What refused the mirror was the rim

A dial can turn the fold over swept a twist tessellation’s two free angles across five tilings and their linear images, and on the sheared square grid it found a window. Between pleat angles of −0.348 and −0.161 the collapse’s signed area is negative: the twists fold to a mirror image of their own flat arrangement. It ended on the question that finding raised and could not answer. Every pattern in the sweep satisfies every condition at every vertex, and the folded positions of its panels compose without disagreement; but a folded object is positions and an order, and nobody had asked whether these positions can be put in an order at all. If they cannot, the window is a region where the flat-folding model describes paper passing through itself. If they can, it is a family of objects nobody has folded.

The question is finite for a patch, and it has been asked. The answer is not either of the two the question expected. The mirrored patches are refused — and so are the unmirrored ones, at every pleat angle tried, on every patch size. The refusals have one shape, a small ring of flaps that cannot be stacked whichever way it is tried, and on this grid every one of them sits at the rim of the sheet. Remove the rim and every patch stacks, mirrored or not. The mirror was never the obstacle.

Every drawn patch is refused, and quickly #

A stacking is a choice, for every pair of panels that overlap in the folded state, of which lies above. It has to keep three kinds of rule. The two panels of a crease lie on the side of each other that the crease’s letter says. A panel that lies across a crease — overlapping both of the crease’s panels where they meet — is above both or below both, since it cannot pass between two pieces of paper joined along that line. And two creases folded in the same place cannot interleave. A set of choices that keeps every rule and closes no cycle is a folded state, and the layers can be put in that order.

The search for one places panels one at a time, walking outward across creases, and at each step tries both choices for every placed panel the new one overlaps. It checks each rule the moment every panel it mentions is down. When it runs out of choices it has proved that no stacking exists, because every choice was tried. When it runs out of budget it has proved nothing, and says so. The same search found folded states for all 144 rules of eight creases close the circle too, and on those the hard cases took more than two million steps.

On the sheared square grid it takes a few dozen. A patch with twists a sixth of the sheet apart has about two hundred panels, and at a pleat angle of −0.25, inside the mirrored window, the search exhausts every possibility in 61 steps. At −0.3 it takes 104 and at −0.2 it takes 54. At −0.5, well outside the window, where the collapse keeps its hand, it takes 399. A refusal in sixty steps is a refusal that never got far from where it started, and that means the contradiction is local: a handful of panels near the first one placed, not a property of the patch as a whole.

So the question became which handful. Taking the panels in the order the search places them, the shortest prefix it refuses is a set that cannot be stacked; dropping its panels one at a time and keeping each drop the search still refuses leaves a set from which nothing can be removed. On every sheared patch tried that set has three panels.

Two flaps that both need to be on top #

The three are a strip and its two neighbours across opposite edges. The two creases joining them carry the same letter, so both neighbours fold onto the same face of the strip, and in the sheared geometry each neighbour is long enough, once folded, to reach past the strip’s far edge — across the crease that joins the other neighbour to it.

The rule about a panel lying across a crease does the rest. The first flap lies across the crease between the strip and the second flap, so it must be above both of them or below both; it is above the strip, so it is above the second flap. The second flap lies across the first one’s crease, and by the same rule it is above the first. Each must be above the other, and no order of three panels does that. This arrangement is called a pinch here: two flaps folded onto one face of a panel, each lying across the other’s crease. The overhang on the pinch drawn is large — each flap reaches across the other’s crease by a quarter to two thirds of the strip’s own area — so this is not a grazing contact that a slightly thicker pencil would remove.

There is a picture of it that needs no rules. Close the strip’s two creases one after the other. The first flap comes down and covers the strip’s far edge, so the second flap’s hinge is now under it; to lie flat the second must pass over the first at its hinge and under it at its far end, which is one flap passing through another. The other order fails the same way.

No condition at a vertex can see it #

Every vertex of these patches satisfies Kawasaki’s angle condition, Maekawa’s count and the smallest-sector lemma, which are the conditions a single vertex is checked against. A pinch passes all of them, and the reason is where it lives. A pinch’s two flaps are joined to the strip along its two long edges, and those edges share no vertex. Nothing at any one point of the sheet sees both flaps, so no condition stated at a point can forbid them meeting.

