Which vertices are the random ones

Hardness is about the worst one made half of this point from the other end. A hardness result is about the worst instance a reduction can construct, the instances a reduction constructs are engineered rather than typical, and so a worst case says nothing about the case actually in hand.

This is the same observation turned on the site itself.

Every sentence here of the form over 373 random degree-four vertices is a statement about a distribution, and no essay has ever said which one. There is no canonical way to pick a crease pattern at random. This repository builds a vertex by cutting half a turn into random pieces twice and interleaving them, which is defensible and is not the only choice, and a designer working on a grid meets an entirely different population. Almost every pattern fails is a sentence about a distribution too, and so is every statement on this site with the word almost in it.

Four populations, each written down on purpose #

Cut twice at random. A random composition of half a turn into n / 2 parts, twice, interleaved — so Kawasaki holds by construction and nothing else is arranged. This is the population every census on this site has used.

Whole multiples of 45°. The same construction restricted to a grid: every sector a whole multiple of forty-five degrees. This is the population a box-pleated design contains, and it is not a rounded version of the first — it is a different set of vertices.

Whole multiples of 30°. The same for a triangular grid, which admits more sector sizes and therefore more vertices.

A named vertex, jittered. One of the vertices this site names — a preliminary base, a waterbomb — with every sector moved by up to a quarter of itself and Kawasaki repaired. This is what “a working pattern nudged until it fails” means, and an earlier essay used exactly that operation.

All four produce developable, Kawasaki-exact vertices. Every one of them is a legitimate answer to “give me a random vertex”. They are not the same population and they do not contain the same vertices.

They do not even contain the same kinds of vertex #

The cheapest way to see that the populations are genuinely different is to count how many arrangements of smallest sectors each reaches — the combinatorial invariant that decides how many letterings a vertex admits.

At degree six: cutting at random reaches thirteen arrangements in sixty vertices. The thirty-degree grid reaches thirteen. Jittering a named vertex reaches four. And the forty-five-degree grid reaches one.

One. Every degree-six vertex a forty-five-degree grid admits has the same arrangement of smallest sectors, so every one of them admits the same number of letterings, so the whole apparatus of conditions cannot tell any of them apart.

A census over that population and a census over the random one are not two estimates of one quantity with different error bars. They are measurements of different things.

The answers, and how far apart they are #

At degree six, per vertex:

Letterings that pass the conditions, letterings that fold, the share of vertices carrying a gap between the two, and the share of letterings whose decision branches:

The number of admissible letterings differs by a factor of 3.75 between the extremes. The number that fold differs by 2.4. The share of vertices carrying a gap between the two runs from nothing to everything. The share of letterings whose decision requires a search runs from nothing to sixty-nine per cent.

Those are not margins. A reader given the first row and a reader given the second have been told different things about the subject.

And it reaches a sentence this site has published #

The uncomfortable case is at degree four.

Crimp it away and ask again states that at a vertex of degree four the four conditions are the whole answer: an assignment passes them exactly when it folds. That is measured, it is checked in the site’s own gate, and on the population it was measured over it is exactly right — over degree-four vertices cut at random, no lettering passes and fails to fold, on every trial.

On the forty-five-degree grid, 42 per cent of degree-four vertices carry at least one. On the thirty-degree grid, 20 per cent.

The mechanism is not mysterious and the site has already published it. Where the lemma says nothing found that a degree-four vertex whose two smallest sectors are equal admits eight letterings under the conditions and folds in six. A tie is an equality between continuous quantities, so a random vertex never has one, and a vertex drawn on a grid is made of them.

So the sentence “at degree four the conditions are the whole answer” is true of a population that contains no ties and false of the population origami is drawn from. The earlier rung was right about its measurement and general about its wording.

Why the grid populations are the extreme ones #

The two grid rows are the ones furthest from the free sampler in every column, and the mechanism is a single fact repeated.

A grid draws its sectors from a small set: at forty-five degrees, from {45°, 90°, 135°}; at thirty, from six values. A degree-six vertex has six sectors, so at forty-five degrees six values are being drawn from three, and by the pigeonhole principle at least two of them coincide. Every degree-six vertex on a forty-five-degree grid has a tie, necessarily.

