The edge was not what made it hard

A sheet’s shape is usually a design decision — a square because that is what paper comes as, a rectangle because a proportion is wanted, a shape chosen to suit a base. An earlier essay in this thread argued that it is also the thing that decides how hard a crease pattern is to reason about, and built a careful case for it: five families of pattern all cost one search step per panel, one family does not, and four candidate explanations for the exception are each killed by one of the five.

The case was good and the conclusion was wrong in two places. The exception is not off the line in the direction claimed, and the rim is not doing what the account says.

What was measured then, and what is measured now #

The earlier measurement was taken while the search still chose which letter to try first by a coin. On the tessellation patches that choice was worth three orders of magnitude — one patch ran from eighty-six steps to past fifteen thousand depending on nothing but the seed — so what the patches showed was an enormous variance and no clear level.

Variance is what “the one family that does not” was about. The five families on the line were flat and repeatable; the patches were not. That is a real difference and it is a difference in the instrument’s behaviour on those patterns rather than in the patterns’ cost.

The coin was removed. Under a fixed letter order every pattern gives the same number every time, and the patches’ number can be read.

It is between a half and two-thirds of a step per panel.

Below the line #

The five published patches cost 0.53, 0.47, 0.51, 0.52 and 0.51 steps per panel. Sixteen more, over four tilings at four sizes, cost between 0.52 and 0.67 and never rise with size.

The grid, the leaf, the Miura and the crumples cost 1.00.

So the tessellation patches are the exception, and they are the cheapest object in the collection per panel of paper. Not the hardest — the least work, per unit of pattern, of anything here.

That reverses the sentence the earlier essay was written to support. It does not reverse the observation underneath it, which was that these patterns behave differently from the others; they do, and the difference is a factor of two in the right direction.

It is worth being fair about how the earlier reading came about, because it was not careless. Under the coin the patches’ worst case was fifteen thousand steps and their best was eighty-six; the grid’s was two hundred and fifty-six either way. Any comparison of worst cases put the patches far above the line, and a family whose cost is unpredictable is reasonably described as the hard one.

What changed is that the unpredictability turned out to belong to the search rather than to the paper, and once it was gone the level underneath it could be seen for the first time. This essay is the second half of that discovery: the first half removed the variance, and the second reads what is left.

What the rim actually contributes #

The mechanism the earlier essay proposed is half right and the half that is wrong is the important half.

The proposal: a vertex near a rim has creases running off the paper, so its conditions constrain less and the propagation stalls there; on a patch most of the pattern is edge, so the region where propagation is weak is most of the sheet, and that is where the search’s work is.

The first two clauses are true. What does not follow is that weak propagation means work. A crease nothing constrains can be given either letter and neither choice can ever be wrong, so a decision about it costs one step and is never revisited. The rim is not where the search struggles; it is where the search cannot fail.

The controlled comparison settles it. Take one rectangle of a twist tessellation and read it two ways — as a patch with an edge, and with its opposite sides joined so that no crease is divided. Same drawing, same vertices, same conditions.

Cut out of the plane, at four periods: forty-eight steps. Joined up: fifty-six thousand seven hundred and seventy-two.

The rim as slack #

The right account is short and it is the opposite of the old one.

A twist tessellation is periodic, so a crease running off the right-hand side of a rectangle is the crease running on at the left. On the pattern those are one crease with one letter, and whatever the conditions at one end demand, the conditions at the other end must agree with — across the whole sheet, with nothing local able to see both.

Cut the sheet and the requirement vanishes. Two ends, two creases, two letters, no reconciliation. Four such creases per period of edge, and each was a constraint spanning the pattern.

So a cut converts long-range constraints into local ones, which is exactly the transformation that turns a search into a reading. The rim’s underdetermined creases are the trace of constraints that have been removed, and removing constraints is what makes the problem easy.

The mechanism, checked against a prediction #

An account that merely reverses a sign is not much better than the one it replaces, so it is worth noting that this one makes a prediction the old one does not, and that the prediction holds.

If the rim’s contribution is removing long-range constraints, then how many it removes should matter little once it removes them all, and where it removes them should not matter at all. Every cut of a rectangle severs the creases crossing each side once per period, whatever position it is in.

If instead the rim’s contribution is creating badly-constrained regions where propagation stalls, then which creases it happens to cut ought to matter a good deal, since different cuts leave different half-panels and different arrangements of weakly-determined vertices.

Sliding the same rectangle across a whole period gives twelve patches with different panel counts, different crease counts and different rims. Their costs run from twenty-five steps to thirty-three. A factor of one and a third, against the factor of more than a thousand for whether the cut exists — which is the prediction the new account makes and the old one does not.

Why the four refutations still stand #

The earlier essay killed size, disorder, irregularity and construction, and every one of those killings is still good.

Size is killed by the grid, which reaches two hundred and fifty-six panels and stays at exactly 1.00. Disorder is killed by the crumples. Irregularity is killed by the quadrilateral meshes. The construction is killed by the corrugations.

What is left standing is still the rim. The correction is entirely about which way it points: the four alternatives do not explain why the patches differ, the rim does, and what the rim does is make them cheaper.

That is a satisfying outcome for a piece of reasoning by elimination. The elimination was sound and the sign was assumed, and the sign was the one thing the available comparisons could not fix.

The second correction #

There is a second thing the earlier essay attributes to the rim which belongs elsewhere, and it is worth separating because otherwise this reads as though the rim explains the whole difference.