The comparison with the rule that breaks the count is close enough to be worth drawing out. There a hundred and twenty of the waterbomb’s repeating rules sent four panels round a vertex in a circle, each lying above the next, and every one of those circles broke Maekawa’s count at the vertex it went round: a cycle of panels round a vertex is a thing a vertex condition catches. A pinch is the same kind of cycle turned the other way out — a cycle of flaps round a panel rather than of panels round a vertex — and nothing at a vertex can catch it. Vertex and panel are the two kinds of cell a crease pattern has, and the local theory of flat folding is a theory of the first.

That is also why the construction let pinches through. Its letters are found by propagating the vertex conditions from crease to crease, and then tested for a forced cycle — a loop of panels whose creases alone put each above the next, which an earlier lettering of the square twist’s patch was caught carrying round twenty-eight panels. A pinch is not one of those cycles: its contradiction needs the folded outlines, and the creases’ letters alone never mention it. The letters passed every test the construction had, and the folded positions passed every check the sheet draws in crooked made of them.

The rings live on the rim #

The pinch drawn above sits one crease in from the edge of the sheet, and that turns out to be the rule rather than the example. Scanning a patch for every pinch directly — every panel, every pair of its neighbours on one face, each tested for reaching across the other’s crease by clipping its folded outline exactly — finds seven on the patch above, and every one of the seven includes a panel on the rim or a panel one crease from it. The patch is six creases deep. Nothing deeper than one crease is involved.

Drawn finer and finer on the same sheet, the patch grows from 51 panels to 715 and from three creases deep to twelve, and the count of pinches grows with it — from five to fifteen at −0.25, from two to eleven at −0.5. But it grows the way the rim grows, not the way the area does. Fourteen times as many panels bring three times as many pinches, and at no size and neither angle does a pinch reach a panel two creases in. A drawn patch of this grid has a ring of pinches round its edge and a clean interior.

The interior is clean because it is lettered by the conditions at its vertices, and away from the edge those conditions settle the strips: the strip at the middle of every pinch, at every size and both angles, lies within two creases of the rim, and deeper in the patch the strips of the same shape carry one mountain and one valley — a pleat, folding one neighbour up and the other down, where the two cannot meet. Near the edge the vertices that would have settled a strip are cut off by the sheet, the propagation is free to give both its creases one letter, and on the sheared grid it sometimes does. What the rim was doing found that a sheet’s edge decides how hard its lettering is to find. Here the same edge decides which lettering is found.

Cut two creases in, and the mirror stacks #

If the refusals are all at the rim, the patch inside the rim should stack. That is a precise experiment, because the rules a stacking keeps mention only panels: cut the sheet along the creases that separate the rim panels and their neighbours from the rest, and the rules of the smaller sheet are exactly the rules of the larger one that mention none of the removed panels. Every stacking of the larger sheet restricts to one of the smaller, so the smaller sheet stacking is a necessary condition for the larger, and it is the question the window asked, with the rim’s letters set aside.

Every one of them stacks. At twists a sixth of the sheet apart, the cut patches keep 79 to 83 panels and the search finds a stacking in between about four thousand and twenty-eight thousand steps; at twists an eighth apart they keep 167 to 199 panels and it takes up to a quarter of a million. Six of the eight are inside the mirrored window, with the collapse’s signed area between −0.029 and −0.044. Their layers have an order, and the order keeps every rule. The mirrored twist tessellation on the sheared square grid is a family of foldable objects — positions that paper can occupy, stacked in a way paper can be stacked, in the idealisation the rules describe.

A ring is the whole of the refusal #

The pinch is the case of two flaps, and nothing in the argument against it depends on there being two. Flaps folded onto one face of a panel, each lying across the crease of the next, must each lie farther from the panel than the next — and if the chain closes, there is no first flap to place. Three flaps arranged that way make a pinwheel, and four or five make longer rings. Any ring of flaps round a panel cannot be stacked, and that half of what follows is an argument rather than a measurement.

The other half is a measurement. The sheared square grid is one of five tilings the earlier essays swept, each folded as drawn and under the same shear, and for every patch the scan for rings and the search for a stacking can be run side by side — on the whole patch, and on the patch with its outer two rings of panels cut away.

