Walking between two foldings

Every question this collection has asked about the letterings of a vertex has been a question about how many. How many assignments the conditions admit, how many of those fold, how the count grows with degree, how it depends on which sector is smallest. A count is a fact about a set, and a set has no shape. Almost every pattern fails is a statement of that kind too, and so is every sentence here that begins with a fraction.

Here is a different question about the same object. Given one lettering that folds and another that folds, can the first be turned into the second one small change at a time, with every lettering along the way folding too?

That is not a curiosity. It is what a folder at a table does — push one crimp the other way and see whether the model still closes — and it is what a program does when it improves a solution rather than enumerating them. Both start from something that works and try a change. If the foldings are joined up, a local search reaches all of them. If they are not, a search that starts inside one piece never leaves it, and nothing about the piece it is in says the others exist.

The smallest move is two creases #

At an interior vertex the letters are not free to move one at a time. Maekawa requires the number of mountains and the number of valleys to differ by exactly two, so flipping a single crease changes the difference by two and leaves the admissible set immediately. Not sometimes — always, at every vertex of every degree.

So the smallest change the theorems permit is two creases flipped together, and there are two versions of it worth keeping apart.

The first allows any two creases, wherever they sit round the vertex. That is what a program with the whole vertex in front of it can do.

The second allows only two creases that are neighbours round the vertex, which is the change a pair of fingers makes: push on one crimp and the two creases bounding it swap. It is the local move in the sense a folder would recognise.

The generic folding set is a cube #

There is a reason the connectivity comes back positive on every generic vertex, and it is structural rather than a run of luck — the set being walked has a shape, and the shape is the plainest one there is.

At a vertex of degree $2k$, Maekawa admits the assignments split $k+1$ to $k-1$, and the smallest-sector lemma forces the two creases flanking each strictly smallest sector to differ. A generic vertex has $k-1$ such sectors, and each of them therefore contributes one letter of each kind to a pair of adjacent creases. That accounts for $2k-2$ of the creases and $k-1$ of each letter; the two creases left over must both carry the majority letter.

So a folding assignment is fixed by $k$ independent binary choices: which crease of each minimum pair is the odd one, and which letter the two leftover creases carry. That is $2^{k}$ assignments — four at degree four, eight at degree six, sixteen at degree eight — and the set is a $k$-dimensional cube.

Which is why even the local move works #

The cube’s edges are exactly the smallest move available, and they are all local.

Flipping both creases of one minimum pair changes that pair’s choice and nothing else. Those two creases are adjacent, because they flank one sector. Flipping the two leftover creases changes the last coordinate, and they are adjacent too, because the pairs occupy the rest of the ring.

So every generating move of the cube is a neighbouring-pair move — the one a folder’s fingers make on a single crimp. The generic folding set is not merely connected under the local move; it is connected by $k$ moves that each change one coordinate, and any assignment reaches any other in at most $k$ of them.

That is a much stronger statement than sixteen of sixteen, and it explains why the measurement never failed on a generic vertex: it could not have.

It also predicts exactly where the local move should start failing, which is the essay’s second bar. A vertex on a forty-five-degree grid has sectors that are equal rather than strictly ordered, so the lemma is silent at some of them, the folding set is much larger than $2^{k}$ — twenty at degree six against eight, a hundred and twelve at degree eight against sixteen — and it is no longer a cube. The extra assignments are the ones with no coordinate to change, and a move that works by changing one coordinate has nothing to offer them.

So the two bars are not two measurements of one property. The first is a cube being walked and the second is whatever a tie leaves behind, and the difference between them is the smallest-sector lemma going quiet.

Any two creases joins everything #

Over sixteen generic degree-six vertices, sixteen of sixteen are in one piece. Over sixteen at degree eight, sixteen of sixteen. Over vertices drawn on a forty-five-degree grid, where the folding sets are much larger — twenty letterings at degree six, a hundred and twelve at degree eight — sixteen of sixteen again.

Every vertex measured, at every degree the enumeration reaches, on every population tried: the letterings that fold are one connected space under the two-crease move.

