A ring and a line

The vertices nobody checks established that a vertex on the edge of the paper is not an interior vertex with fewer conditions on it. It is a different object: its sectors come in a line rather than in a ring, which makes it exactly a one-dimensional crease pattern, and the solver that decides one had been in the collection for a long time before anybody pointed it at a vertex.

That rung counted the boundary vertices — a hundred and five of them on the printed shelf against ninety-two interior ones — and built the witness pattern showing the gap is not vacuous. What it did not do is hold anything fixed. Every boundary vertex it measured had whatever sectors the pattern happened to give it, and every interior vertex it compared them with had different sectors again, so the comparison was between two populations rather than between two arrangements of one thing.

This is that comparison. Take a vertex — a set of sectors summing to a full turn — and ask its letterings twice. Once with the sectors closing round, and once with the same sectors, in the same order, in a line.

What cutting a ring open costs, before anything is measured #

The comparison has a cost built into it that has to be stated before the numbers, because it runs the other way from the result and it is not small.

A ring cannot be opened anywhere except at a crease. Cutting between two creases would leave the paper in one piece and change nothing; cutting along one turns that crease into the strip’s two loose ends. So a degree-d vertex has d creases and 2^d letterings, and the strip made from it has d−1 creases and 2^(d−1).

The line has half as many letterings to choose from. At degree four that is eight against sixteen; at degree six, thirty-two against sixty-four.

Against that, the line loses a condition outright. Maekawa’s theorem says the mountains and the valleys round a flat-foldable interior vertex differ by exactly two, and it is a winding argument — a statement about a closed circuit. On a line there is no circuit, and the condition does not weaken, it does not exist.

The lemma survives, and partially. The big-little-big condition says a sector strictly smaller than both its neighbours must have its two creases lettered differently, and a sector at either end of a line has one neighbour rather than two, so the condition applies to the interior sectors and not to those.

So it is a smaller alphabet against a milder rule, and which way that comes out is a measurement.

Why the ring’s count has no variance and the line’s does #

Before the table, one feature of it is worth predicting, because it is the sharpest thing in the measurement and it is easy to read past: the ring’s counts are 4.00 and 8.00 exactly, over forty vertices each, while the line’s are 5.58 and 14.67. One column has no spread at all and the other does.

The smallest-sector lemma bites once per sector that is strictly smaller than both its neighbours, so the count depends on how many such sectors there are. On a line of four sectors the two interior ones are adjacent, so at most one of them can be a minimum — the count is eight or four and nothing else.

On a ring of four it looks as though two opposite sectors could both be minima, which would give two. Kawasaki forbids it. The alternating sums are equal, so $a_1 + a_3 = a_2 + a_4$; two opposite minima would need $a_1 < a_2$, $a_1 < a_4$, $a_3 < a_2$ and $a_3 < a_4$, and adding the first and third gives $a_1 + a_3 < a_2 + a_4$, which the equality forbids.

So a generic ring has exactly one strictly smallest sector, always, and its count is a constant. The line has no Kawasaki to enforce that, and its count is a random variable — which is the whole reason one column is round and the other is not.

Reading the line’s mean backwards #

That makes the line’s mean informative rather than merely large, because at degree four it can only be composed of two values.

Eight letterings if no interior sector is a minimum, four if one is. A mean of 5.58 therefore says

$$8p + 4(1-p) = 5.58 \quad\Longrightarrow\quad p = 0.395$$

Just under two in five of these vertices have no interior local minimum at all once the ring is cut open — which is a fact about where the smallest sector happened to land relative to the cut, and is the mechanism behind the whole gain.

At degree six the line has four interior sectors and the count is thirty-two over two to the number of minima. A mean of 14.67 gives an average halving factor of 0.458, a shade under one half — so these vertices carry a little over one interior minimum apiece, against the ring’s guaranteed exactly one.

The line is not merely subject to a milder rule; it is subject to the same rule at a lower rate, because cutting the ring open can put the smallest sector at an end, where the lemma cannot reach it. That happens two times in five, and it is worth two extra letterings every time.