At a pleat angle of −0.25, all twenty readings are decided, and in all twenty a patch stacks exactly when it carries no ring. That is the claim the census makes and it is stronger than the square grid’s: nothing else refuses these patches. Every refusal the search found, on every tiling and at either map, has a ring of flaps beside it, and every patch with no ring stacked.

What changes from tiling to tiling is where the rings sit. On the square grid and the elongated triangular tiling as drawn there are none at all. On the triangular grid and the honeycomb as drawn there are a few, all inside the band the cut removes, so the drawn patch is refused and the cut patch stacks. Under the shear the honeycomb carries sixty-four pinches, thirty of them deeper than the cut reaches; the triangular grid carries ninety-six pinches and twenty-six pinwheels, and more than half of them survive the cut. On those patches the rings are not a fringe. They are part of the pattern’s interior, and cutting two creases in leaves a sheet still full of them.

The case the window asked about has a twin, and the twin goes the other way. The elongated triangular tiling under the same shear collapses by the same map as the square grid, to seven figures, which is how a dial can turn the fold over grouped them; it is mirrored at −0.25 too. But it carries twenty-nine pinches and a pinwheel, twelve of the pinches survive the cut, and the cut patch is refused. Two patterns that fold to the same positions, one stacking and one not, which is the clearest possible statement that the collapse map, mirrored or otherwise, does not decide whether a pattern can be folded.

Three flaps, four, five #

The pinwheel is worth looking at, because it is the ring that appears on a tiling with no shear at all. On the rhombille tiling as drawn, each twist polygon at a vertex where three rhombi meet is a small triangle, and its three ring creases carry one letter, so all three neighbours fold onto one face of it.

Folded, each of the three lies across the crease of the next one round. The first must be farther from the triangle than the second, the second than the third, and the third than the first: a cycle of three, and the same contradiction as the pinch. The rhombille as drawn carries forty-two of these and twenty-one rings of four at −0.25, twenty and fourteen of them inside the cut, and the cut patch is refused. That makes the rhombille the one tiling of the five whose twist tessellation fails to stack at this pleat angle with no shear at all — the pattern itself, not its edge.

It also says what the mirrored collapse was standing in for. Whether paper passes through itself is not decided by the collapse at all: it is decided by how far each flap reaches when it folds, which the pleat angle sets directly, and the collapse’s sign is a separate consequence of the same angle. On the sheared square grid the two share a window, and the rings do not care which side of it they are on.

The same verdict at a second pleat angle #

A census at one setting describes that setting, and the dial that decides nothing is a reminder that a dial can move a reading a great deal or not at all. The pleat angle sets every flap’s reach, so the census was run again at −0.5, outside the mirrored window on every tiling.

The verdicts move — the honeycomb sheared now carries one pinch, at the rim, and stacks once cut; the triangular grid as drawn carries a single ring of five — but the rule does not. Eighteen of the twenty readings are decided and all eighteen agree. The two that are not are whole patches with no ring, on the triangular grid sheared and the honeycomb as drawn, where the search ran out of budget before finding a stacking; both stack once cut. Thirty-eight decided readings across two pleat angles and five tilings, and in every one the stacking exists exactly when no ring of flaps does.

What the stacking does not show #

A stacking is an order, not a motion. A patch that stacks has a flat folded state with its layers in a consistent order; whether the sheet can be brought there from flat without its panels colliding on the way is a different question, and for twist tessellations, which fold all at once rather than crease by crease, it is the question a folder actually meets. Nothing here addresses it.

And the converse is measured, not proved. That a ring of flaps cannot be stacked follows from the rule about panels lying across creases. That a patch with no ring can be stacked is a regularity observed on thirty-eight decided cases and contradicted by none; it may fail on a tiling or a setting not tried, through two creases folded in the same place, which these patches never produce and which the search’s third rule exists for.

A refusal is about these letters. The rings at the rim are made by the letters the construction gives the strips there, and those letters are one choice among several the vertex conditions allow. A drawn patch refused here might stack under another lettering of the same creases, and nothing above says it would not.

What the stacking rests on #

Zero thickness and rigid panels. Paper with thickness changes the question in both directions: a ring of flaps that reaches across a crease by a few thousandths of a panel might be relieved by the crease’s own radius, and how many layers of a real sheet can lie in one place is a question the density a paper allows asks of a pattern separately, with an answer in millimetres that this model has no way to give.

