Where you cut hardly matters

A patch is a specimen. The tessellation fills the plane, a square is taken out of it, and where that square falls is an accident of what size sheet was wanted and where the drawing happened to start. This collection has drawn hundreds of them and read numbers off them, and a reasonable worry about all of that is that the numbers are about the square rather than about the pattern — a worry sharpened by the discovery that the counts a patch reports can sit on a knife edge of the construction’s own pitch.

The worry is sharpest for search cost. A cut divides creases, leaves partial panels round the edge, and removes constraints that ran from one period of the pattern to the next — so a cut in a different place divides different creases and removes different constraints. It is easy to believe that the difficulty of a patch is largely a matter of luck.

It is not, and the measurement is direct.

Twelve cuts #

Take one drawing: a square twist tessellation at a fixed turn, pitch and fill. Take a rectangle three periods across and three up. Slide it across one whole period of the drawing in twelve equal steps, so that every distinct position of the rectangle relative to the pattern is represented, and letter each of the twelve patches that result.

They really are different patches. The panel count runs forty-nine, forty-nine, forty-nine, sixty-one, sixty-one, forty-nine, forty-nine, forty-nine, sixty-one, sixty-one, forty-nine, forty-nine, so four of the twelve cut through twelve extra panels. The crease count runs eighty-four or ninety-six with it.

The vertex count does not move at all: thirty-six at every position, because the rectangle is placed where its sides run furthest from every drawn point and there is a range of such placements rather than a single one.

What they cost #

Twenty-five to thirty-three steps.

In order: thirty, thirty, thirty, thirty-one, thirty-one, twenty-five, twenty-five, twenty-five, thirty-three, thirty-three, thirty, thirty. The largest is a third more than the smallest, and the pattern of them repeats — six values, twice — because the drawing has a symmetry that the sweep runs over twice.

A factor of one and a third, across every way there is of cutting this drawing at this size.

It is worth noticing what a factor of one and a third is on the scale this collection usually works at. The panel counts across the twelve differ by twenty-four per cent, so the cost differs by about as much as the amount of paper does. A patch with more panels costs more steps, roughly in proportion, and nothing else about the cut leaves a trace.

That is the null result stated positively: the cost tracks the size of the patch and not the shape of its rim.

Against the other comparison #

Set that beside the difference between cutting and not cutting.

The same rectangle at the same size, with its opposite sides joined so that no crease is divided at all, costs six hundred and twenty-five steps. At four periods the cut version costs forty-eight and the joined version fifty-six thousand seven hundred and seventy-two.

So: a factor of 1.3 for where the cut falls, and a factor of twenty at three periods and more than a thousand at four for whether there is a cut. The two effects are not of the same kind and they are not close to the same size.

The same on three more tilings #

One tiling is not a result, so the sweep runs on four.

The triangular tessellation at three periods: panel counts of a hundred and thirty-three, a hundred and thirty-nine or a hundred and forty-five depending on the cut, a hundred and eight vertices at every position, and costs of seventy-four to ninety-one steps. A factor of 1.23.

The honeycomb: the same panel counts, the same hundred and eight vertices, seventy-five to ninety-two steps. A factor of 1.23.

The elongated triangular tiling: two hundred and seventeen to two hundred and thirty-five panels, a hundred and eighty vertices, and a hundred and twenty-nine to a hundred and thirty-seven steps. A factor of 1.06 — the tightest of the four.

Four tilings, and the whole spread across every way of cutting each of them is between six and thirty-three per cent.

It is also worth being clear that the two are not two points on one curve. There is no continuous parameter running from cut here through cut there to do not cut: cutting anywhere severs every long-range constraint that crosses the rim, and not cutting severs none. The twelve positions differ in which creases they divide and agree completely in that they divide the ones crossing each side once per period.

So the sweep is not measuring a weak version of the same effect. It is measuring a different quantity — which creases, rather than whether — and finding that it does almost nothing.

Why the vertex count is constant and the others are not #

The three counts behave differently under a sliding cut and the reason is worth having, because it explains which quantities a patch reports honestly.

Vertices are points, and a rectangle placed to keep clear of them stays clear of them over a range of positions. So sliding the cut within that range moves no vertex in or out, and the number of conditions the patch asks is genuinely fixed.

