The sheet draws in crooked

Every twist writes an equilibrium ended with a prediction, which is a better place to end than a conclusion. The reasoning was short. A pleat is two parallel creases, so folding it slides the twist beyond it relative to the twist before it; the size of that slide is set by the pleat’s width, which is the length of the twist’s side, which is the edge’s own length times the weight the pattern puts on it. A tiling whose weights are all equal therefore contributes the same multiple of every edge to the collapse, and one whose weights differ does not — so the second kind should fold to a different shape rather than to a smaller copy.

Folding it flat is one similarity had measured the first kind and found one answer for all five tilings as drawn: multiply the plane by a scale, turn it by an angle, and that is where every cell of paper goes. Those five all carry equal weights. The patterns that now exist on the sheared and stretched tilings mostly do not, so there was something definite to look for.

The looking was worth doing twice over. The first half of the prediction is right, and the finding is the one this line of argument has been pointing at since the loops closed on tilings that would not fold. The second half — the part that said which patterns — is wrong, and it is wrong in both directions at once.

Measuring a collapse rather than describing one #

What a collapse does to the plane can be read off the pattern without any theory about pleats. Each twist polygon keeps its orientation through the fold, because crossing a pleat is a reflection in one crease followed by a reflection in a parallel one, and two such reflections compose to a translation. So a twist that sits at one place on the flat sheet sits at another place in the folded one, and the collapse is the map between those two sets of places.

Fitting it needs one precaution. A tiling with two kinds of vertex puts its two kinds of twist at different offsets from their own lattice points, so a single map fitted across both kinds reports a residual that is about the offset rather than about the map. Restricting the fit to twists of one class removes that, and what is left is either a linear map or nothing. It is a linear map. Across the twenty patterns the fit misses no twist by more than a part in $10^{13}$ of the patch’s own width, which is the accuracy of the folded positions and not a tolerance: the collapse of a twist tessellation is exactly affine, and that is a fact rather than an approximation.

Having a matrix, the two questions are its singular values — how far the sheet draws in along each of two perpendicular directions — and their ratio. A ratio of one is a similarity.

The five as drawn agree, again, and more exactly than before #

On the five tilings as drawn, the general construction’s collapse is the same map on all five: principal factors of 0.5016 both ways, a turn of $-30.320^\circ$, and an area that comes to 0.25159 of the flat sheet. Ten figures of agreement across five tilings whose vertices have degrees three, four, five and six.

That reproduces folding it flat is one similarity and extends it in the way that matters. That essay’s number was 0.410373441 at a turn of 36.62°, measured on the older construction with its own scaling rule. The number here is different because the construction is different — the older rule fixes the side distances up to a common factor and then scales them to fill the available room, while this one fixes them outright from the edge lengths and the pleat angle. Running the older construction at the same turn gives 0.462916, a third number. What survives all three is not the value but the coincidence: whichever rule places the polygons, the five tilings collapse by the same map as each other, and only the shared value moves.

That is a stronger statement than the original, because a coincidence that holds under three different placement rules is unlikely to be about the rule.

It also says something about where the coincidence comes from, by saying where it does not. The five tilings agree with each other and not with themselves across constructions, so the shared quantity is not the polygons, which differ between the three rules, and not the turn, which is the same in two of the three and gives different answers. What the five share is the tiling’s own symmetry: every vertex of each of them is equiangular, which every twist writes an equilibrium found to be exactly the condition under which the pattern writes the same weight on every edge. That looked like the explanation until the images were measured.

Ten images that fold crooked #

The images are where nothing measured here has been before. Ten of the fifteen collapse by a map whose two principal factors differ, and the differences are not small: 2.818 on the sheared honeycomb and the sheared rhombille, 3.542 on two of the general images, 4.414 on the sheared square grid and the sheared elongated tiling, and 5.386 on the stretched honeycomb and the stretched rhombille, where the sheet draws in to 0.150 of its length one way and 0.809 the other.

These are the first patterns measured here whose folded form is a different shape from the flat one. Every fold looked at until now — accordions, Miuras, waterbombs, twists — has either kept a direction alone or drawn in by a pair of factors along fixed axes, and a shrink is two numbers is the essay that separated those cases. A twist tessellation was the family member that drew in equally both ways. That is now a property of the twist tessellations that had been built rather than of twist tessellations.