A patch, clipped. Every pattern here is a square sheet cut out of an unbounded drawing, as cutting a patch out of a plane set out, and the rim is where the cut falls. A different placement of the same drawing moves the rim, and with it every pinch on this grid; the interior’s verdict would not move, because the interior is the same pattern wherever the cut falls. An overhang smaller than a ten-thousandth of the panel it lies on is not counted as a reach, which removes one contact on the sheared square grid at −0.25 measured at under two millionths of its strip — a coincidence of float arithmetic at a clipped corner, not a flap.

And the cut is two creases: every panel on the rim and every panel sharing a crease with one. One is not enough on the sheared square grid, where some pinches lie wholly one crease in; two is the smallest that clears every ring found on that grid at every size.

Whose patterns these are #

Twist tessellations are Shuzo Fujimoto’s, who worked them out in the 1970s and published them as patterns to fold, and they were being made independently elsewhere within a decade. The construction of one on an arbitrary tiling — a twist polygon at every vertex, a pleat along every edge — is the one Robert Lang and Alex Bateman analysed when they showed in 2018 that a tiling carries a flat twist tessellation exactly when it is a spider web. Their theorem is about the folded positions: that the pleats close. Whether the positions stack is not part of it, and nothing here contradicts it. The pleat angle as a free dial is the freedom one number where the corners wanted four set out, and it is that freedom which lets the sheared images fold at all.

The three rules a stacking keeps are the standard conditions of flat-folding theory for ordering the layers of a folded state, and Marshall Bern and Barry Hayes showed in 1996 that deciding whether a crease pattern has any such order is NP-hard in general — which is why a refusal in sixty steps was the clue it was, since a hard instance does not give itself up that fast.

How each answer was checked #

Every refusal is exhaustive, the search having tried every choice; a run that stopped on its budget is reported as undecided, never as a refusal, and two such runs appear in the second census. Every stacking found keeps every rule, checked as each panel is placed.

The scan for rings is independent of the search. It reads folded outlines and crease positions and clips one against the other exactly; the search reads only the list of rules. The two were run side by side on every patch above and the census requires them to agree wherever the search decided. They were checked against each other a second way before any figure was drawn: the smallest set the search refuses on the sheared square grid was found by deletion, without the scan, and it is a pinch.

The pinch’s own conditions are required of each drawing — one letter on every crease joining a flap to the middle panel, a positive overhang across the next — and the rim’s rule is required at every stop of the first figure’s dial, from −0.85 to −0.05.

Still open: a lettering for the rim, and the converse #

The cut patches stack, which settles the question about the mirror. It leaves the drawn patch. A lettering of the rim’s strips that makes every one a pleat — one mountain and one valley — would remove every pinch on the sheared square grid, and whether such a lettering satisfies the vertex conditions at the rim’s own vertices is a finite question with a small number of free creases in it. Drawing letterings at random from the whole patch is not the way to find one: sixty independent draws of the sheared square patch at −0.25 all forced a cycle among the creases’ own orders before a pinch could even be looked for. A search that fixed the interior’s letters and varied only the rim’s would be.

The second lead is the converse. Thirty-eight decided readings say that a patch with no ring of flaps stacks, and the argument says only that a ring forbids it. A proof that rings are the only obstruction, for twist tessellations, would turn the census into a test that needs no search at all — scan the folded outlines, find no ring, and know the patch stacks. The place it would fail, if it fails, is two creases folded in the same place, the one rule no ring involves.

Sideways from here, the rhombille is the open case. Its twist tessellation is refused as drawn, with no shear and no rim to blame, by pinwheels round its triangular twists at a pleat angle of −0.25, and a metamaterial with no edge found the way to ask a repeating pattern about itself with the rim removed entirely: glue the edges. A glued rhombille patch would say whether the pinwheels are the pattern’s or the patch’s, and the pleat angle at which they first appear would be a boundary on the dial that the collapse’s sign knows nothing about.

The habit worth carrying is about suspects. When an object is refused and it has one obvious peculiarity, check the refusal on a case without the peculiarity before blaming it. The mirrored collapse was the strange thing about this window and the obvious cause; the unmirrored patches next to it were refused just as fast, for a reason at the edge of the sheet that the mirror had nothing to do with.