The edge is what makes it hard

A sheet’s shape is usually treated as a design decision: a square because that is what paper comes as, a rectangle because a proportion is wanted, a shape chosen to suit a base. It is not usually treated as the thing that decides how hard a crease pattern is to reason about.

It turns out to be exactly that, and the evidence is a list of families that all behave the same way and one that does not.

Five families that behave identically #

The orthogonal grid a box-pleated design lives on, at nine sizes from four panels to two hundred and fifty-six: one step per panel, no backtracking anywhere.

Six crumples of deepening severity, produced by folding a sheet at random and flattening: one step per panel less a constant, no backtracking.

The tapered leaf at four widths, the four corrugation families — Miura, leaf, Yoshimura, waterbomb — and eight fold-and-cut patterns: the same, every one of them.

Forty-six patterns from five constructions, spanning a factor of thirty in size, from the most regular object the collection can draw to the least, and the number of decisions any of their searches has ever taken back is zero.

It is worth naming the alternative explanations before dismissing them, because each of the five families is in the list to kill one.

Size is killed by the grid, which reaches two hundred and fifty-six panels — larger than any tessellation patch here — and stays exactly linear.

Disorder is killed by the crumples, which have no repeating structure, no symmetry and no design, and which behave identically to the most regular object in the collection.

Irregularity of the vertices is killed by the quadrilateral meshes, built so that no two of their vertices are alike, whose letters are found in five or six steps apiece.

The construction is killed by the corrugations, which come from the same body of ideas as the patches and are drawn by the same machinery.

Four candidate explanations, four families that refute them, and what is left standing is the property none of the five has.

One family that does not #

The tessellation patches. Five printed ones and a hundred and twenty over a grid of parameters, and on these a search’s cost is genuinely variable: one of them runs from eighty-six steps to fifteen thousand depending on nothing but where it starts, nine of them have no consistent lettering at all, and three more needed a quarter of a million steps to establish that.

They are not the largest patterns here — the sixteen-division grid has two hundred and fifty-six panels against the rhombille patch’s hundred and fifty-seven. They are not the least regular; a crumple has no repeating structure at all and a patch has one everywhere except at its rim. They are not built by a stranger construction; the corrugations and the patches come from the same family of ideas.

What they have that nothing else here has is a boundary that cuts the pattern. A grid, a crumple, a corrugation and a fold-and-cut pattern all fill their sheet: their creases end at the paper’s edge because the construction put them there, and the vertices near the edge are vertices the construction intended. A patch is an infinite tessellation with a square cut out of it, and its rim runs through the middle of the pattern’s own structure.

Why a rim changes the search #

The mechanism is short. The search propagates: writing a letter on one crease forces letters on others through the conditions at a shared vertex, and the forcing sweeps outward.

A vertex in the interior of a pattern has all its creases present, so its conditions are fully constrained and the propagation through it is decisive — a single known letter usually determines the rest. A vertex near a rim has creases that leave the paper, so the conditions there involve fewer creases and constrain less; and a crease with an end on the rim answers to one vertex instead of two.

So the interior of any pattern propagates cleanly and the boundary does not, and a search’s real work is concentrated entirely near the edge. On a grid the edge is a thin frame around a large interior. On a tessellation patch, most of the pattern is edge — the rim cuts through polygons and pleats, leaving a ring of partially-present vertices all the way round — and the region where the propagation is weak is most of the sheet.

The prediction this makes #

If the account is right, then a patch’s difficulty should fall as its rim becomes a smaller fraction of it — a finer pitch, more twists on the same sheet, proportionally less boundary.

It does, though not cleanly, because changing the pitch also changes the pattern’s size and the count of polygons on the sheet moves in integer jumps rather than smoothly. What can be said is that no patch anywhere in the sweep has a difficulty that comes from its interior: every search cost above one step per panel, on every patch measured, is attributable to decisions taken among the creases the rim has left underdetermined.

The stronger version of the prediction is available and has not been tested: a tessellation on a sheet with no boundary — a torus, or a patch with its opposite edges identified — should search at one step per panel like everything else. Building one is straightforward and it is the obvious next measurement.

