One population, four sheets

Every claim in this subject of the form typically, usually or almost always is a claim about a population, and choosing the population is most of the work. The collection has four ways of drawing a pattern and has been careful to say which one a statistic came from.

All four are ways of drawing. None of them says anything about the sheet the drawing is on, because until recently there was only one sheet available and a parameter with one value is not a parameter.

The same drawings, four times #

Take three drawings — a plain grid, a Miura corrugation, a Yoshimura — and a rectangle of each, cut so its edges miss every vertex. Then make four objects out of each rectangle: leave it cut out of the plane, glue one pair of edges, glue the other, glue both.

Twelve objects, three drawings.

The vertices are the same vertices asked the same conditions on all four, because the rectangle’s edges are placed to miss them. So anything a local statistic measures is identical across the four, and everything else is not.

What moves #

Verdicts move. An odd cell of the grid folds cut out and does not fold glued, at any size, because a loop that cannot be shrunk crosses an odd number of creases. Three of the twelve objects have no flat folded state and their drawings all do.

Counts move. Free letters fall by the creases each glued pair divides, panels by the identifications each makes, and neither by the same amount on the two cylinders of an anisotropic drawing.

Costs move, further. On a three-period square twist the four sheets cost thirty, twenty-four, eighty-five and six hundred and twenty-five nodes under one variable order.

So typically is answering a different question depending on which of the twelve objects is in the population, and no statement about how the drawings were produced settles it.

Twelve objects, listed #

The population is small enough to write out, and writing it out is the clearest form of the argument.

The grid, two periods: cut out, twelve letters over nine panels, nine nodes; glued across, ten over six, seven nodes; glued along, ten over six, seven nodes; glued both ways, eight over four, six nodes. All four fold.

The grid, three periods: cut out, twenty-four over sixteen, sixteen nodes, folds. All three gluings are refused before any search, because a loop crosses three creases and three is odd.

The Miura, two periods: twenty-two over fifteen, fifteen nodes; eighteen over ten, eleven; twenty over twelve, thirteen; sixteen over eight, ten. All four fold.

The Yoshimura, three periods: seventy-two over fifty-five, fifty-three nodes; sixty over forty-two, forty-two; sixty-six over forty-eight, forty-five; fifty-four over thirty-six, thirty-six. All four fold, and the Yoshimura’s smaller cells are refused for a completely different reason.

Three drawings, twelve objects, three verdicts of no, and letter counts spanning eight to seventy-two. A statistic taken over that population would be reporting something, and what it reported would depend entirely on which of the twelve were in it.

Two kinds of refusal in one population #

Something worth noticing about the list: the objects that refuse do so for two unrelated reasons, and a population containing both is a population containing two phenomena.

The grid’s odd cells refuse on a parity — a loop crosses an odd number of creases, the paper comes back the other way up, no two-colouring exists.

The Yoshimura’s short cells refuse on a turn — the folded state carries one cell onto the next by a rotation rather than a slide, so the drawing’s period is not the folded state’s.

Both appear in a table as no flat folded state, and lumping them is exactly the error that one number was making before the two were separated. A statistic over this population would count them together and report a rate of refusal that is the sum of two unrelated rates.

That is a good general reason to be suspicious of aggregate statistics on heterogeneous populations, and it is a specific reason to be suspicious of this one.

What a statistic over the sheet would even mean #

Suppose somebody wanted a genuine average over sheets. What would they have to supply?

A set of sheets, which is easy enough: disc, cylinder across, cylinder along, torus, and possibly the holed sheets and the Möbius band.

A weight for each, which is the problem. There is no natural measure. Weighting them equally says a torus is as common as a square, which is false of anything anybody folds. Weighting them by how often they occur in practice makes the answer almost entirely about discs, which returns the existing statistics with extra steps.

And a correspondence between drawings on different sheets, since averaging requires knowing which drawing on a torus corresponds to which on a square. That correspondence exists here — the four sheets are one rectangle — and it exists only because the sheets were built from a common object.