A second thing changes with them and it has been counted before. The period nobody measured found that a repeating pattern has two periods — the one it is drawn at and the one it folds to — and that the second had never been recorded anywhere, although on one family the two differ by a factor of three. A crooked collapse makes that worse in a specific way: the folded period is no longer one number times the drawn one, because the two directions of the unit cell are multiplied by different factors and turned by different amounts. A pattern whose collapse has a ratio of 5.386 has a folded cell whose proportions are nothing like its drawn cell’s, and quoting one number for how much smaller it got would be quoting an average of two things that are five times apart.

The area tells a second story. The determinant of the collapse is how much of the flat sheet’s area survives, and it falls as the anisotropy rises — 0.2516 on the similarities, 0.1827 at a ratio of 2.818, 0.1217 at 5.386. A crooked collapse is a tighter one, which is not obvious and is worth a designer’s attention: the same tiling sheared draws in more paper for the same turn and the same pleat angle, and it does so unevenly.

The prediction, and where it breaks #

The weights were supposed to say which patterns these are. They do not.

The stretched triangular grid writes weights spread six to one and collapses by a similarity — to ten figures, the same 0.5016 and $-30.320^\circ$ as the unsheared case. So unequal weights do not force a crooked collapse.

The stretched honeycomb writes the same weight on every edge and collapses at a ratio of 5.386 — the largest anisotropy in the table. So equal weights do not force a similarity.

Those are the two directions of the claim and both fail on a measured pattern. The failure is not a matter of degree, either: the two most extreme entries in each table belong to the opposite entry in the other. Whatever the weights govern, it is not this.

The reasoning that produced the prediction can be located precisely, which is more useful than abandoning it. It treated the collapse as a sum of per-edge displacements and assumed each displacement was the edge times its own weight. The first part is right — the fold is a composition of translations, one per pleat crossed. The second is not, because a pleat’s translation is perpendicular to its creases rather than along its edge, and the creases run at the pleat angle to the edge rather than along it. So each edge contributes a displacement turned out of its own direction by an amount that depends on where the sheet is being crossed, and summing those is not the same as summing weighted edges. An argument that gets the mechanism right and the direction wrong produces a prediction that is testable and false, which is the best kind to have made.

It is worth saying plainly that the prediction was published before it was tested, and that this is the outcome that makes publishing it worthwhile. A prediction kept private is indistinguishable from a hunch; one written down with the arithmetic that produced it can be located when it fails, and the locating is where the understanding is. The mechanism — a fold is a composition of translations, one per pleat — survives intact and is used below. What did not survive is a claim about the direction of each translation, which was never checked because it was never noticed as a claim.

What the measurement does say #

Three regularities survive, and all three were found by looking at the table rather than by being looked for.

The triangular grid is a similarity under every map tried — as drawn, sheared, stretched and under a general matrix, all four at 0.5016 and $-30.320^\circ$ to ten figures. No other tiling here does that, and nothing in the account above suggests why one should.

The anisotropy depends on the map and on almost nothing else. The square grid and the elongated triangular tiling return identical values under all four maps; the honeycomb and the rhombille return identical values under all four maps, different from the first pair’s. Agreement to ten figures between two tilings whose vertex degrees are four and five, and between two whose degrees are three and a mix of three and six, is not a coincidence anybody should accept quietly.

The collapse’s turn moves with its anisotropy. On every similarity here the rotation is $-30.320^\circ$; at a ratio of 2.818 it is $-22.728^\circ$, at 4.414 it is $-17.747^\circ$, and at 5.386 it is $-15.695^\circ$. The turn and the ratio move together across all twenty patterns, monotonically, which is one more regularity than a fitted matrix had any obligation to supply.

And the value is not a property of how far the map moves things. A shear of 0.3 gives 4.414 on the square grid and 2.818 on the honeycomb; a stretch of 1.5 gives 1.000 on the square grid and 5.386 on the honeycomb. The same map is the strongest distorter of one pair and has no effect at all on the other.

What the fit rests on #

The construction is the general one. Every pattern measured here places its twist polygons with one side distance per incident edge, which is what one number where the corners wanted four established the corner conditions actually allow. On the five tilings as drawn that rule returns the same polygons the older one does, so the similarity measured there is the older construction’s too; on the images the older construction has no pattern to measure, since closing the loops is not folding is the record of its failing every one of them.

