Cutting a patch out of a plane

A tessellation has no edge. The construction that turns any tiling into a twist pattern takes a tiling of the whole plane, puts a polygon at every vertex of it and a pleat along every edge, and returns something that goes on for ever. A sheet of paper is a square.

So every picture of a tessellation is the answer to a question the construction does not ask: where does the paper stop, and what happens to the creases there? The question sounds like presentation. It is not. Two answers to it are drawn here, and one of them produces a pattern that cannot be folded while satisfying every condition the subject checks.

Assembling, and why it seems right #

The first answer builds the patch out of whole units. Take every twist polygon that fits entirely on the sheet, draw it, and draw the pleat between any two such polygons that are neighbours in the tiling. That much is unambiguous.

The trouble is the polygons at the edge of the patch, which have neighbours off the paper. Their outstanding pleats have to go somewhere. Leaving them out is not an option: a corner of a twist has four creases and Kawasaki holds there because of how they alternate, so removing one leaves a vertex of odd degree, which fails immediately and for a reason that has nothing to do with the paper’s edge.

Drawing them and stopping where the missing neighbour would have been is not an option either. That leaves a crease ending in the middle of the sheet with nothing at its end — and a fold line that stops mid-sheet is one the paper cannot obey, since the material has to turn about the line and past its end there is nothing to turn about.

So the pleat is extended along its own line until it reaches the rim, where a crease may legitimately end. Every vertex of the result satisfies developability, Kawasaki, Maekawa and the big-little-big lemma, on all five tilings, with zero failures.

What the stub runs through #

The extended pleat travels through the ground the missing neighbour would have occupied, and beyond it. Whatever is on the far side of that ground is in its way.

On a square grid it meets nothing, and the reason is a fact about directions rather than about luck. The square twist’s pleats leave in the sheet’s own two directions, so every stub reaching a given edge of the paper is parallel to every other stub reaching that edge, and parallel lines do not meet. On the other four tilings the stubs arrive at an edge in two or three different directions, and near the corners of the sheet some of them cross.

The counts are twelve crossings on a triangular grid, eighteen on a honeycomb, twelve on a rhombille, five on the elongated triangular tiling, and none on the square. Those crossings are vertices the crease list does not have, and a crossing has no flat folded state whatever its letters are.

The consequence is not local. Composing the reflections around the panels of those four patches, the two routes to a panel disagree by 1.73, 2.00, 1.86 and 1.73 sheet widths — the same refusal a ring of paper gives, at the same scale, meaning there is no consistent placement of the panels in the plane at all. The paper cannot be put down.

Five tilings, and the one that escapes #

The family is worth reading across, because the crossings do not scale with the size of the patch or with the number of creases in it.

The rhombille has the most creases of the five — two hundred and two against the square’s eighty-four — and twelve crossings, the same as the triangular grid’s hundred and two creases produce. The honeycomb has a hundred and twelve creases and eighteen. The elongated triangular tiling has ninety-six and five, the fewest of the four that fail; its rows of squares alternate with rows of triangles, so along two of the four rims the pleats do leave in one direction and only the other two rims produce anything.

What the count tracks is the corners of the sheet. A stub crossing another stub happens where two rims meet and their two families of parallel lines run into each other, so the number scales with how many directions arrive at a corner rather than with how much pattern there is.

Clipping instead #

The second answer inverts the order of two operations. Instead of choosing the units and then dealing with the edge, it generates the tessellation over a region comfortably larger than the sheet — every polygon and every pleat, all of them whole — and then clips the drawing to the square.

A crease that leaves the paper is shortened to the rim and ends there, which is what a crease is allowed to do. A crease entirely off the paper is not drawn. Nothing is ever extended, so nothing runs through ground the construction did not fill, and there is no stub to cross anything.

The counts after the change are: no crossing on any of the five, and panels that place to within a part in 10¹⁴ on all of them. Every vertex condition still passes, exactly as it did before — which is the point worth keeping. Nothing this collection checked changed. Both cuttings were green all the way through, and the difference between them is a pattern that folds and a pattern that cannot.

