Which pair is glued

A rectangle has two pairs of opposite edges. Glue either pair and the result is a cylinder — a sheet with two edges rather than four, Euler number nought, half the rim gone.

A cylinder is one name and it covers two objects, and the two are different in two independent ways.

The first difference: the drawing is not isotropic #

Most patterns look different along their two directions, and the counts say so immediately.

Take the elongated triangular tiling’s twist tessellation and cut one period of it. Glue the left and right edges: forty-two free letters over twenty-two panels. Glue the top and bottom instead: fifty letters over thirty panels. The same rectangle, the same drawing, and eight letters and eight panels between the two answers.

The reason is not subtle. What a glued pair of edges saves is the creases that pair divides, and a drawing that crosses one crease per period in one direction and four in the other divides four times as many with one pair of edges as with the other.

So half the rim is a statement about the topology and not about the amount of anything the search cares about. On the Miura the two halves are two letters and four; on the elongated tiling they are two and ten.

The second difference: the same counts, different cost #

The first difference disappears on a drawing symmetric between its two directions. The square twist tessellation is one: its two cylinders have exactly the same numbers of vertices, creases and panels at every size.

At three periods, one of them settles in twenty-four nodes and the other in eighty-five.

Nothing about the objects differs. Same drawing, same rectangle, same tables at the same vertices, same number of letters to choose. What differs is which creases have been identified with which, and therefore which decisions the search makes in which order.

Why an identification changes the order #

A search picks a crease, gives it a letter, and propagates. Which crease it picks next depends on what the propagation left undecided, and that depends on the structure of the object.

Identifying the left and right edges makes certain creases the same crease. A letter set on one of them is set on both, so the propagation wave that would have stopped at the right-hand edge now reappears at the left and carries on. Identifying the top and bottom instead makes a different set of creases the same, and a different wave carries on.

The two searches therefore visit the object in different orders, meet their first contradiction at different depths, and back up different amounts. That is the whole mechanism and it has nothing to do with either sheet being intrinsically harder.

Which of the two effects is bigger #

On the drawings measured here, the ordering effect dominates and it is not close.

The counting difference between the two cylinders is at most a factor of about one and a half in letters. The cost difference reaches a factor of three and a half on the square twist at three periods, and a factor of eighteen on the triangular tiling at two — six hundred and seventy-four nodes one way, thirty-seven the other, on a cell whose two directions differ by four letters.

So the honest summary is that the two cylinders of one cell differ a little in what they are and a great deal in what a particular search does to them, and the second is the number a reader is likely to see quoted.

The numbers, laid out #

Since the argument turns on comparing pairs of node counts, here they are without the figures around them.

Square twist, symmetric drawing. One period: five nodes cut out, four each way glued, three both ways. Two periods: thirteen, eleven, eleven, nine. Three periods: thirty cut out, twenty-four one way, eighty-five the other, six hundred and twenty-five both ways.

Triangular twist. One period: twelve, nine, ten, eight. Two periods: forty-six cut out, six hundred and seventy-four one way, thirty-seven the other, four hundred and fifty-five both ways.

Elongated triangular twist. One period: seventeen, twelve, sixteen, eleven. Two periods: fifty-eight, sixty-three, fifty-seven, one hundred and sixty-two. Three periods: one hundred and thirty-two, one hundred and thirty, one hundred and thirty-three, and a torus that exhausted its budget.

Two things to read off that. The small cells are all flat and all similar. The interesting rows are the ones where a single cylinder jumps — six hundred and seventy-four against thirty-seven on the triangular cell at two periods, which is the largest gap here and is between two sheets differing by four letters.

And the direction of the jump is not consistent. On the square cell at three periods the first cylinder is the dear one; on the triangular cell at two periods it is the other. There is no rule here for which direction is worse.

Why there is no rule #

That last observation deserves its own paragraph, because it is the reason this essay is a warning rather than a result.

If the difference between the two cylinders were a property of the drawing — if one direction were genuinely more constrained than the other — then the dear direction would be the same at every size and on every tiling with the same anisotropy. It is not.

What decides it is where the search’s first bad guess falls, and that is decided by an interaction between the identification and a variable order that knows nothing about either. Change the order and the two costs change; there is no reason to expect them to change together, and no reason to expect the ranking to survive.

So the correct reading of eighty-five against twenty-four is not that one sheet is three and a half times harder. It is that a particular search, on two sheets that differ in one respect, took three and a half times as long on one of them, and that the difference is attributable to the identification only in the sense that nothing else varied.

What a cylinder is, physically #

It is worth remembering that one of these two objects is a thing people manufacture and the other usually is not, because that decides which asymmetry a designer meets.

A corrugation glued across its pleats is a tube: the folds run round it, and pushing its two ends together collapses it. That is the shape of most engineered folded tubes, and it is the one with a parity condition on how many facets go round.

The same corrugation glued along its pleats is a ring of bellows: the folds run round the ring the short way and the ring closes on itself. That is a real object too — a folded gasket, a concertina closed into a loop — and it is rarer.