It comes out one way, at every degree, on every population #

Over 234 vertices, drawn from three populations at two degrees, the line never admitted fewer letterings than the ring. Not once. And on the mean it admitted substantially more.

The first column is a check rather than a result. A generic vertex of degree d folds in exactly 2^(d/2) ways — four at degree four, eight at degree six — and that is the count this site published before any of this was measured, arrived at by a completely different route. It comes back exactly.

The last two columns are the result. At degree six with angles cut at random, an eighth of the ring’s letterings fold and nearly half of the line’s do. The absolute counts nearly double and the shares more than triple, because the shares are divided by an alphabet that halved.

The gridded vertex is the strange one again #

The grid rows do something the generic rows do not, and it is the fourth time this site has met the same effect from a new direction.

At degree six on a forty-five degree grid, thirty per cent of the ring’s letterings fold and eighty-seven per cent of the line’s do. That is a vertex where nearly every arrangement of letters is acceptable once the ring is opened, and it is the population with the most coincidences in it.

The mechanism is the one that has come up in the vertex catalogue, in the cost of deciding and in the connectivity of the folding set. Big-little-big only bites where a sector is strictly smaller than both neighbours, and a grid produces ties, and a tie is where the lemma falls silent. Take away Maekawa as well and a gridded fan is very nearly unconstrained.

Which means the vertex a box-pleated design is made of is the one whose behaviour changes most when it arrives at the edge of the sheet — and a box-pleated design is exactly the kind that puts a great many creases out to the paper’s edge.

Half the alphabet is not half the freedom #

The two numbers pull in opposite directions and it is worth saying which one wins and by how much, because “the strip is freer” is a sentence that could mean either.

Take degree six with generic angles. As a ring: 64 letterings available, 8 fold. As a line: 32 available, 14.67 fold on the mean. The absolute number of foldings goes up by 83 per cent while the number of candidates goes down by half.

So the strip is freer in both senses at once, and the halving of the alphabet is more than paid for. The ring’s condition is not a mild restriction that the strip relaxes slightly — Maekawa is the binding constraint at a vertex, and removing it more than compensates for having one crease fewer to letter.

That is worth holding onto when reading what a cut buys. A slit turns interior vertices into boundary ones, and the licence it grants is measured there as a factor on a whole pattern’s lettering count. This is the same licence measured at the vertex itself, and the two numbers are different because a slit does two things at once: it releases the vertices at its ends, and it removes the crease it was cut along from every constraint that mentioned it.

Where the paper still closes round #

There is a boundary vertex where the paper does close round, and it is the case that makes the ring-and-line distinction genuinely about arrangement rather than about quantity.

A slit tip — the point at the end of a cut that removes no paper — has a full turn of paper at it, exactly as an interior vertex does. Walk round it and the paper is continuous the whole way. But the sectors still come in a line, because the cut is a boundary and the walk starts on one side of it and finishes on the other.

So the amount of paper at a point and whether the conditions apply are independent. An interior vertex and a slit tip both have 2π of paper. One is subject to four conditions and the other to one, and the difference is a cut of zero width.

That is the sharpest way to state what this essay is about. The theorems of flat folding are not about how much paper is at a point. They are about whether a walk round the point returns to where it started, and a cut of no width destroys that without removing anything.

How much paper a vertex can have #

The span of paper at a boundary vertex is not fixed, and the range it takes on a real sheet is worth writing down because it bounds everything above.

At a corner of a square sheet the fan spans ninety degrees. Along an edge, a hundred and eighty. At a slit tip, three hundred and sixty. At an interior vertex, three hundred and sixty in a ring.

The span matters for what can be drawn rather than for what folds. The letterings a fan admits depend on the relative positions of its creases, not on the absolute span — a strip is a strip whatever its length — so the corner is not more constrained than the edge in the sense this essay is measuring. What the corner is short of is directions: on a forty-five degree grid, an interior vertex has eight crease directions available, an edge vertex three, and a corner exactly one.

So a corner on a box-pleated design can carry a single crease and nothing else, and that is a fact about the grid rather than about the theorems. It is also why the printed shelf’s corners are almost all bare.