What it costs, and what it does not #

Clipping puts more pattern on the same paper, because the units at the edge are now drawn as far as the rim rather than omitted. The triangular patch goes from a hundred and two creases to a hundred and forty-two, the honeycomb from a hundred and twelve to a hundred and forty-two, the rhombille from two hundred and two to two hundred and eighty-two. In folding length — the measure that says what an evening costs — the same three go from 16.2 to 17.4, 17.1 to 17.5 and 21.5 to 24.2 sheet widths.

Two things do not change, and both are worth stating because they are the numbers other essays here quote.

The shrink is identical. How much smaller the twisted paper is than the sheet it came from is computed over the twist units that sit wholly on the paper, and the two cuttings keep exactly the same set of those: 2.363 on the square, 1.713 on the triangular, 1.799 on the honeycomb, 1.635 on the rhombille, to every digit.

The square grid’s patch is unchanged, crease for crease. Sixty-four vertices, eighty-eight edges, forty-nine panels, both ways. Its stubs already ran where the clipped pattern’s creases run, because its pleats are parallel to the rim they meet. So the tiling this collection has drawn most often was never affected, and the tilings that were are the ones nothing was printed from.

The condition underneath is untouched #

None of this reaches the construction itself, and it is worth saying which part is which.

A twist tessellation folds because of a condition on the tiling: the two sides facing each other across a pleat have to be the same length, which fixes the size of one twist polygon from its neighbour’s and propagates across the tiling. On a tiling whose vertices are all alike that condition is satisfied by giving every vertex the same size and is invisible; on the rhombille it forces a ratio of three to one between the hexagon’s twists and the triangle’s, and nothing else folds.

That condition is about the plane. It holds on both cuttings, because both draw the same polygons at the same sizes; the clip changes which parts of them appear on the paper and nothing else.

So the fault was never in the mathematics of the pattern. It was in the sentence that turned an infinite object into a picture — which is a sentence every figure of a tessellation in this collection contains, and which none of them used to state.

What the repair reveals #

Placing the panels was not the last question, and clearing it lets the next one be asked for the first time.

A lettering can force a loop of panels — a cycle in which each panel must lie above the next, all the way round — and a loop is a proof that no ordering exists, available in one pass over the crease list. That test needs the panels to have been placed. On the four patches that could not be placed, it could not be run at all.

Run on the clipped patches, it says something new and unwelcome: every one of them still has no ordering. The triangular patch’s propagated letters force a loop of fourteen panels, the honeycomb’s a loop of thirty-six, the elongated tiling’s sixteen, the rhombille’s twenty — and redrawing each of them a dozen times with the branch order randomised produces no clean lettering on any of the four. The square patch, as before, comes back clean about one draw in six.

That is progress of the only kind available here. The refusal has moved from this drawing is not a crease pattern to this crease pattern’s letters contradict themselves, which is a question about lettering, on an object that now exists — and the second is a question the redraw machinery is built for, while the first was not.

What is still not claimed #

No patch of these four is claimed to fold. Each of them places, each of them satisfies every vertex condition, and each of them has letters that force a loop; the honest verdict is no ordering found and none proved impossible for the pattern as such, since a loop refutes a lettering rather than a drawing.

A tessellation patch is still not a tessellation. The unit is verified and the plane is not, and clipping does not change that: it makes the picture a legitimate crease pattern on a square, which is a smaller claim than the pattern tiling the plane and folding.

And the clipped patch has more boundary in it, not less. Every unit at the rim is now cut, so a larger share of what a reader sees is edge — which is a fact about patches in general and is measurable in its own right, rather than a defect of either cutting.

The rule this leaves behind #

Two operations, and their order is the whole of it: generate, then clip — never choose, then finish.