That is a claim with no proof behind it here, and the honest statement is exactly that. It has never once failed on a vertex this collection has been able to ask, and no argument is offered for why it should hold in general — which is the same footing the commuting of crimps was left on, and for the same reason: a measurement over every case a machine can reach is evidence and is not a theorem.

Two neighbouring creases does not #

Restrict the move to neighbours and the picture comes apart.

Of sixteen generic degree-six vertices, seven stay in one piece and nine do not — and on all nine the two pieces are exactly a set and its reverse: every lettering in one piece is the other piece’s lettering with every mountain and valley swapped. At degree eight it is sharper: four of sixteen stay in one piece, ten of the rest split into exactly two pieces related by that swap, and two split into four.

Swapping every letter is a real operation and not a bookkeeping trick: it is the model folded from the other side of the paper, which an earlier rung showed every theorem here is blind to. So what the measurement says is that a folder pushing crimps one at a time can reach every folding of the vertex that shows the same face of the paper, and can never reach the mirror family without turning the model over.

And the gridded vertex does not come apart #

The result that makes this rung worth writing is the one that runs the wrong way.

A vertex drawn on a forty-five-degree grid is the hard case everywhere else in this collection. Its sectors coincide, so the big-little-big lemma has nothing to forbid; the crimp reduction loses its forced move; the four conditions stop being sufficient; more letterings pass and more of them need a search. On every measure of difficulty the site has, a grid vertex is worse.

Under the neighbour-only move it is better. Sixteen of sixteen gridded degree-six vertices stay in one piece, against seven of sixteen generic ones; at degree eight, sixteen of sixteen against four.

The mechanism is not mysterious once the two facts are put beside each other. A grid vertex has ties, ties mean the lemma forbids nothing, and what it stops forbidding is precisely the letterings that would have been the gaps in the walk. The set is bigger, and it is bigger in the places that were disconnecting it.

So the coincidences that make a vertex harder to decide make it easier to walk, and the two are the same fact seen from opposite ends: a condition that rules letterings out is a condition that cuts the space up.

What the two pieces are #

A split that produced two arbitrary heaps would be a fact about nothing. This one produces a heap and its mirror image, on nine of the nine generic degree-six vertices that split, and that is a structure worth naming.

Turning a model over exchanges every mountain with every valley. It is the one operation that is guaranteed to take a folding to a folding — the folded object is the same object seen from the other side, and no theorem in the subject can tell the two apart. So the folding set always has that symmetry, and the only question is whether a walk can cross it.

Under the two-crease move it can. Under the neighbour-only move it cannot, and the reason is a parity: at a degree-six vertex the reversal changes all six letters, a neighbour move changes two adjacent ones, and the sequence of neighbour moves that would compose to the reversal has to pass through letterings the lemma refuses.

That gives the result a physical reading. A folder working one crimp at a time is exploring one face of the paper. Every folding they can reach shows the same side outward; the mirror family is a fold away and is not a crimp away, and the fold in question is the one that turns the model over in the hand.

Where the count and the walk disagree #

The two measurements can be put side by side, and they order the populations in opposite directions.

How many letterings fold at each vertex, and how many of sixteen such vertices stay in one piece under the neighbour-only move:

Read down the first column and the grid looks like the difficult case: more letterings pass, so more work to sift. Read down the second and it looks like the easy one: whatever a folder has in hand, the rest is reachable.

Neither column is wrong and neither is the whole answer. What they measure are two different questions about the same set — how large it is, and how well a local method covers it — and a subject that only ever reports the first has no vocabulary for the second.

Which theorem was checked, and how #

Membership in the set is not decided by the move machinery. Whether a lettering folds is decided by an exhaustive stacking search over the vertex — every ordering of the sectors tried against the two non-crossing conditions — which knows nothing about moves, neighbours or connectivity. So the graph is assembled out of verdicts taken one at a time, and its connectivity is a property of those verdicts rather than of the procedure that collected them.

Three checks make the measurement mean what it says.