There is a way of putting a number on it that makes the mechanism concrete. On a sixteen-division grid, the vertices whose creases all lie on the sheet are two hundred and twenty-five of two hundred and twenty-five — every interior vertex is complete, and the boundary contributes creases that end on the rim and nothing else. On a tessellation patch, the vertices near the rim have polygons and pleats cut through them, and the ring of such vertices is several deep because a twist polygon is several creases wide.

So the ratio the essay is really about is not perimeter to area. It is the number of vertices whose conditions are fully constrained against the number that are not, and on a grid that ratio is enormous while on a patch it is close to one. Where the paper stops established that most of a patch is edge; this is the consequence of that for anything that has to reason about the pattern.

The prediction has a rate, and the rate discriminates #

“Not cleanly” is the right report of what was measured and the account can be pushed to something sharper, because it says which quantity the excess should track.

If the work is done at the rim, the excess above one step per panel should scale with the rim’s unit count rather than the interior’s. A patch $n$ units across has a rim going as $n$ and panels going as $n^2$, so

$$\frac{\text{excess steps}}{\text{panels}} \;\propto\; \frac{1}{n}.$$

Doubling the pitch should therefore halve the excess per panel while roughly doubling the excess in total. That is a rate rather than a direction, and it is measurable on the sweep that already exists.

Or an exponent, if the rim’s decisions interact #

The same account permits a second and much worse scaling, and separating them is what the measurement would be for.

Linear growth assumes the rim’s underdetermined creases are decided independently — each a local choice, contradicted locally, costing a bounded amount. If instead a wrong letter at one rim vertex is only contradicted by a wrong letter somewhere else on the rim, the search must explore combinations, and the excess grows as $2^{cn}$ rather than as $n$.

The rhombille patch already leans toward the second. Its cost runs from 86 steps to 15,000 on one pattern with nothing changing but a seed — a factor of 174, or between seven and eight binary decisions’ worth of tree — which is a great deal of exploration for a set of choices that were supposed to be contradicted where they were made.

So the ladder’s next measurement is a slope, not another patch. Excess per panel against pitch, on the sweep already run: a straight line through the origin on a reciprocal axis confirms independent rim decisions, and anything curving upward says the rim talks to itself, which would explain the tail rather than merely locating it.

What this says about choosing a sheet #

For a designer the reading is narrow and useful.

A pattern that fills its sheet by construction — a corrugation designed at the size it will be folded at, a box-pleated base drawn on the grid it occupies — has no reasoning difficulty in it at all. Its letters can be found mechanically and its difficulties are elsewhere: in the layers, in the thickness, in the hands.

A pattern cut out of something larger has a rim’s worth of underdetermined structure, and everything about it is harder to establish: its letters may not exist, finding them may cost, and the vertices where the trouble lives are exactly the ones no theorem fully constrains. The vertices nobody checks is where that was first noticed as a gap in the checking; here it appears as a cost.

That is an argument for designing a tessellation to a sheet rather than clipping one to it, and the argument is not aesthetic. A tessellation laid out so that its unit cell divides the sheet has no cut vertices at all, and the resulting pattern is in the first category.

The same reading, in three older results #

Once stated, the rim turns out to have been the answer to three things this collection had already recorded separately.

The crossings. An earlier construction drew a patch by extending each pleat to the rim when its neighbour was off the paper, and four of the five tilings acquired between five and eighteen places where two creases passed through one another with no vertex there. Every vertex condition passed at every listed vertex, because a crossing is not a listed vertex. The fault was entirely at the boundary.

The fragments. A clip that catches a pleat within a hair of a corner leaves a crease eight millionths of a sheet long, and ten patches of a hundred and twenty carry one. Again, entirely at the boundary, and again invisible to every check.

The region with no letterings. Nine patches have no consistent lettering at all, and while the mechanism is about which sector at a vertex is smallest, the patches are patches — the same construction on a boundaryless plane has a lettering by symmetry, since the tessellation’s own unit cell provides one.

Three findings, three different instruments, one location. That is a stronger reason to believe the account than the search costs are, because none of the three was looking for it.

What would settle it #

The measurement that would turn this from a well-supported reading into a result is short to describe and has not been done.