What the shelf’s own boundary vertices do #

The populations above are sampled, and the printed shelf is not. Its boundary vertices are whatever the patterns gave it, and it is worth reading them beside the sampled numbers because they are the fans a reader can actually hold.

There are a hundred and five of them, against ninety-two interior vertices — a fact recorded when the reduction was first made and still the most surprising number in this anchor, because the subject’s entire apparatus is about the smaller group. Twenty-seven carry two creases or more, and the rest carry one.

A fan with one crease folds, always: a strip with one crease is a sheet folded in half, and there is nothing for a condition to refuse. So the interesting population on the shelf is the twenty-seven, and every one of those folds as well. Nothing published here was ever wrong — the gap the reduction opened is real and the shelf happens not to fall into it, which is a fact about the shelf.

That distinction is worth keeping sharp. The witness pattern built for the earlier rung is two patterns differing in one letter at a boundary fan, both of which this site’s checker passes, one of which paper refuses. The gap is not hypothetical. It is simply not somewhere any published figure stands.

The solver that answers both #

Nothing in this essay needed new machinery, which is the part worth recording.

The ring is decided by the exhaustive stacking search this site uses on vertices. The line is decided by the one-dimensional layer solver, which knows nothing about vertices at all and was written for a strip of stamps. Neither knows the other exists, and the comparison is between two independent answers about one set of sectors.

That independence is the reason the generic column’s exact 4.00 and 8.00 are evidence and not bookkeeping. If both sides were computed by one routine with a flag, agreement with a published count would say only that the flag was set correctly.

Why cutting a ring open never costs anything #

The measurement is a mean over vertices and the claim under it is stronger than a mean: on every one of the 234 vertices tried, the line’s count was at least the ring’s. Not “on average”, not “with two exceptions” — never fewer.

It is a proof rather than a tendency, and the proof runs in the direction opposite to the one a reader first reaches for. Do not try to extend a line’s folding to a ring’s; restrict a ring’s folding to a line’s, and count how many restrictions can collide.

Take any lettering of the ring that folds. The paper is flat folded; cut it along one of its creases and nothing moves, so what is left is a fan of the same sectors, in the same order, folded flat. That is a folding of the line. So restriction is defined on every ring folding and lands on a line folding every time — no exceptions, no conditions, and no arithmetic.

Now ask whether two ring foldings can restrict to the same line folding. They would have to agree on every crease but the one that was cut, and differ there. Maekawa says the ring’s mountains and valleys differ by exactly two; flipping a single crease changes that difference by two, so if one of the pair has a difference of two the other has nought or four. Neither is two. At most one letter on the cut crease is available, so no two ring foldings share a restriction.

A map that is defined everywhere and never collides cannot land in a smaller set than it starts from. The line’s count is therefore at least the ring’s, at every vertex, at every degree, for every arrangement of sectors — and the 234 vertices are a check on the machinery rather than the evidence for the claim.

What the proof does not give is the size of the gap. It says the line has at least as many and it says nothing about eighty-three per cent, which is a fact about how many line foldings fail to extend and depends on the sectors. The measurement is what supplies that, and the grid rows are where it is largest — a gridded fan is nearly unconstrained, so almost everything it admits is something the ring refused.

What the boundary is worth to a designer #

Three practical readings, and the third is the one a folder can act on.

Paper at the edge is cheap in letters as well as in area. The corner is worth four times the middle as a statement about how much paper a flap consumes; this is the same boundary being cheap in a second currency. A crease taken out to the sheet’s edge is a crease with more letterings available to it.

A design with its creases running to the rim is a design with room in it. That is the counterpart of the buried-crease rule: a crease with an interior vertex at each end has its letter settled when the pattern is drawn, and a crease that reaches an edge does not.

And the conditions are a property of the topology, not of the paper. Nothing about the fibre, the thickness or the size of the sheet appears anywhere above. What decides whether Maekawa applies is whether a small circle round the point stays on the paper, and that is a question a drawing answers.

Two of the conditions need a whole ring of paper round the point before they can be stated at all. The third does not: the big-little-big lemma asks only about one sector and its two neighbours, which is why it is the condition that follows the pattern out to the rim and survives into a strip.