A tie is where the big-little-big lemma falls silent, so it constrains nothing, so more letterings pass. It is also where the crimp reduction loses its forced move, so more decisions need a search. And it is where the conditions stop being sufficient, so more vertices carry a gap.

All four columns of the grid rows are consequences of one arithmetic fact about how many sector sizes the grid offers. That is why the thirty-degree grid sits between the other two: six values rather than three, so ties are common rather than certain.

The pigeonhole has a threshold #

The argument that every degree-six vertex on a forty-five-degree grid must have a tie generalises, and generalising it turns the ordering of the four populations into arithmetic.

A grid of step $s$ offers the sector sizes that are whole multiples of $s$ strictly between nothing and a straight angle — three of them at forty-five degrees, five at thirty. Call that count $v$. A vertex of degree $d$ draws $d$ sectors from those $v$ values, so a repeat is unavoidable once $d > v$.

At forty-five degrees the threshold is three, so ties are certain from degree four upward. At thirty degrees it is five, so ties are certain from degree six upward. A fifteen-degree grid would offer eleven values and not force one until degree twelve.

And the free sampler has $v$ infinite, so it never forces a tie and never produces one, since a coincidence between two continuous quantities has probability zero.

Which predicts the ordering before any counting #

That threshold reproduces the table’s shape without measuring anything.

At degree six, the forty-five-degree grid is three past its threshold and the thirty-degree grid is exactly at it — so ties are certain on both, though the forty-five-degree grid has more of them per vertex. Both rows are far from the free sampler and the forty-five is further, which is what the table says.

At degree four, the forty-five-degree grid is one past its threshold and the thirty-degree grid is two short of it. So ties are compulsory on one and merely possible on the other, and the measured shares of vertices carrying a gap — 42 per cent against 20 — fall in exactly that order.

The whole disagreement between the populations is one inequality, $d$ against $v$. Everything the essay attributes to “how many sector sizes the grid offers” is that comparison, and the four rows are four positions relative to it.

It also says what a fifth population would do. A design on a fifteen-degree grid — finer than box pleating and coarser than free angles — would behave like the free sampler at degree four and six, and only start to diverge at degree twelve, which is past anything a folder draws. So the disagreement this essay is about is a property of coarse grids specifically, and it disappears as the grid refines, well before the grid becomes a continuum.

Which theorem was checked, and how #

The comparison would be worthless if the populations were one population with different labels, so the first check is that they are not.

A grid sampler emits only sectors on its grid, checked sector by sector: no angle it produces is off a whole multiple of its step. And the free sampler emits none of them: over forty vertices, not one had every sector on the forty-five-degree grid. Between them those two say the populations are disjoint rather than nested.

Each row is then an exhaustive count over the vertices the sampler actually produced, and the count of those is reported with every row — a sampler that rejects much of what it makes is describing a smaller population than it was asked for, and hiding that would be the first way to get this wrong.

And there is a population with no randomness in it at all for comparison: the complete catalogue of vertices a forty-five-degree grid admits, six of them, each counted once. It gives an answer of its own again — thirty-two letterings admitted per vertex, twenty-eight folding, half the vertices carrying a gap — which is a fifth answer to the same three questions.

What a reader should do with a number from this site #

The rung is self-critical and it should end with something usable rather than with a caution, so here is the rule it implies.

A number about vertices at random is a number about generic angles. It describes what happens when nothing coincides, and it is the right number for a question about what is true almost always.

A number about a printed pattern is a number about that pattern. The site prints eight and censuses them individually; those numbers are not estimates of anything and they are the ones closest to a reader’s own paper.

A number about a grid catalogue is a complete count. Six vertices at forty-five degrees, thirty at thirty: those are enumerations of finite sets and there is no sampling in them at all.

The three kinds appear throughout this site and are not always distinguished in the prose. Where a sentence says almost every, it is the first kind. Where it names a pattern, it is the second. Where it says the catalogue, it is the third — and the third is the only one that is both general and exact.