A patch, clipped. The twists measured are those lying wholly on a sheet cut from a larger drawing, so the fit is over between eleven and forty-two positions depending on the tiling and the map. A fit over fewer than eight returns a matrix that moves with the clipping, and is refused rather than reported — which is why the patches here are finer than the ones the patterns are usually drawn at.

One class of vertex. Where a tiling has several, the largest class is used and the others are discarded. That is what makes the residual meaningful, and it means the map is fitted to a sublattice of the pattern rather than to the pattern.

One turn and one pleat angle. Everything here is at a turn of 0.42 radians and a pleat angle of $-0.5$. The principal factors certainly depend on both; whether the ratio does is untested, and it is the first thing to check before quoting any of these numbers as a property of a tiling.

And the folded positions come from composing reflections, which is the flat-folding model and not a simulation of paper. A pattern that passes every vertex condition and composes to consistent panel positions has a folded state in that sense; whether the layers can be ordered without two of them passing through each other is a separate question that nothing here asks.

What the ellipses cannot show #

They do not say the anisotropy is useful. A sheet that draws in five times as much one way as the other is a description, not a material. Turning it into one would need the folded thickness, the stiffness along each axis and the behaviour part-way through the fold, and none of those is a question about a flat-folded state.

They do not identify the principal directions with anything in the tiling. The matrix has two of them and they are reported only through their factors. Whether they line up with the map’s own axes, with the tiling’s edges, or with neither, is visible in the matrix and has not been read out.

They say nothing about intermediate states. The map measured is between two states, flat and fully closed. A rigid-folding path between them would have its own map at every angle, and a sheet that draws in unevenly at the end may do so evenly on the way or not at all.

And the five similarities among the images may not be five. Three of them are the triangular grid’s, and two are stretches of tilings built on a square lattice, where the stretch leaves every angle a right angle. That is a small and structured set, and a sixth map would be a better test of it than any argument about the five.

Still open: which map, and why these pairs #

The measurement leaves one question sharp enough to be worth stating as a computation rather than as a direction.

The anisotropy is a function of the linear map and of which of three groups the tiling falls in, and the groups agree to ten figures across tilings with different vertex degrees. That is a strong enough regularity to be derivable, and deriving it means writing the collapse as a product of the per-pleat translations rather than fitting it — which is the calculation the failed prediction was reaching for and got the geometry of wrong. The per-pleat translation is perpendicular to the pleat’s creases and twice the perpendicular distance between them; both of those are known exactly from the side distances the construction solves for; and composing them round a face of the tiling is a finite sum with no fitting in it. What comes out would say which map, and it would say why the triangular grid is exempt.

A second computation is available and cheaper. The dial that decides nothing ran the turn from one fence to the other and found every measurable property of a twist tessellation moving while the count of its letterings stayed put. The collapse is now one more measurable property and it has two dials rather than one — the turn and the pleat angle. Whether the anisotropy is a function of the tiling and the map alone, with both dials only setting the overall scale, is settled by running the same table at a second turn. If it is, the ratio becomes a number that belongs to the pair (tiling, map), which is the form in which it could be looked up rather than computed.

Sideways from here, the anisotropic patterns are the ones a designer would want and nobody has drawn. A twist tessellation is chosen for a deployable surface partly because it draws in the same way in every direction, which is convenient and is also a constraint; a family that draws in five to one is a different component. The patterns exist, they satisfy every condition applied to a crease pattern here, and they have never been folded out of paper by anybody. One vertex, repeated is the argument that a tiling is a way of making a material rather than a way of making a picture; a material that contracts unevenly is a different material, and the route to it here is a shear applied to a tiling somebody already folds.

There is also a question about the found patterns. Patterns nobody designed is about the ones a sheet arrives at on its own, by buckling, without anybody choosing them — and a crushed cylinder’s diamonds are not equiangular, because the cylinder’s proportions decide them. Whether a found pattern’s collapse is a similarity is therefore not something to assume, and it is measurable by exactly the fit used here.

The habit worth carrying is about predictions that come out of a mechanism. A mechanism gets the list of contributions right and the shape of each one wrong very easily, and the two errors look identical from inside the argument: both produce a sum over the same terms. The way to tell them apart is to check a case the mechanism says nothing about — here, a tiling whose weights are extreme and whose collapse should therefore have been extreme — and the check is cheap exactly because the prediction is specific.