The Yoshimura folds flat at a few heights

Patterns nobody designed drew the Yoshimura pattern at three proportions and read a rule off them. Taller than the equilateral height, every vertex fails the big-little-big lemma. Shallower, the sectors go four large and two small, no sector is strictly smallest, and the lemma has nothing to forbid. The admissible band is bounded above and open below, it concluded, and that became the standing account: the row height is a dial, and anywhere under the top is allowed.

Everything in that sentence is about conditions at a vertex. Half the waterbomb’s rules have no stack asked the repeating rules of the waterbomb and the Yoshimura for a folded state and found that passing the vertices promised much less than had been assumed; it asked at one height, the equilateral one, since that is the height the rules were drawn at. The obvious next thing is to turn the dial and ask again.

One lettering, every height #

The Yoshimura lettering everybody draws gives every course one letter and every zigzag the other. Write $\alpha$ for the base angle of its triangles, sixty degrees at the equilateral height and smaller below it, and write it as $\pi/n$ with $n$ not necessarily a whole number. At every $n$ from 3 upward, the lettering passes all three vertex conditions at all interior vertices. So ask the search, on each of seven patches from three by three up to five by five, at sixty values of $n$.

The picture does not look like a band. The sheet has a folded state at $n = 3, 4, 5, 6, \dots$ and nowhere between them, until a point past which it stacks at every height: $n > 5$ on three rows, $n > 6$ on four, $n > 7$ on five. On four rows, sixty degrees stacks and fifty-nine and a half does not. Forty-five degrees stacks and forty-seven and a half does not. Every verdict in that drawing is a proof for the patch it names, since the search either returned a stacking or ran out of possibilities.

The proportion the earlier essay drew as the allowed shallow case, three quarters of the equilateral height on six columns and five rows, is proved to have no folded state. It passes every vertex; its sectors are as drawn; no paper could take the shape.

What the panels do when folded #

The reason is visible as soon as the panels are drawn where folding puts them.

The sheet folds into a rosette. A panel’s folded place is the composition of reflections in the creases between it and a fixed panel, and two reflections in lines that meet at $\alpha$ are a rotation by $2\alpha$. So every panel that has not been turned over sits at a whole number of steps of $2\alpha$ about one centre, which is a thing that can simply be checked, and it holds on every frame of the slider. At $\alpha = \pi/n$ a run of $n$ steps comes back to the start, the triangles close on a ring, and further rows land precisely on panels already there. At $n = 3$ the ring has shrunk to one triangle, which is why the equilateral sheet piles every panel on a single triangle. At $n = 4$ it is a square and at $n = 5$ a pentagon.

None of this depends on the letters. A folded place is fixed by the creases and not by which way they go, which is what thirty-two rules, one object found on the waterbomb, and the same check here finds a rule with quite different letters putting every panel in exactly the same place. The rosette belongs to the pattern. The letters only decide what order its layers are in.

Between whole numbers the steps do not come back after one turn. At $n = 3.5$ they come back after seven steps and two turns, and panels that were one exact pile are now seven directions overlapping each other partly. Panels that used to meet along their whole edges now cross. That alone does not decide anything, since other rules stack on this same arrangement, but it means there is suddenly far more for the letters to get wrong.

Below $\pi/(\text{rows} + 2)$ the usual lettering stops being refused at all. That threshold is measured rather than derived. It is set by rows and not by columns on every patch drawn, and it is what a small patch has been hiding: on three rows, the whole region below thirty-six degrees looks fine, and a sheet one row taller loses the part from thirty to thirty-six.

Seven panels refuse every other height #

The refusal is not diffuse. Remove panels from a refused patch as long as what remains stays refused and the process stops at seven.

The seven are a central triangle, two diamonds of two triangles each flanking its apex, and an outer triangle beyond each diamond. Listing their $5040$ orders directly, with the crossing conditions recomputed from the folded positions by a separate sampling and not taken from the search, gives none that works. There are eighteen crossing conditions on six creases, and all of them are conditions of a panel lying across another pair’s crease.