Where the model stops #

Nothing here is a claim about sampling. There is no estimator, no error bar, no confidence statement and no distributional assumption anywhere in this essay. Every number is an exhaustive count over a stated finite population of crease patterns, and the argument is a comparison between populations that were each written down on purpose. The word sample is not the right word for any of it, which is why it does not appear.

The four populations are also four, and there are more. A population of vertices taken from published designs would be different again and is the one a reader might most want; assembling it means reading crease patterns out of other people’s work, which this site does not do.

And the essay does not say which population is right. There is no such thing. What there is, is a requirement to say which one a number came from, and that is a requirement about how results are reported rather than about how they are produced.

What the picture cannot show #

A table of four rows cannot show that the rows are measurements of different objects rather than different measurements of one. A reader looks at four numbers under one heading and reads them as estimates that disagree, which is exactly the wrong reading — there is nothing they are estimates of.

Nothing shows a population. Each row stands for sixty vertices and the figure has room for none of them, so the object being compared is invisible in every figure of this essay. That is unavoidable and it is the reason the sampler descriptions are given in words rather than drawn.

The jittered population, which is the surprising one #

Three of the four rows are explicable and one is not, and the odd one out deserves its own paragraph because it is the population an experimenter is most likely to build without thinking.

Jittering a named vertex gives eight admissible letterings per vertex, eight that fold, no gaps anywhere and no branching. It is the tamest population of the four, tamer even than cutting at random — and it is built from a vertex that is itself full of ties.

The reason is the repair. Kawasaki does not survive a random jitter, so it has to be restored, and restoring it moves the sectors off each other: whatever ties the named vertex had are destroyed by the jitter and not recreated by the repair. So a jittered preliminary base is a generic vertex that happens to be near a special one, and it behaves like a generic vertex in every column.

That is a warning about a common experimental move. “Take a working example and perturb it” sounds like a way of exploring the neighbourhood of a real case, and for this subject it is a way of leaving the interesting set immediately — because the interesting set is defined by coincidences, and a perturbation is precisely an operation that removes them.

The generalisation #

The useful statement is about what a measured “typical” case is, and it applies wherever a subject has instances.

An average is taken over a distribution. If the distribution is not stated, the average is not a fact about the subject — it is a fact about the procedure that produced the instances, and two procedures that are both reasonable can give answers that differ by a factor of four.

That is worth holding beside the standard complaint about worst cases. A worst case is a statement about a maximum, which needs no distribution and is therefore honest and pessimistic. A typical case is a statement about an average, which needs a distribution and is therefore only as meaningful as the distribution is declared. Neither is the useful one on its own; what is useful is knowing which of the two a number is.

The version of that for this site is a rule about wording rather than about method. A measurement over random vertices is a measurement over random vertices, and it is a claim about the vertices origami actually contains only if somebody has checked. On the one occasion this site did check, at degree four, the two answers were different.

Who found it, and when #

The distinction between worst-case and average-case behaviour is old and belongs to the study of algorithms, where the difficulty of specifying an input distribution is well understood and much discussed. Nothing about the idea is new here.

What is new is applying it to this subject’s own numbers, and the reason it has not been done is the reason such things usually are not: the site that produces the numbers is the site that would have to check them, and a census that undermines an earlier census is not what a census is usually for. The degree-four case is the one that makes it worth doing anyway — a published sentence turns out to be true of a population and not of the paper.

Where the ladder goes next #

The immediate continuation is to go back through the site’s own measurements and say which population each of them is over. Most will be the free sampler and most will be unaffected; the ones that matter are the ones whose conclusions are about designs rather than about vertices, and there are more of those than this essay has room to check.

The other direction is the population nobody has: vertices from real designs. It cannot be built from other people’s crease patterns here, and it could be built from the grammar of design methods — box pleating, twenty-second-grid work, the tree method’s ridge and hinge creases — each of which generates vertices by a rule that could be sampled from. That would be a fifth population and the closest anybody could honestly get to the one that matters.