Panels are regions, and every position of the cut slices a different set of them. Forty-nine or sixty-one on the square, depending on whether the sides happen to pass through the middles of twist polygons or between them.

Creases are segments, and the cut divides whichever ones it crosses. Eighty-four or ninety-six.

So a patch is honest about how many questions it asks and unreliable about how much of the pattern it holds — which is the same conclusion the essay on what a rim is worth reaches by a different route, arriving here as a statement about how much any of it varies.

Why the cost varies as little as it does #

The obvious model predicts more variation than there is, and it is worth seeing why it is wrong.

A cut removes constraints — one per divided crease — and the twelve positions divide either twelve or eighteen creases. If the cost fell steeply with the number of constraints removed, the four positions that divide eighteen would be markedly cheaper than the eight that divide twelve, and they are not: the twelve-crease cuts cost twenty-five to thirty and the eighteen-crease ones cost thirty-one to thirty-three, which is the wrong way round and by very little.

What that says is that the cost is not sensitive to how many long-range constraints are removed once enough of them are. Removing all of them at every period of the boundary — which any cut does — is sufficient to make the pattern read rather than search, and the residual differences are about which handful of extra decisions the particular arrangement of half-panels leaves.

That is a reassuring shape. It means a patch’s cost is a property of the pattern-with-a-rim rather than of the particular rim, and the numbers this collection has read off patches from the beginning are about the right object.

The shape of the twelve, which is the drawing’s #

The costs come in a pattern rather than a scatter — thirty, thirty, thirty, thirty-one, thirty-one, twenty-five, twenty-five, twenty-five, thirty-three, thirty-three, thirty, thirty — and the pattern is worth reading, because it is the strongest evidence that the sweep is measuring the cut rather than measuring noise.

Six values appear and then repeat. That is the drawing’s own symmetry: a square twist tessellation has a half-period translation that maps it to itself with the twists rotated, so sliding the rectangle half a period gives a patch that is congruent to one already seen. Twelve positions across a full period visit six distinct arrangements twice each.

A sweep measuring anything random would not do that. A sweep measuring a systematic drift — a bug that made later positions slightly different from earlier ones — would not do that either. What it does instead is reproduce a symmetry that was never told to it, which is the arithmetic confirming that the twelve patches are twelve cuts of one drawing.

The same repetition appears on the triangular tiling and the honeycomb, and on the elongated tiling it does not, because that tiling’s period is tall and its half-period is not a symmetry of the same kind.

The population this is and is not #

There is a population argument here and it is worth stating the limits of it plainly.

Twelve cuts of one drawing at one size on four tilings is a population of cuts, and it settles a question about cuts: they do not matter much. It is not a population of patterns. The turn, the pitch and the fill are held fixed, and moving those is a different sweep with a different answer — ninety-six patches over a grid of tiling, turn and fill, on which the cost varies far more and nine of which have no lettering at all.

So the two populations answer two questions. Across constructions, a tessellation patch’s behaviour varies a great deal. Across cuts of one construction, it barely varies at all. Which is the useful way round: the parameter a person chooses on purpose matters, and the one that falls out of where the paper happened to be does not.

The thing that does depend on the cut #

One quantity here is exquisitely sensitive to the cut, and it belongs in this essay as the exception.

The bottom of the layer stack — which panels have nothing below them — exists only because of the cut, sits only on the cut, and would vanish if the cut were removed. Slide the rectangle and a different set of panels becomes minimal, every time, because the minimal panels are precisely the ones whose neighbours below were cut away.

So the honest summary is not that a cut’s position is irrelevant. It is that the position affects the quantities that belong to the paper and leaves alone the quantities that belong to the pattern, which is exactly the division a reader wants and had no way to check before the two objects could be built side by side.

What this licenses about the older measurements #

The practical value of a null result is what it permits, and this one permits a good deal.

Every measurement this collection has published about tessellation patches was taken on one placement of one rectangle: how much of a patch is edge, what its crease length comes to, how its conditions behave, how much it shrinks. Each of those was, strictly, a statement about that placement.