The difference between a closed and an unclosed height is in those conditions, and at the top of the range it can be counted directly. At $n = 3$ the same seven panels have no crossing conditions at all, since panels lying exactly on one another meet edge to edge, and three conditions of the other kind, about creases sharing a line; at $n = 3.05$ they have eighteen crossing conditions. At $n = 4$ they have six, and at $n = 4.05$ eight. A proportion that closes has fewer constraints than one a hair away, since exact coincidence removes crossings, which also explains why the verdict is discontinuous and why the strip above has single dark cells with pale cells either side.

That is a strange thing for a dial to do. The bending of the sheet varies continuously, the sectors vary continuously, every vertex condition varies continuously or not at all, and the existence of a folded state is present at a set of isolated points.

Which rules care about the height #

The usual lettering is one of twenty-six rules that pass the vertices, and the others behave differently. The four zigzag classes can be split evenly in three ways, by band, by lean or crosswise, and which way they split decides everything.

Sixteen rules do not care. Those whose zigzags split by band or by lean stack at every height drawn, on the same rosette, through the same partial overlaps that refuse the usual lettering. Those are the rules that behave the way the earlier account assumed every rule did. Four rules, with zigzags split crosswise and courses that agree, stack at no height at all, and these are the four the census at the equilateral height refused. Four more with crosswise zigzags and differing courses stack only at exactly sixty degrees, where that census counted them among the twenty-two; at every other height drawn they have no state, so that twenty-two is itself a property of one special proportion. Away from the equilateral height, then, the count on a tall enough patch is sixteen, plus the two usual rules where the ring closes.

The most recognisable lettering on the most recognisable found pattern is thus the one member of its family whose folded state depends on number theory.

Why some rules never stack #

That census left a question: the crosswise rules were refused on the equilateral sheet by something that needed sixteen panels, with no short argument to hand. There is a simple measurement that goes most of the way. Read each crease as an arrow from the panel its letter puts lower to the panel it puts higher, and ask for the longest chain of arrows.

Under the rules that never stack, the letters alone put nearly the whole sheet on one staircase. Every panel but a handful is forced above the one before it, before any overlap has been looked at. A folded sheet has panels returning to the same place over and over, and a single rising chain leaves almost no freedom in how they interleave. That is why deleting panels finds no small refusal: the argument runs along the whole chain. The crosswise rules with differing courses have a chain of between two fifths and a half of the sheet, and those stack only on the one proportion with the fewest crossings. The rules split by band have a chain of four at any size, since their letters reverse the climb every other panel.

Chain length is not a complete account, and the table shows where it stops. The usual lettering’s chain is shorter than that of the rules split by lean, and it refuses where they stack. Its refusal is the seven-panel one, about a particular arrangement rather than about the sheet being wound up. So there are two different failures on this one pattern, a local one that depends on the height and a global one that does not.

The ring and the can #

A can crushed underfoot does not collapse flat onto a triangle. The cylinder the pattern chooses found that a Yoshimura tube’s courses must close round the tube exactly once, a condition on columns rather than on height, and the rosette here is a different closure about a different axis: it is the rows that come round, in the plane of the fold. So nothing in this account says what an actual buckled tube does at its proportion, and the shell problem patterns nobody designed set aside stays set aside. A thin wall does not need to reach a flat state to be a Yoshimura pattern; it has only to fold part way and stop against itself, which a spiral would do. Crushed cans are not flat, which this result would predict and did not cause.

What it does change is any account that treats the flat state of this pattern as something a designer can have for the asking. In practice the proportions drawn most often are the equilateral triangle and the right isosceles triangle, sixty and forty-five degrees. Those were presumably chosen for being easy to divide on square paper. They also turn out to be the two steepest heights at which that lettering folds flat at all, and the equilateral is the only height where every panel lands on one triangle.

Assumptions and what cannot be seen #

A folded state here is a stack of panels of no thickness, flat, with a stated order. A real sheet at fifty-nine degrees might wrinkle, tear or bow a little and be called folded, and with thickness the isolated points must become small intervals whose width depends on the paper. The exactness of $\pi/n$ is a statement about the ideal and is not a tolerance for a folder.

