A dial can turn the fold over

The sheet draws in crooked measured what a twist tessellation’s collapse does to the plane on five tilings and their linear images, and found that ten of the fifteen images fold by a map with two different principal factors — a circle on the flat sheet folds to an ellipse, as much as 5.386 times longer than it is wide. It found the images falling into three groups that returned the same numbers to ten figures, and it found the triangular grid folding by a similarity under every map tried. Then it named the cheap test that would decide what kind of number the ratio was: run the same table at a second turn. If the ratios did not move, they would belong to the pair of a tiling and a map, and could be looked up.

The table was run at a second turn, and at a second pleat angle, and then both dials were swept. The ratios move, and not by a little. On the sheared square grid the ratio that read 4.414 reads 2.735 at a slightly larger turn, 11.08 at a smaller one, and 383.5 at a pleat angle of −0.35. And the sweep found something the table could not have: across a stretch of pleat angles the collapse’s area changes sign, which means the twists fold to a mirror image of their own flat arrangement.

The same table at a second turn #

The first measurement was taken at a turn of 0.42 radians and a pleat angle of −0.5, and those two numbers were the only settings the collapse had been read at. The turn is how far each twist polygon is rotated against the vertex it sits on; the pleat angle is how far each pleat’s creases lean from the edge of the tiling they run beside. Both are free in the construction one number where the corners wanted four set out, and both were fixed for the census by convenience. The older construction, which closing the loops is not folding found failing on every one of these images, had no pleat angle to set; it is the extra freedom that made the images foldable, and it is the same freedom the sweep below runs.

At a turn of 0.5 every anisotropy in the table falls. The sheared square grid goes from 4.414 to 2.735, the sheared honeycomb from 2.818 to 2.196, the stretched honeycomb from 5.386 to 3.460 and the general images from 3.542 and 5.060 to 2.395 and 3.318. So the ratio is not a property of the tiling and the map. It is a function of four things, and the two that were held fixed are doing as much work as the two that were varied.

What did not move is as informative. The five tilings as drawn are still similarities, and still the same similarity as one another — the coincidence folding it flat is one similarity first measured on an older construction, which now holds at every setting of both dials, with only its factor moving: 0.414 at a pleat angle of −0.35, 0.502 at −0.5, 0.962 at −1.1. The triangular grid is still a similarity under every map. The square grid and the elongated triangular tiling still agree to every figure printed, and so do the honeycomb and the rhombille. The grouping belongs to the tilings; the numbers belong to the dials. That is the opposite of what every twist writes an equilibrium hoped the edge weights would supply — a number per tiling — and it fits what the weights turned out to do, which was predict nothing about the collapse at all.

The same table at a second pleat angle #

A smaller change to the other dial does something more violent.

At a pleat angle of −0.35 the sheared square grid’s ratio is 383.5: its folded sheet draws in to 0.706 of its length in one direction and to 0.0018 in the other. That is not an extreme distortion of an ellipse. It is very nearly a line. The stretched honeycomb reads 16.70 and the general images a little over thirteen, so the whole family has moved toward the same edge, and the sheared square grid has nearly gone over it.

A ratio of 383 is the kind of number that means the measurement has found a singularity and is standing close to it. The way to see what is there is to stop sampling the dial at two points and run it.

The area of a collapse can be negative #

The census reports the ratio of the two principal factors, and a ratio cannot tell a map that squashes the plane flat from one that is about to turn it over. The determinant of the fitted map can. It is the area the folded twists span for each unit of area they spanned on the flat sheet, and it carries a sign: positive when the collapse keeps the plane’s handedness, negative when it reverses it.

Swept, the sheared square grid’s area falls from 0.63 at a pleat angle of −0.9, reaches zero at −0.348, goes negative, and comes back through zero at −0.161. Between those two angles every twist of the pattern is folded to a place such that the folded arrangement is the flat arrangement reflected. At the two crossings the collapse is singular: every twist of one class folds onto a single line, and the ratio the census reports is infinite.

The honeycomb group under the same shear dips to about 0.09 near −0.3 and rises again without crossing. The triangular grid’s curve does not dip at all, and it is not a curve belonging to the triangular grid alone: at every setting checked it is exactly the curve of every tiling as drawn, because under this shear the triangular grid folds by the very map the unsheared tilings fold by.

