The proportion everyone draws is the hard one

A population nobody chose made a point about samples. Five crease patterns had been measured over and over because somebody had drawn five, and a grid of ninety-six, built without regard to what would be interesting, found things the five could not. One population, four sheets added that a population has a second half, the sheet each drawing is put on, and that this half had never been chosen either.

Both of those sampled across patterns. There is a third thing a population quietly fixes, which is the proportions within a pattern. Each family has dials, and each family was measured with its dials where they are usually drawn. The Yoshimura’s row height is a clear case. It is drawn equilateral, every panel a triangle of sixty degrees, since that is the proportion that divides a sheet evenly and is easiest to draw, and every count this subject holds of a Yoshimura came from there.

The Yoshimura folds flat at a few heights found that the height decides whether its usual lettering has a folded state at all. That raises a plainer question for rules that stack at every height. If the height does not change the verdict, does it change anything else?

A cost that falls off a cliff #

Take three rules and ask a search for a stacking of a four-by-four sheet at each height, writing the base angle as $\pi/n$. It is the same search with the same restarts, on sheets that differ only in where their zigzags meet.

The rule with zigzags split by band costs 268,650 steps at the equilateral height and 808 a quarter of a step away, three hundred and thirty-two times fewer. Nothing has been simplified between those two sheets. They have the same panels, the same creases and the same letters; the zigzags meet at about $55^\circ$ rather than $60^\circ$ and the sheet is a little squatter. Going on along the dial, the costs wander within about a factor of three, with small bumps at $n = 4$ and $n = 5$, and nothing else comes close to the point everybody draws.

And the drop is not a slope. A thousandth of a step from the equilateral height the same rule costs 810 steps, and a hundred-thousandth of a step away it still costs 810. There is no region in which the sheet is nearly equilateral and the search is nearly as slow. The expensive sheet is one point on the dial, and it has no neighbourhood.

That rule is the largest case, and the choice of rule matters. A single rule on a single sheet is subject to whatever order the search happens to take, as the difficulty was in the coin found, so a figure like three hundred and thirty-two should not be quoted as a constant. Taken across the sixteen rules that have a stacking at every height, the medians are steadier and they say the same thing.

On every sheet measured the equilateral height is the dearest of the five, by three times on twenty-one panels, fourteen on thirty-six and thirteen on forty-four. An early draft of this argument expected that premium to grow steadily with the sheet, and it does not: fourteen to thirteen is flat within what these medians can resolve. What can be said is that the premium is real on every sheet and that the costs at the other four heights sit close together, at around twenty-five hundred steps on the largest. The equilateral sheet costs a dozen times what its family costs.

The larger sheet shows why single figures need care. The band rule there costs 143,259 steps at the equilateral height, less than half what it cost on a smaller sheet, because a search that stops at its first stacking can stumble on one early. The shape does not change: a spike at $n = 3$, bumps at the other whole values and level ground between them.

The cost follows the number of states #

So what is different about the equilateral sheet? Listing every folded state on the smallest sheet answers it, since on three by three the list can be finished.

A band rule has 9,790 folded states at the equilateral height and 92 a quarter of a step away. At $\pi/4$ it has 1,578, with 38 just below; at $\pi/5$ it has 230, with 68 just below, and by $\pi/6$ the two have merged at 34. Between one whole value and the next, on this sheet, the count does not move at all, and three separate heights give exactly the same number. Each whole value is a sudden multiplication of the ways the same paper can be stacked. The other rule shows less, 180 against 16 at the equilateral height, and at the next whole value it does not rise at all; so the size of the multiplication depends on the letters, but it is always largest at sixty degrees and never negative.

The cost tracks this, though not one for one. The band rule’s states multiply by a hundred and six at the equilateral height and its cost on the same sheet by forty-four; the lean rule’s states multiply by eleven and its cost by three. Both move the same way and the cost moves less, which is what a search that stops at its first success ought to do: more arrangements to wade through, and also more stackings to stumble upon.

This is a surprising direction for a count of folded states to go. The equilateral sheet looks like the most tightly packed flat state the pattern has, with every panel on a single triangle, and a reasonable first guess is that the panels are so crowded that few orders can work. The reverse holds. A hundred times as many orders work there, because of what coincidence does to the rules.

Coincidence removes the conditions #

Two kinds of condition bind a stack of panels. A panel that reaches across a crease cannot lie between the two panels that crease joins. And two creases on one folded line must nest and not interleave. Count them at each height.

At the equilateral height no panel lies across any crease: there are no conditions of that kind at all. A quarter of a step away there are 1,024. Each panel lands exactly on the others, so all the edges coincide, and no edge is left inside a panel for anything to be caught on. What remains are the conditions of the other kind, 338 of them on this sheet, about creases sharing a line. Move a little and panels are shifted against one another, each edge now running across the interior of several other panels; the first kind arrives by the thousand and the second falls to 53. At $\pi/4$ and the other whole values the same exchange happens on a smaller scale, and the dips in the first curve are where the spikes in the cost and in the states are.

So the cost has a plain account, and it runs against the first instinct about hard instances. The search is slow at the equilateral height because the sheet is free, not because it is tight. A condition of the first kind decides something immediately: this panel is not between those two. A search presented with a thousand of them is told what to do at nearly every step. A search presented with none has to find out by trying, and what it tries is a far larger set of arrangements in which most fail only much later, through the creases on the shared line. The equilateral pattern is an under-constrained instance whose constraints are of the kind that bite late, which is the classic shape of a costly one, and it is the instance that gets drawn.

A refusal does not notice the sheet #

The other side of the same picture is what happens where there are many conditions and they conflict.