The patches cost half the line and the other families cost the line. Half of that gap is not about the rim at all: it is about how much the conditions at a vertex decide. A twist polygon’s corner admits four labellings where a grid’s, a leaf’s, a Miura’s and a crumple’s admit eight, because the lemma about a strictly smallest sector applies at a twist corner and not at theirs.

So the patches are cheap for two independent reasons — their vertices are the most decided in the collection, and their rims have removed every long-range constraint — and the first of those is measurable on its own across nine families.

The rim’s contribution is the larger of the two and the harder to see, because it can only be measured by taking it away.

What a designer should take from it now #

The practical advice changes and it becomes simpler.

The earlier advice was, in effect, that a pattern cut out of a larger design is harder to reason about than one drawn to fill its sheet. That is not so. A cut makes a pattern easier, and a designer who takes a tessellation and cuts a square out of it has produced a pattern whose letters are less constrained and more quickly found than the tessellation’s own.

The caution that replaces it is about claims rather than about work. A property established on a patch is a property of a pattern-with-a-rim, and the rim is not a neutral frame around the interesting part. It has removed constraints, and a statement of the form this tessellation is easy to letter proved on a square is a statement about the square.

That is a narrower and more useful thing to know than the old advice, and it is why the position of the cut turns out not to matter while the existence of one does.

What a sheet’s shape does decide #

The thread this essay belongs to is about sheet shape, and it is worth restating what shape is now understood to control, because the list is different from the one it started with.

Where the paper is worth most. A corner of a square is worth four times the middle for a designer packing flaps, and that is entirely about geometry rather than about letters.

How much cheap paper there is. A hole is cheap paper and a notch is not a hole: the shape decides which regions of the sheet can be spent freely.

Which constraints survive. This essay. A shape cut out of a periodic pattern severs the constraints crossing its boundary, and that is a fact about the pattern rather than about the paper — a corrugation drawn to fill the same shape severs nothing, because there was nothing outside to sever.

The first two are about what a sheet can hold and the third is about what a pattern on it can be asked. Only the third depends on where the pattern came from, and it is the one that had been read backwards.

What is unchanged #

Most of the earlier essay, and it is worth saying so.

Five families do cost one step per panel and do not backtrack. The propagation on a corrugation is decisive. A patch is mostly edge. The vertices near a rim are asked less. Each of those measurements stands exactly as recorded, and the ones this essay revises are the two sentences that put them together.

The revision needed an object that did not exist when the essay was written, and the essay says so in its own last section: the comparison it wanted was one tessellation built three ways, and the third did not exist because every piece of machinery here assumes a disc of paper with a rim.

The habit this is an instance of #

Two corrections have now been made to the reading of these patterns, and they have the same shape, which is worth naming.

The first found that a heavy tail in the patches’ cost was manufactured by the search’s own randomisation. The second finds that the level underneath that tail is below the line rather than above it. Both were reachable only by holding something still that had never been held still — a coin in one case, a rim in the other — and in both the collection’s account was a correct reading of a measurement that included the instrument.

The habit that produces this is not carelessness; it is that a measurement over a family is much easier to make than a measurement of one variable within a family. Five families with one property each is what the collection could build; one family with one property varied is what it could not, until the object with no rim existed.

So the useful lesson is about which comparisons to want rather than about which conclusions to distrust. A property established by eliminating alternatives across families is a property whose sign has not been measured, and the sign is the part that turned out to be wrong both times.

Which theorem was checked, and how #

The two readings of the rectangle come out of one set of segments, computed once from the construction before anything is cut, so the comparison is between two treatments of one drawing rather than between two drawings.

The vertex counts are checked to be identical on both sides of every comparison, and the figure refuses to draw if they are not. That is the claim the whole correction turns on: if the cut removed vertices as well as constraints, the two objects would differ in more than one thing and nothing could be attributed.

The costs are measured under a fixed letter order on both sides, so neither number is a sample. And the patches’ level is measured at sixteen combinations of tiling and size rather than at the five published patches, so that “between a half and two-thirds” is a range over a population rather than a spread over five examples.

The claim, restated #

For a reader who wants the whole correction in four lines:

A tessellation patch costs about half a step per panel, at every tiling and size measured, which is below every other family here and not above.

The rim is why, and it makes the patch cheap rather than dear: it severs the constraints that ran from one period of the pattern to the next, four per period of edge, and each of those was a constraint spanning the sheet.

The patch’s vertices are also the most decided here, admitting four labellings rather than eight, which accounts for the other part of the gap.

And the earlier account’s four eliminations stand. Size, disorder, irregularity and construction are still refuted by four families; only the sign of what remains has changed.

What the picture cannot show #

A chart of nodes per panel does not show what a node is, and the whole correction depends on it. A node is a decision the propagation did not make, and a decision that is never withdrawn costs exactly one — so a chart of costs is a chart of how many free choices each pattern has, and the free choices are invisible in every drawing of a crease pattern.

Nor does anything here show the earlier essay being wrong in the act. Its measurements are reproduced; only the sentence joining them is replaced, and a picture of a replaced sentence is not a picture.

And nothing here settles what happens to a sheet cut in a shape that is not a rectangle aligned with the pattern. Every patch in this collection is such a rectangle, and a sheet cut at an angle, or with a notch, or with a hole in it, severs a different set of constraints in a different arrangement. The account predicts that it would still be cheaper than the pattern and says nothing about how much, and no such patch has been searched.

Nor does anything here say what a sheet would have to be for the joined object to be foldable by a person. It is not a piece of paper and it is not a proposal for one; it is the pattern, written down without the finite square that every previous description of it carried. A reader who wants to hold one of these tessellations still holds a patch, and every number a patch gives is still the number that patch gives.