The law is measured. The rule that a sheet stacks exactly at whole $n$ and past rows plus two is an observation over seven patches and sixty heights, all of them decided, with whole $n$ tested further up to twelve on patches up to five by four. It is not proved for every patch and every height. In particular, the grid of $n$ is in tenths, and very near a whole number a sheet might behave differently from a tenth away, though the refusal is proved at the single value tested each time. A stack is not a motion: having a folded state says nothing about whether the paper can reach it, and this pattern famously cannot fold rigidly. The rosette is a drawing of positions with no order implied by its shading, and the picture cannot show which layer is on top, only that layers overlap.

The chain is drawn at the equilateral height, though the arrows come from the letters and depend on the height only through which panel a crease calls near. Attribution stops at the pattern. The pattern is named for Yoshimura, who described it in buckled tubes in 1955, and its letters and proportions belong to no folder in particular. The closure at $\pi/n$ was found here by search, then recognised, and may well be familiar to anyone who has folded it at unusual proportions; no citation for it was found, which is not the same as there being none.

What a closed ring costs #

Closure is not free, and the two steepest heights show the trade. At sixty degrees every panel lands on one triangle, which is the most compact flat state this pattern has and also the deepest pile, since the whole sheet’s layers sit over one small area; a panel is not the unit of depth names the Yoshimura as one of the uniform piles for exactly this reason. At forty-five degrees the same panels are spread round a square, at thirty-six round a pentagon with a hole in it, and each further whole number gives a wider ring with the layers shared out further round. A designer picking a proportion for a flat package is choosing a polygon, and is choosing it from a list: there is nothing between the triangle and the square, however slightly the angle is moved.

So a stated tolerance on the base angle reads differently here than it does elsewhere. A tolerance is a direction found on a solved mesh that error in one direction costs far more than in another, and it could describe that cost as a length. Here there is no length to report in the ideal, because the state exists at one angle and not beside it. What a real sheet does in that gap is the content of the next section, and it is the question the ideal cannot answer.

An unusual dial #

The unexpected thing is what kind of quantity this turned out to be. Essentially everything else on this pattern is a smooth dial: stiffness, the height of a tube, the energy of a course. Here a continuous proportion controls a yes-or-no with a rational condition inside it, of exactly the sort that decides which regular polygons a compass can construct, and a pattern found by stepping on a can answers to whether a number is the reciprocal of a whole one. The two are not formally related. It is the same shape of answer, though, turning up where nobody had asked a question of that kind, because every earlier test of the pattern was a test at one vertex and a vertex cannot count how far round a sheet has gone.

It also fits a pattern across these essays. A unit that folds is not a tessellation found a small patch hiding the vertices at which a rule fails, taught with a wrong reason found a verified unit hiding a forced loop, and here a short patch hides a whole range of heights, by not being tall enough to go round.

Still open: a sheet that leaves the plane #

A partly folded Yoshimura is where the unclosed heights could still live. A rosette that cannot pass itself in the plane might stop short of flat, with the triangles still at an angle. Whether the seven-panel refusal becomes a stopping angle, and what that angle is at each proportion, needs a kinematic model of a pattern that is known not to fold rigidly, so it needs an account of where the bending goes. Patterns nobody designed asked what the real proportion of a crushed can is, and this would be the first prediction to test against one.

The threshold wants a proof. Rows plus two is a clean statement with a plausible count behind it, and an argument that the fan can always be stacked until it closes would turn this drawing into a theorem. Other rosettes are likely. Any pattern whose reflections compose to a rotation should have them: the waterbomb’s survivors were found at one proportion, and its base angle has only ever been drawn at forty-five degrees, so a stretched waterbomb cell is the obvious next sheet to ask.

The habit worth carrying is about dials. A parameter that every local test accepts has been tested locally. Before calling a range admissible, walk along it and ask the global question at each step, because the answer need not depend on the parameter in anything like the way the local ones do.