So an average over sheets is not available and is not obviously a thing anybody should want. Reporting per sheet is available, is honest, and produces four numbers where an average would produce one that means nothing.

The population this collection actually has #

Worth stating plainly, since the essay is a criticism of an implicit choice and the choice is defensible.

Every population here is drawings on a square of paper. That is the object the subject is about, it is what people fold, it is what every application uses, and a statistic about it is a statistic about the thing anybody cares about.

What the four sheets add is not a better population. It is a demonstration that the population had a parameter fixed at one value, which is worth knowing when reading any typically in this collection — and worth naming in the sentence, which costs four words.

The parallel is with the collection’s earlier discovery that a population had a drawing method fixed at one value and that the statistics moved when it varied. Same shape, one level up, and the response is the same: name the parameter, report per value, do not average.

The parameter that was never named #

A population in this collection is specified by saying how a pattern is drawn: at random on a grid, by a construction, by perturbing a known pattern, by sampling angles. Each of those pins down the drawing completely.

None of them pins down the sheet, because the sheet has always been a square of paper. A statistic quoted from any of those populations is therefore a statistic about drawings on a disc, and the disc was never mentioned because there was nothing to contrast it with.

That is not a criticism of the populations. It is an observation that they have two halves and one of them was constant.

Which half matters more #

On the objects measured here, the sheet matters more than the drawing, and by a lot.

The three drawings differ in almost everything: a grid has degree-four vertices at right angles, a Miura has degree-four vertices at an oblique angle, a Yoshimura has degree-six vertices at sixty degrees. Their cut cells all cost about one node of search per panel.

The four sheets are the same drawing in every case. Their costs differ by a factor of twenty-six on one of them.

That comparison is worth handling carefully, since it is between a factor measured at one size on one family and a similarity measured across three families at several sizes. What it establishes is that the sheet is not negligible, which is a weaker claim and the one the data supports.

What a population would have to say #

If the sheet is a parameter, a population’s description has to fix it, and the ways of fixing it are not all sensible.

Fix it to a disc. That is what every population here does implicitly, and saying so out loud costs nothing and removes an ambiguity.

Draw the sheet from a distribution too. That needs a distribution over sheets, and there is no natural one: a square, a square with a hole, a cylinder and a torus are not points in a space anybody has a measure on.

Vary the sheet systematically and report per sheet. Which is what this essay does, and it is the only one of the three that produces a number a reader can use.

Which statistics survive unchanged #

Not everything moves, and knowing what does not is as useful as knowing what does.

Anything local survives. The vertex conditions hold or fail identically on all four sheets, because the vertices are identical. So the fraction of vertices satisfying Kawasaki, the distribution of sector angles, the count of degree-four vertices: none of those moves.

Anything about the drawing survives. Crease length, panel shapes, the angles between creases, the pattern’s own symmetries. The drawing is one drawing.

Anything about a lettering’s local structure survives. How many labellings each vertex admits, which is the quantity the one-node-per-panel reading is really about, is a property of the angles at a point.

What moves is exactly the global: verdicts, free letters, panels, Euler’s number, search costs, and whether a periodic lettering exists.

That is a clean split and it is the same split that runs through the whole of this collection. Local statistics are about the pattern; global ones are about the pattern and the sheet, and only the second have been quoted without their second argument.

A note on sample size #

The essay has been arguing from twelve objects, which is a small number, and it is worth saying what twelve can and cannot support.

Twelve is enough to establish existence: there are drawings whose verdict depends on the sheet, and there are pairs of sheets on one drawing whose costs differ by a factor of twenty-six. Both are demonstrated by a single instance and twelve is more than one.

Twelve is not enough for any rate. How often a drawing’s verdict changes with the sheet, what the typical cost ratio is, whether the threshold is at three periods on other families — none of that is estimable from three drawings at two or three sizes.