The one-crease move must join nothing. It is not a move at a vertex, and the graph must say so by finding no edges at all rather than quietly falling back on a larger move. Over a degree-six vertex with eight foldings, the one-crease move leaves eight pieces of one.

The reversal must map pieces to pieces. Swapping every letter takes a folding to a folding, so it permutes the pieces; whether it fixes each piece or exchanges two is the thing being reported, and a run where it did neither would mean the pieces were computed wrongly.

Both populations must be asked the same question. The samplers differ in one thing — whether the sectors are cut at random or drawn from a grid’s small set of sizes — and every other parameter is held: the same degree, the same number of vertices, the same move, the same enumerator.

Why the move a folder makes is the restricted one #

The two moves differ by a fact about hands rather than about theorems, and it is worth saying plainly why the restricted one is the interesting one.

Changing two creases that are neighbours round a vertex is a single physical act: the sector between them is crimped one way instead of the other, and the paper on both sides stays where it was. Changing two creases on opposite sides of the vertex is not one act. It is two, and between them sits a lettering that violates Maekawa — a configuration the paper cannot hold even for a moment, because the layers have nowhere to be.

So a walk under the unrestricted move is a walk through states that exist on paper only at its endpoints. That is perfectly good as a statement about a search over labels, and it is not a statement about folding anything. The neighbour-only walk is the one whose every step is a fold, and it is the one that comes apart.

The reduction that decides a vertex works on exactly the same move, which is why the two results sit together: crimping away a sector is the operation that decides whether a vertex folds, and swapping the crimp is the operation that moves between the letterings that do.

Where the model stops #

Degree four is trivial and says nothing, which is the degree most of the subject is taught with. Four letterings fold, every one differs from every other in two creases or in four, and the graph is complete on both moves. The question only becomes a question at degree six, which is where the enumeration is still exhaustive and the answer has room to be either.

Ten creases is out of reach. Deciding one lettering means enumerating stackings, and the vertex is refused above degree eight rather than sampled — so nothing here is a statement about a vertex of degree ten, and the trend from four to six to eight is three points.

Nothing here is about a sheet. One vertex is being walked, and a vertex is the one place in this subject where the conditions are the whole answer. Whether the letterings of a pattern are joined up is a different question with a different answer, and it is the next rung.

What the picture cannot show #

A graph drawing shows which letterings are joined and cannot show why. The two pieces of a split vertex look like two piles of labels, and what separates them is a fact about which sector is smallest and which creases bound it — geometry that has been discarded by the time the letterings are drawn as dots on a circle.

The layout is also not the object. The dots are placed in a ring because a ring is legible, and their positions carry no meaning at all: two letterings adjacent on the circle are not adjacent in any sense the essay is about. Only the lines mean anything.

The idealisation, named #

The vertex has no thickness and its creases are lines, so a lettering is a pure combinatorial object — six letters, nothing else — and two vertices with the same arrangement of sector sizes have the same folding set. Real paper does not work like that: pushing a crimp the other way on a sheet that has already been folded meets a crease that remembers, and the move that this essay treats as free costs something at the table.

That matters for the reading, not for the result. The claim is about which letterings are reachable in principle by a local change, and a paper crease’s memory makes the walk harder rather than shorter.

The generalisation #

The statement worth carrying is about search rather than about paper.

A set of solutions and a space of solutions are different objects, and only the second one supports a local method. Counting solutions tells nobody whether a hill-climb finds them; that depends on which changes preserve validity, and the answer can invert under a change in the move that looks like a detail. Any two creases joins everything; two neighbouring creases joins a half or a quarter. Nothing about the count predicts that, and the count is what every table in this collection reports.

The second half is the inversion. The instances that are hardest to decide can be the easiest to search, because both difficulties come from the same source — a condition that constrains — and a constraint removed makes deciding harder and moving easier at the same time.

Where the ladder goes next #

The obvious continuation is the sheet. A crease pattern has several vertices, its letterings are constrained at all of them at once, and a move that repairs one vertex disturbs its neighbours — so there is no reason to expect the answer above to survive, and it does not.