps 0 and 1 at the other — and every sequence of whole-pile folds that makes both folds leaves stamp 6 outside that crease rather than inside it. That is a description of the pile rather than a proof about the machine; the measurement says the nesting is never produced, on any of the fourteen markings, and says nothing shorter about why.

It does explain why seven is where the misses begin. The tuck needs an accordion for the last stamp to wrap round and a fold at its far end to slide into, and with five stamps in the accordion and one at each end, seven is the first strip on which that arrangement exists and the machine fails to make it.

Eight stamps, and nothing new #

At eight stamps there are 1,392 foldings and the machine misses 64 of them. The obvious question is whether eight brings new kinds of miss.

It does not. Take a stamp off either end of the strip — stamp 7, or stamp 0 with the rest renumbered — and every one of the 64 missed piles becomes the missed pile of seven. Conversely, every pile of eight stamps that becomes the missed pile of seven when an end is removed is itself missed. The two sets are the same sixty-four piles, checked one for one rather than by their totals. And no pile the machine reaches reduces to a missed one.

So a miss is inherited. An eighth stamp added to the tucked pile, wherever it is added, leaves a pile the machine still cannot make; added to any other pile of seven, it leaves one the machine can. The sixty-four fall into four classes under turning — two with the new stamp at stamp 6’s end, two with it at stamp 0’s — and those four are the one obstruction of seven with a stamp on.

Two descriptions of the machine’s reach #

Where the machine catches up gave the stamp-folding numbers a second definition up to six stamps: the number of distinct results a sequence of whole-pile folds can produce. That definition failed at seven, and the failure now has a size and a shape.

Up to seven stamps, the piles a guillotine brake can make are all the foldings except one class, and at eight, all the foldings except those containing that class. Whether the pattern continues — whether every miss at nine stamps contains the tucked pile of seven, or whether nine brings an obstruction of its own — is the question the machine’s reach now reduces to, and the enumeration here stops at eight because the layer solver it checks against stops there.

The two outcomes would say very different things. If every miss contains the tucked pile, the all-layers machine is characterised by one forbidden pattern, the way a class of permutations is characterised by a pattern it avoids, and its reach at every length is countable from that one pile. If nine brings a new obstruction, the characterisation is a list that grows, and the machine is harder to describe than its misses at seven and eight suggest.

Why the grouping was the right one #

It is fair to ask why turning a pile should be the operation to group by, rather than the operations that act on markings — swapping every letter, or reading the strip from the other end — which are the ones the count counts labels uses to count objects rather than labelled piles.

The marking symmetries were tried first, and they do not collapse the misses. Swapping letters and reading the strip backward group the fourteen markings into four families — of four, four, four and two — which is what a list of unrelated exceptions looks like. Turning a pile does not act on markings at all: it changes which stamp is at the bottom, so it moves a pile of one marking to a pile of a different marking, and it is the only operation of the ones tried that relates the fourteen markings to one another. That it groups them into exactly one class is the finding; that it could have done so is visible only after the stamps are drawn as a pile rather than listed as letters.

There is also a physical reading. A pile of equal stamps has no preferred bottom — laid on a table, any of its outer stamps could be the one touching the table — and turning is what happens to the list when the same folded object is described from a different outer stamp. The fourteen missed states are one folded object described fourteen ways, which is a much smaller thing for a machine to be unable to make.

What a folder would see #

On a table the missed object is easy to describe and easy to make by hand. Fold the middle five stamps into a tight accordion. The first stamp hangs off one end of it; fold it under. The last stamp hangs off the other end; bring it round that end, under the accordion, and push it into the fold between the first stamp and the accordion. A pair of hands does this in a few seconds, and the tuck at the end is the move every folder makes without naming it.

A brake cannot push anything into a fold. It can only turn a side of the pile over, and everything on that side goes with it. That is the plain-language version of the measurement, and it is also a reminder of why the machine models here are worth keeping apart: the one move the brake lacks is the move a hand finds most natural.

What the classes assume #

The stamps are equal and the pile is flat. Turning is a symmetry only because every stamp covers the same interval; on an unevenly creased strip a stamp at the bottom of the pile is not interchangeable with one at the top, and the uneven strips are where the machine reaches nothing at all.

A pile is a labelled list. Stamps are told apart by their position along the strip, so a pile and its reverse are two lists; that is the convention under which the counts are 462 and 1,392, and under which turning and reversal act without fixing anything.

The machine is the ideal simple fold. It takes every layer crossing a line at a stamp boundary, turns one side over, and never tears; the brake it models is exact, and a real brake with any slack in it is a different machine.

What the measurement does not settle #

It does not prove that turning preserves the machine’s reach. The symmetry of foldings has the three-sentence argument above. The symmetry of the reachable set is measured on every pile from four stamps to eight and has no argument here; a proof would presumably show that a fold sequence producing a pile can be rewritten to produce the turned pile, and nothing like that has been written.

It does not prove the heredity beyond eight. The inheritance from seven to eight is exact and complete, and it is two sizes.

And it describes the missed pile without deriving it. The tuck — the last stamp round the accordion and into the fold that holds the first — is what the pile looks like. That the all-layers machine cannot nest two wraps that way round is the measured fact, and a proof of it would be the first argument in this line that explains a miss rather than finding one.

Still open: whether nine stamps has a pile of its own #

The obvious next measurement is nine stamps, and it needs a layer solver that can enumerate nine segments; the one used here refuses above eight by design. The count to aim at is known in advance: nine equal stamps have 4,536 foldings in 252 classes of eighteen, so the answer is a number of classes, and the prediction from heredity is that the missed ones are exactly the classes that reduce to the tucked pile when an end stamp is removed.

Sideways from here, the tucked pile is a candidate for something this subject rarely has: a single forbidden configuration for a folding machine. The flat-folding theorems forbid local patterns at a vertex; a machine forbidding a global pattern of wraps would be a different kind of result, and the natural place to test it is the crimping machine, whose reach on equal stamps could be grouped into the same classes and whose misses could be checked for a pile of their own.

The habit worth carrying is about exceptions that come in numbers. When a count of failures is a multiple of the size of something, look for the symmetry that makes it one. Fourteen misses on a strip of seven stamps is twice seven, and twice seven is exactly the size of a class under turning and reversal; the count was a single pile announcing itself, and it took grouping the piles to hear it.