At $\pi/3.5$ the usual lettering is refused in exactly 523 steps on every sheet from twenty-one panels to sixty-five, and a crosswise rule in exactly 57. The cost does not grow at all, because the search reaches the seven panels that refuse it in the same order whatever surrounds them, and it never looks further. A rule with a stacking, meanwhile, grows steadily with the sheet, since a stacking has to account for every panel.

The rules that never stack make the point most sharply, since they are refused on both sides of the line. A quarter of a step from the equilateral height such a rule is refused in 34 steps, on twenty-one panels, thirty-six and forty-four alike. At the equilateral height the same rule on the same sheets takes 2,099 steps, then 24,805, then 50,402, and that cost grows with the paper. Half the waterbomb’s rules have no stack found the same rule refused there only by a set of sixteen panels, which is the same fact seen from the other side: with no conditions of the first kind available, a refusal cannot be local. Its proof is short where the conditions are many and long where they are few, and the difference between those two proofs is a change of about one degree in a drawing, or of much less.

That reverses what the conditions would be expected to do for a designer. Anyone checking a pattern would rather have a sheet that says little and leaves many options open. A checker prefers one that says a great deal: in this family the sheets with the most conditions are the ones that give an answer in dozens of steps, whichever answer it is, and the one with the fewest is where the same answer takes tens of thousands.

What this does to the earlier measurements #

The placement of a cut and a population nobody chose measured cost over families of twists with their own dials, and those are not in question: they sampled across a parameter, and what they found holds where they sampled. The case here is narrower. Anything said about the cost of a Yoshimura, the depth of a Yoshimura or the number of its states on the assumption that the equilateral sheet represents the pattern is a statement about the single most exceptional member of that family. The twenty-two rules that stack there are sixteen away from it, and eighteen where the ring closes. The 9,790 states are 92 a quarter of a step away. And anything like one marking, many objects, whose point is how many folded objects a marking makes, needs the proportion stated beside its count, since on this family the proportion moves that count by a factor of a hundred.

This is also an argument about how a family gets sampled in the first place. Regular patterns are drawn at the proportions where things coincide, because those are the ones that are simple to construct, and the things that make them simple to construct are the same things that remove their conditions. So the convenient instance is systematically the unconstrained one, and a survey of what is drawn over-reports both freedom and cost relative to the family it stands for. That does not make such a survey wrong. It does make it a survey of a corner.

The stated limits #

Steps are one search’s steps. They count nodes of a particular order with particular restarts, and the 332 and the 123 belong to single rules on single sheets. A better order might well reduce the premium, and which choice the cost lives in is the reason for not over-reading any one row. The medians are firmer, and it is the medians that carry the claim. Only one family is measured, and only one of its dials. That the same thing happens on a stretched waterbomb, or on a Miura whose two angles are made to coincide, is a prediction and not a finding. The sheets stay small. Forty-four panels is what can be finished across sixteen rules and five heights, and whether the premium stays at about thirteen or changes on much larger sheets is unmeasured, since the steps grow past what a figure can recompute.

The states are counted on three by three only, and the exact count of 9,790 is for a sheet with rim panels; a tall patch would have many more, and the ratio could differ. The picture cannot show why the search takes the route it does, only how long it took, and it says nothing about paper: a stacking is an order of ideal layers, and in a folded sheet of real thickness there is no such thing as panels landing exactly on one another.

Which instances the rest of the family stands for #

If the equilateral sheet is the exception, the family’s typical cost is the level ground, and that has a tidy shape of its own. Away from whole values of $n$ the conditions of the first kind number in the hundreds or thousands, the search reaches a verdict in thousands of steps at most on these sheets, and on three by three a count of folded states holds exactly constant between one whole value and the next, 92 from $\pi/3.25$ to $\pi/3.75$ and 38 from $\pi/4.25$ to $\pi/4.75$. Between coincidences the combinatorics is one fixed object as far as these counts can see, and a real number’s exact value makes no difference to it. The dial is doing almost nothing almost everywhere, and then at isolated points it does everything.

That shape is common in things that depend on incidences. A generic position has a fixed combinatorial type, and only special positions change it. What is unusual here is that the special positions are the ones people choose. The folder wanting a clean drawing and the engineer wanting a single repeating triangle were both reaching for one of those isolated points, each for good reasons.

Freedom as the expensive thing #

The unexpected part is the sign. Discussions of hardness usually look for what makes an instance tight, and a sheet with every panel on one triangle looks as tight as a sheet can get. Yet that is the point with the fewest conditions and the most solutions, and the search suffers from it. In satisfiability the hard instances sit at a critical ratio of constraints to variables, with too few constraints to prune and too many to give solutions away easily. This is not the same phenomenon, since nothing here is a random ensemble and the sheet has plenty of solutions, but it is a relative of it: a problem can be expensive because it says too little.

Still open: a family with several coincidences to choose from #

The Yoshimura has one dial and its coincidences are isolated. A pattern with two independent dials would have coincidences along curves, and the question of whether cost follows them, or whether only the intersections of the curves are expensive, is the next measurement. The Miura’s two angles are the natural place to start, and the Miura’s cost has only been measured where a single step per panel left nothing to measure, so a rule whose letters leave it room is what would be needed.

The other direction is a better order for the free sheet. With no conditions of the first kind, everything a search knows comes from creases sharing a line, so an order that handles one folded line at a time, as a book is filled one page at a time, might remove much of the premium. That would turn this from a fact about instances into a fact about what the search had been looking at.

The habit worth carrying is about what gets drawn. The instance chosen for being easy to draw is seldom typical of what it stands for. Before quoting a cost, a count or a depth from it, move its parameters a very little and measure again.