What a mirrored collapse is #

The mirror sounds impossible and has a one-dimensional version that is not. Take a strip of paper with panels along it and a pleat between each pair. Folding a pleat slides the paper beyond it back toward the paper before it, by twice the pleat’s width. If the pleats are narrow, each panel ends up a little closer to its neighbour than it was and the panels keep their order along the strip. If each pleat is wider than the spacing of the panels, each panel slides back past the one before it, and the folded strip has its panels in reverse order. Exactly at the width where the slide equals the spacing, every panel folds onto the same spot.

A twist tessellation does this in the plane. Each twist polygon keeps its face and is only turned, since crossing a pleat is two reflections in parallel creases and two reflections compose to a slide. So the twists are carried rigidly and their centres move by the sum of the slides of the pleats between them. Where the slides in one direction take back more than the spacing of the twists in that direction, the arrangement comes out reflected. At the crossing they take back exactly the spacing, and a whole row of twists folds onto one spot, which is the singular collapse the sweep found.

The dial makes the point with two dots. On the flat circle, point 1 is to the right and point 2 above, so going from 1 to 2 the short way is anticlockwise. At pleat angles more negative than −0.348 the ellipse is traced the same way. Inside the window the folded 2 lies clockwise of the folded 1, and at −0.25 the ellipse’s two axes are 0.670 and 0.065 of the circle’s radius: nearly a line, and backwards.

A reading of the two principal factors #

Any two-by-two map can be split into a part that treats every direction alike — a scale and a turn — and a part that picks out a direction: a reflection in some line, scaled. Call their sizes $c$ and $d$. Then the two principal factors of the map are exactly

$$\sigma_1 = c + d, \qquad \sigma_2 = \lvert c - d \rvert,$$

and its signed area is $c^2 - d^2$. A similarity is a map with $d = 0$. The anisotropy is $(c + d)/\lvert c - d\rvert$, which is infinite at $c = d$ and comes down on either side; and the collapse is mirrored exactly when $d > c$, when the part that prefers a direction outweighs the part that does not.

That reading makes the sweep legible. On the sheared square grid at a pleat angle of −0.5 the two parts are 0.481 and 0.303, so the factors are 0.784 and 0.178 and the ratio 4.414. At −0.3 they are 0.320 and 0.363: the direction-picking part has overtaken, the factors are 0.683 and 0.042, and the area is negative. The shear does not decide how anisotropic the fold is; it supplies a direction-picking part, and the dials decide how that part compares with the rest. The triangular grid is the tiling for which a shear supplies none at all, which is a restatement of its exemption rather than an explanation of it.

The turn does not turn it over #

The other dial, swept the same way, bends the same curves without sending any through zero.

At a pleat angle of −0.5 the sheared square grid folds from a turn of 0.3 upward, and its area rises from 0.086 there to 0.43 at a turn of 0.7, while its ratio falls from 11.08 to 1.47. The honeycomb and rhombille under the same shear do not fold at all below a turn of about 0.32. Over the range where these patterns exist the turn only moves the collapse toward and away from the singular edge, and the anisotropy fades as the turn opens: at a turn of 0.7 every sheared pattern draws in within a factor of one and a half of evenly.

The contrast between the two dials is real and worth noticing. The dial that decides nothing swept the turn from one fence to the other and found every measurable property of a twist tessellation moving while its count of letterings stayed put. The collapse is one more property that moves with the turn — and the pleat angle, which that essay did not sweep, moves it further and across a boundary the turn does not reach.

Other maps come close and turn back #

The window of mirrored collapses is not a property of every image.

Under the stretch that doubled the honeycomb’s anisotropy at the census’s settings, the honeycomb group’s area dips to about 0.022 near a pleat angle of −0.3 — a ratio near nineteen — and turns back up without crossing zero, and somewhere between −0.7 and −0.8 the stretched honeycomb’s patterns stop folding at all. The square grid’s group lies on the drawn tilings’ curve, since a stretch along the grid’s own axes leaves every angle a right angle and the grid folds as a similarity. Under the general map, sampled at every hundredth of a radian, the square group comes within about a seven-hundredth of zero near −0.25 and does not cross.

So the mirror belongs to particular images, and a close approach to the singular edge is common. The census’s single setting, a pleat angle of −0.5, happened to sit well away from every one of these edges, which is why nothing in the first table looked singular.