Stated that way it applies past twist tessellations. Any construction that fills a region with a repeating unit has to decide what happens at the edge of the paper, and it has exactly the three options this one had: omit the outstanding creases and leave odd vertices, stop them where their neighbours would be and leave creases ending in the middle of the sheet, or extend them and put creases through ground that was never filled. All three are wrong. The fourth option is not to make the decision at all — to fill more than the paper and then cut.

The cost is visible in that figure and it is not a defect. A patch cut from the plane has cut units at its rim by construction, and at the sizes these figures are drawn at, between two-fifths and three-fifths of the units on the paper are cut ones. A reader looking at a tessellation figure is looking at a boundary as much as at a pattern, which is a fact worth knowing whichever cutting produced it.

One test catches all three #

The rule at the end lists three wrong ways to finish a patch, and they look like three separate mistakes needing three separate vigilances. They are not. All three are caught by a single check, it is cheap, and naming it is more useful than the rule.

Take the crease list as drawn and planarise it: split every segment at every point where it meets another, and collect the resulting vertices. Then compare that vertex set against the one the construction declared.

Omitting an outstanding pleat leaves a declared vertex whose degree has dropped by one, so a vertex of odd degree appears in the planarised graph. Stopping a crease where its missing neighbour would have been leaves a vertex of degree one, which is a crease with a loose end. Extending a stub through unfilled ground produces a vertex the construction never declared — a crossing, appearing in the planarised graph and in no list.

Three faults, three signatures, one pass. The check is: planarise, and require that every vertex of the result was declared, that none has degree one, and that none has odd degree. A drawing passing all three is a crease pattern; a drawing failing any of them is not, whatever its declared vertices satisfy.

Which is a check on the drawing rather than on a point #

That is why it was missing, and the reason is worth stating in the form that transfers.

Every condition this subject has is evaluated at a declared vertex: read the sectors, count the letters, compare the alternating sums. Such a check cannot notice a vertex that was not declared, because it never visits one — and it cannot notice a crease with nothing at its end, because a crease with nothing at its end declares no vertex there to be visited.

The planarisation check is of a different kind. It does not evaluate anything at a point; it asks whether the set of points the drawing implies is the set the construction thinks it made. That is a question about the whole crease list at once, it needs no angles and no letters, and it costs a sweep over the segments.

So the collection had, and has, a full complement of instruments for asking whether each vertex is right, and until now none for asking whether the vertices are the vertices. Both are necessary and only the first was automatic — which is the shape of every fault this essay is about, and the reason the fault survived a suite that was working correctly the whole time.

It generalises past patches, too. Any drawing assembled from parts — a graft, a molecule filling a polygon, a pattern with a cut in it — implies a vertex set, and the assembling code believes it knows what that set is. Comparing the belief against the planarisation is one function call, it has no false positives, and it is the only check in this subject whose subject is the drawing rather than the paper.

Why it was invisible #

The fault lasted as long as it did because every instrument pointed at those patches was pointed at their vertices, and the fault is not at a vertex. It is at the two ends of a crease and in the ground between them.

That is the same shape as the other repairs this collection has had to make to its own drawings: a lettering that satisfies every condition and folds nowhere, a caption refuted by the numbers in its own figure, a tick label that was never drawn at all. In every case the checks ran, passed, and were about something else.

There is a second reason, and it is about how the patches were used. A tessellation patch here has always been a figure — a picture of what the plane version looks like, drawn beside an argument about the construction — and the arguments it illustrated were about the tiling underneath, the polygon angles, the pleat widths and the twist angle’s two ends. Every one of those is true of the assembled patch and of the clipped one alike, so nothing an essay said about them was wrong, and nothing in the figures looked odd. A crossing between two nearly parallel stubs at the corner of a patch of a hundred and twenty-five panels is not visible at the size a page draws it.

The general lesson is cheap to state and expensive to notice: a construction that has to decide where to stop is making a claim about the paper, and the claim needs a check of its own. Every condition in this subject is about a point. Where the paper ends is not a point.