Take one tessellation, at one turn angle and one pitch, and build it three ways. Clipped to a square sheet, as every patch here is. Fitted to a sheet whose side is an exact multiple of the unit cell, so the rim falls between units and cuts nothing. And periodic, with opposite edges identified so there is no rim at all.

The three have the same interior, the same vertex template, the same number of polygons to within one, and differ only in what happens at the edge. If the account above is right, the first should show a spread in its search cost and the other two should not, and the third should be exactly one step per panel.

That is a controlled comparison rather than the family-by-family argument this essay makes, and it is cheap on the four uniform tilings. The reason it is described rather than reported is that the third construction does not exist here — identifying opposite edges of a sheet is a change to what a sheet is, and every piece of machinery in this collection assumes a disc of paper with a rim. A sheet with a hole in it was the last time that assumption was disturbed, and it took a considerable amount of work.

Which theorem was checked, and how #

The claim that the five easy families never backtrack is asserted rather than observed: every pattern in every ladder is required to cost no more than one step per panel, and the assertion fails on the first that does not. That is stronger than a straight line through the points, because a search can visit exactly n nodes while taking a wrong turn and recovering.

The patches’ behaviour is asserted from the other side: on the one patch that has a spread, the coin’s worst run is required to be many times its median and a constant order’s cost is required to be identical on every seed. Both halves, since either alone would be an advertisement.

Every lettering counted anywhere is written back onto its pattern and put past the four conditions at every interior vertex and a folded sheet rebuilt from scratch.

What the picture cannot show #

The boundary’s effect directly. Nothing here isolates it — the comparison is between families that have a cutting rim and families that do not, and those families differ in other ways too. What makes the reading credible is that every other candidate explanation has a family that refutes it: size is refuted by the grid, disorder by the crumples, irregularity by the meshes, and construction by the corrugations, which come from the same place as the patches.

That is a reasonable standard of evidence and it is not a controlled experiment. The controlled experiment is the boundaryless tessellation, and it has not been run.

Nor does “harder” mean hard. The worst patch in the collection is settled in eighty steps by a search taking its decisions in a fixed order. The difficulty being described is a difficulty of a factor of two hundred within a range that is entirely tractable, and the reason it is worth naming is that it was invisible until somebody asked which patterns were different rather than how large the numbers were.

What a folder already knows #

There is a hand version of this and it is completely ordinary, which is the best evidence that the reading is right rather than clever.

Ask anybody who has folded a tessellation what the difficult part is and they will say the edges. The middle of a tessellation is rhythmic — the same collapse, repeated, with the paper telling the hands where to go — and the rim is where the pattern is cut off mid-unit, where a pleat has nowhere to go, and where the folder has to decide what to do with paper the pattern did not plan for.

That is the same statement as everything above, in the register a folder uses. The interior is determined and the boundary is not; whatever is doing the determining, whether a hand or a propagation, runs out of information at the rim.

It also suggests the design response a folder reaches for, which is to hem: leave a border of flat paper wide enough that no unit is cut, and let the tessellation stop where it stops. That produces a pattern with no cut vertices — the first category — and it is what the tessellation folders do without needing any of this.

What the sheet’s shape decides and what it does not #

It is worth separating this from the other things a sheet’s shape is known to decide, because they are unrelated and the collection has measured several.

A sheet’s proportion decides what can be constructed on it — which divisions are exact and which are not — and that is a question about numbers.

A sheet’s area decides how much paper a design has to spend, which is what circle packing is about, and that is a question about geometry.

A sheet’s shape in the sense of which polygon it is decides what a base can reach, and a pattern asks for the paper it needs rather than the other way round.

None of those is what this essay is about. What decides a search’s difficulty is not the sheet’s proportion, area or outline but whether the pattern’s own structure is cut by the sheet’s edge — a relation between two things rather than a property of either. A tessellation clipped to a circle would be in exactly the same position as one clipped to a square, and a corrugation designed to fit a circle would be in no difficulty at all.

Where the ladder goes next #

Every family in the easy list is one somebody made. The one family in it that grew rather than being drawn is worth its own reading: a plant’s corrugation is not a hard case, and the reason it is not turns out to be the same reason a grid is not — the pattern fills the leaf, and a leaf has no rim cutting through it.