The sweep says the placement contributes at most a third to the search cost and, for the counts, exactly the amount the panel count moves — so a quantity read off one patch is a quantity about patches of that size and construction, plus a knowable few per cent.

The exceptions are the two identified above and one more. Anything that is a count over the rim — the divided creases, the partial panels, the panels with nothing below them — depends on the placement directly, and anything that is a count over the interior does not. That is a rule a reader can apply to any number here without re-measuring it.

What a designer takes from it #

A short and unglamorous thing. A tessellation patch is not luckier or unluckier depending on where the sheet was cut.

Somebody laying out a twist tessellation on a square of paper chooses the pitch and the turn deliberately and the registration — where the pattern sits on the sheet — more or less by eye. This says the second choice does not affect how hard the pattern is to letter consistently, to within a third at worst and six per cent at best.

That is a small piece of freedom and it is worth knowing it is free. The registration can be chosen for how the model looks, for how the edges finish, for where the rim falls relative to the twists — and it will not make the pattern harder to get right. What it will change is which pieces of paper end up at the bottom of the finished stack, since the bottom of the stack is entirely a fact about the edge — a difference a folder can see and a search cannot.

Why the answer had to be measured rather than argued #

There is an argument that the cut position should not matter, and it is not good enough, which is why the sweep exists.

The argument goes: the pattern is periodic, so every position of the rectangle is a translate of every other as far as the interior is concerned, and only the rim differs. Since the rim is a vanishing fraction of a large patch, the cost should converge.

Two things are wrong with it. The rim is not a vanishing fraction at these sizes — most of a patch is edge at every size this collection draws — and the whole finding of this thread is that the rim is where the search’s freedom is. So an argument that the rim contributes little would prove too much: it would predict that cutting and not cutting are nearly the same, which is false by three orders of magnitude.

What the sweep establishes is narrower and could not have been guessed from that reasoning: the rim matters enormously and its position does not. Those are compatible and neither implies the other, and only a measurement separates them.

Which theorem was checked, and how #

Every one of the twelve patches at every position is built out of the same set of segments, computed once from the construction before anything is cut. So the twelve are genuinely twelve cuts of one drawing rather than twelve drawings, and any difference between them is the cut.

Each is then read independently: its own crease list, its own faces found by walking the planarised graph, its own conditions applied at its own interior vertices. Nothing is carried over from one position to the next, so a systematic error in the sweep would show as a systematic drift across the twelve, and what shows instead is a six-value pattern repeating twice, which is the drawing’s own symmetry.

The costs are measured under a fixed letter order, so each number is the number that patch gives every time rather than one sample from a run-to-run distribution. That matters here more than usual: a randomised order once produced a three-order-of-magnitude spread on these very patterns, and a sweep measuring cut positions under a coin would have measured the coin.

One quantity the cut does change, exactly #

The sweep is a null result about cost, and it is worth pairing with the quantity where the cut’s position is not neutral but is exactly predictable.

A cut divides creases, and each divided crease is counted twice. So a patch reports more crease than the pattern holds — fifty per cent too many at one period of the square tessellation, twenty-five at four, seventeen at nine — and the error falls as the reciprocal of the size without ever reaching zero.

The crease length per unit area is exact at every size and every cut, because the two halves of a divided crease add back up. One measurement survives the cut and the other does not, and which is which is decided by whether the quantity is a sum over the paper or a count of objects.

What the picture cannot show #

Twelve points on a chart do not show which creases each cut divided, and the interesting negative result is that it does not matter. A figure that showed the twelve patches side by side would show twelve visibly different drawings and would suggest, wrongly, that the differences are consequential.

Nor does the sweep say anything about a cut that is not a rectangle. Every patch here is a rectangle aligned with the drawing’s own periods, which is the shape the whole collection uses; a sheet cut at an angle to the tessellation, or in a shape with corners in it, is a different question and one nothing here has asked.

And twelve is twelve. The sweep steps across one period in twelfths, which is enough to catch the repeating structure and not enough to rule out a narrow position between two of them where something else happens. Nothing in the construction suggests such a position exists — the counts move in steps as the sides pass through polygons, and the twelve catch both states — but a finer sweep is a thing that has not been run rather than a thing that came back clean.