What this measurement cannot show #

Whether the mirrored patterns can be folded out of paper. The collapse is read from the flat-folding model, which composes reflections panel by panel and asks that the result be consistent. It does not ask whether the layers can be stacked without passing through one another, and a mirrored arrangement is the case where that question bites hardest: every pleat in one direction has to carry its twist back past a neighbour, which is a great deal of paper stacked in a small place. The patterns in the window pass every vertex condition applied here. Whether a layer order exists for them is untested, and nothing here should be read as saying they fold.

Why the tilings fall in these three groups. The square grid and the elongated triangular tiling agree to every figure at every setting of either dial, and so do the honeycomb and the rhombille; the triangular grid’s images fold by exactly the map the unsheared tilings fold by. The sweep confirms those facts across the dials and explains none of them.

What happens between the flat sheet and the folded one. The map measured is between two states. If the twists’ arrangement stayed an affine image of the flat one all the way through a motion, a mirrored collapse could not be reached without passing through a state where a row of them lines up; whether a rigid folding path does that, or whether no rigid path exists at all, is a question about the motion that nothing here addresses.

And two dials, not every construction. The side distances are the ones the general construction solves for at a given turn and pleat angle. A different rule for placing the twist polygons could move every crossing reported here.

What the fit rests on #

A patch, clipped, with the collapse fitted to one class of twist. Every figure uses the fit the sheet draws in crooked introduced: the twists wholly on a sheet cut from a larger drawing, the largest class of them, and a two-by-two map fitted to their positions before and after folding, refused unless it rests on at least eight twists and misses none of them by more than a hundred-millionth of the patch. Near a crossing the map is nearly singular and still fits exactly; what grows there is the ratio, not the residual.

One period. The patterns are drawn with twists about a sixth of the sheet apart, and the period nobody measured is the reminder that the period a pattern folds to is a separate number from the one it is drawn at; a mirrored collapse makes the folded period’s direction reversed as well as its length changed. The collapse map does not depend on the period, since a smaller period is the same pattern at a smaller scale, but whether a pattern folds at all can depend on how a clipped patch meets its own rim.

And the crossings are found by halving on the sign of the fitted area between two sampled pleat angles, to far below a millionth of a radian, and reported to three figures.

How the claims were checked #

The groups are checked, not assumed. At three stops along every sweep the square grid and the elongated triangular tiling are fitted separately and required to give the same map to seven figures, and likewise the honeycomb and the rhombille; and the triangular grid under the map is required to fold by the same map as the square grid as drawn. Each sweep draws one member of each group and would stop at a stop where the pairs disagreed.

The mirrored window is required to exist where it is claimed: on the sheared square grid the sweep must find exactly two sign changes with negative area between them, and the crossings printed are the ones found.

The dialled ellipse is required to lie inside its circle, both principal factors below one at every stop, which is what keeps its canvas from moving; and every stop of the dial is a pattern that folds, since a stop that did not would be a slider with a broken frame.

Still open: whether a mirrored pattern has a layer order #

The claim this essay most needs checked is the one it cannot make. A mirrored twist tessellation is a crease pattern that satisfies every vertex condition and whose panels compose to a consistent folded position; whether its layers can be ordered is a finite question, and a layer-ordering search answers it for small patches. A patch of the sheared square grid at a pleat angle of −0.25, three twists by three, is the obvious first case, and the answer decides whether the window is a family of foldable objects nobody has made or a region where the flat-folding model’s positions describe paper passing through itself.

The derivation the previous essay named is still owed, and now it has a sharper target. Composing the per-pleat slides round a face of the tiling gives the collapse with no fitting in it; what it must reproduce is a crossing at a pleat angle of −0.348 on the sheared square grid, a number with no free parameter in it, and three groups that agree at every setting of both dials. A derivation that gets those right would say why the triangular grid is exempt.

Sideways from here, one vertex, repeated argues that a tiling is a way of making a material, and a material whose folded form can be tuned from even to lopsided to reflected by one angle is a component with a control on it rather than a fixed sheet. A shrink is two numbers is the essay that first treated how much smaller a folded sheet gets as a pair rather than one factor. A pair with a sign is a third thing — the folded sheet can be smaller, differently shaped, or reflected — and the Miura and the Yoshimura, whose shrinks that essay measured, have dials of their own that have never been run toward an edge.

The habit worth carrying is about measurements taken at one setting. Before reporting a number as a property of an object, run every dial the object has, and look for where the number stops existing rather than only where it changes. A ratio of 4.4 at one setting read as a fact about a sheared grid; swept, it was one point on a curve that runs to infinity twice.