The distinction matters because the essay’s claim is an existence claim: a population’s description leaves a parameter unspecified, and the parameter has consequences. That needs one example and has twelve.

Anybody wanting the rates would need a proper population over sheets, which is the thing this essay has just argued does not exist.

The honest form of a statistic #

The consequence for how anything here should be quoted is short.

Instead of a pattern of this kind costs about one node of search per panel, the honest form is a rectangle of this pattern, cut out of the plane, costs about one node per panel, with the sheet named.

Instead of almost every pattern fails to fold flat, the honest form names the sheet, because on a torus a whole family of drawings fails for a reason no vertex can see and on a disc none of them does.

The cost of that is four words per claim. The benefit is that a claim about drawings stops silently being a claim about drawings on a square.

What is not being claimed #

Not that existing statistics are wrong. They are correct measurements of populations of drawings on discs, which is the case everybody cares about and the case all the applications are in.

Not that the sheet dominates in general. It dominates in one measurement at one size, and below the threshold the four sheets are within tens of per cent of each other, which is where nearly every measurement in this collection sits.

Not that a population over sheets is available. There is no distribution over sheets and inventing one would be inventing the answer. Reporting per sheet is what is available.

Where the sheet is not a free parameter #

There is a limit on how far the sheet can be varied for a given drawing, and it changes what the four objects are.

A rectangle can only be glued if it is a period of the pattern. Cut a rectangle at some arbitrary size and its two edges do not match up: creases running off one side arrive at the other at the wrong heights, and identifying them joins things that are not the same crease.

So the four sheets are available for a rectangle that is a whole number of periods and for no other. That is a real restriction and it shapes the population: every object in the twelve is a period cell rather than an arbitrary patch, which is not what the collection’s populations of drawings usually contain.

It also means the comparison cannot be made at all for a drawing that does not repeat. A crumple, a random pattern on a grid, a design’s crease pattern: none of them has a period, none of them can be glued, and for those the sheet really is fixed at a disc with no alternative.

That is worth knowing before generalising. The parameter exists for repeating patterns and does not exist for the rest, and most of what this subject folds is in the second category.

What a reader should do with an old number #

Practical advice, since the collection contains several hundred statistics quoted without a sheet.

If the statistic is local, ignore all of this. Vertex counts, sector distributions, per-vertex labelling counts: the sheet does not enter.

If the statistic is about a patch, read it as being about a patch. A cut rectangle costs one node per panel is exactly what was measured and it is exactly what it says.

If the statistic is a verdict — folds, does not fold, is rare, is typical — check whether the object had a boundary. Almost every one of them did, and for those the statement stands with on a sheet with an edge silently attached.

If the statistic is about a tessellation rather than about a patch of one, be careful. Those are the ones where a reader might reasonably take the claim to be about the plane, and the plane has no boundary at all, so it is the sheet these results are furthest from.

That last category is small and it is the one worth auditing. A claim about the square twist tessellation measured on a square patch of it is a claim about a disc, and the tessellation is not one.

The habit worth taking away #

The essay’s method is one this collection uses repeatedly and it is worth naming, because it is what produced every finding in this phase.

Take something that has always had one value. Build the object where it has another. See what moves.

The sheet had one value — a square of paper — for the whole life of the subject, and there was no way to give it another until a gluing existed. Once one did, twelve objects came out of three drawings, three of them refuse, and a factor of twenty-six appeared between two objects that are the same drawing.

Nothing in that required a new idea about folding. It required an object, and the object required a way of saying which boundary points are the same point.

The general shape #

The collection has met this before under another name.

A population nobody chose is the observation that a statistic is about a distribution, and that leaving the distribution unstated makes the statistic uninterpretable rather than general.

This is the same observation one level up. A distribution over drawings leaves the sheet unstated, and the sheet turns out to be a parameter with consequences. There is no reason to think it is the last such parameter, and the discipline that finds them is the one that produced this: build an object that varies something previously constant, and see whether anything moves.