So a designer meeting the first asymmetry is meeting a real constraint about a real object, and the numbers about letters and panels are numbers about how much freedom their pattern has. A designer meeting the second — the ordering effect — is meeting an artefact of somebody’s search, and it tells them nothing about their tube.

Keeping the two apart is most of the practical value of this essay.

One place the distinction already bit #

The two cylinders were treated as one case in the first version of every table here, on the reasonable-sounding ground that they are the same kind of sheet with the same amount of rim gone.

The counts made that untenable almost immediately: a column headed cylinder held two numbers that had no reason to be equal, and on the elongated tiling they differed by eight. The tables were split.

The costs made it untenable a second time and in a way that splitting the columns does not fix, because two columns holding twenty-four and eighty-five invite exactly the reading that one sheet is dearer than the other — which is a claim about objects, and the measurement supports a claim about one procedure on two objects.

The current state is that the columns are split, the numbers are reported, and the essay saying what they do and do not mean is this one.

Testing that, and what it would take #

The claim above — that most of the difference belongs to the order — is stated as an expectation rather than a measurement, and it is worth being clear about that.

What would settle it is running the same four sheets under a spread of variable orders and looking at the distribution rather than at one number. If the two cylinders’ distributions overlap heavily, the single-run difference is a sample from noise. If they are separated, the difference is a property of the objects after all.

The collection has the machinery for exactly that measurement — cost spreads over many seeds are how the coin’s contribution to a search’s tail was established — and it has not been pointed at this question.

The reason is cost. A spread is dozens of searches, a torus at four periods is fifty-six thousand nodes, and doing that for four sheets at three sizes on five tilings is a different scale of computation from anything here. It is recorded as owed.

The general habit this belongs to #

There is a discipline behind all of this that is worth naming, because the collection applies it repeatedly and this is a clean case.

Vary one thing. The four sheets are one drawing with four identifications, which is why any difference between them is attributable at all.

Say which thing. Half the rim names the topology; across the pleats names the drawing; and the two are different variables that happen to coincide on a symmetric cell.

Do not report a single run as a property. A node count is a measurement of a procedure on an object, and the procedure has as much freedom in it as the object does.

The first two were observed here and the third was not, until the two cylinders of a symmetric drawing came out three and a half times apart with nothing between them to explain it.

What this does to any claim about a sheet #

The consequence for method is uncomfortable and worth stating flatly.

A single node count is not a property of a sheet. It is a property of a sheet, a variable order, a value order and a tie-break, and the collection has known for some time that the order is where most of the cost lives. What this measurement adds is that the same is true between two objects that are supposed to be the same, which removes the last reason to hope that a difference between two counts is a difference between two objects.

The defence is to compare things that differ in one respect and to say which respect. Every figure here holds the order fixed and varies the identification. A claim about which of two identifications is dearer is then a claim about the identifications; a claim that one of them is hard is not supported by anything.

The asymmetry that is real #

None of the above says the two cylinders are the same object, and it is worth separating the part that is solid.

The counts genuinely differ on any anisotropic drawing, they differ for a reason that can be computed from the drawing without running anything, and the difference is additive with the other pair. That is arithmetic and it holds exactly.

And the parity behaves differently in the two directions on some families: the Miura has a parity condition across its courses and none along them, so half its cells refuse one gluing and none refuses the other. That is a difference in what the sheets are, not in what a search does, and it is as solid as the counts.

The collection’s own asymmetries #

Anisotropic drawings turn up repeatedly here and the two cylinders are the sharpest case.

The elongated triangular tiling’s cell is tall, so its two directions cut very different numbers of creases.

The Miura crosses one crease per period across its courses and four along them, which is a structural asymmetry rather than a matter of the rectangle’s shape.

And the square twist is symmetric between its axes, which is why it is the drawing on which the ordering effect can be isolated.

Two objects, or one object twice #

There is a question lurking under the whole essay that is worth asking directly: are the two cylinders genuinely different sheets, or are they the same sheet described twice?

As surfaces they are the same: a cylinder is a cylinder, and there is no invariant of the surface alone that distinguishes them.

As sheets with a pattern on them they are different, and the difference is not a matter of labelling. One of them has ten letters where the other has two, one of them refuses at odd sizes where the other never does, and no relabelling makes those numbers agree.

The confusion arises because the rectangle has a symmetry that the drawing on it does not. Rotating the rectangle a quarter turn exchanges its two pairs of edges, so the two gluings look like one gluing seen from two angles — and they would be, if the drawing rotated with it. It does not, except on the square twist, where it does and where the counts are consequently equal.

So the right statement is that the two cylinders are two different patterned sheets on the same surface, and the surface is what the word cylinder names. That is the same distinction the collection draws everywhere between a pattern and a sheet, arriving in a place where the two are easy to conflate because the sheet is simple.

Where the anisotropy comes from #

The counting difference has a cause in the drawing, and knowing it means the two cylinders’ counts can be predicted rather than measured.

Every pattern here is built from courses or pleats running in one direction, with something crossing them. A loop of the cell running along a course crosses whatever the course carries; a loop running across the courses crosses the courses themselves.

On the Miura, one course per period is crossed going across, and four crease pieces per period are crossed going along, because a period is two rows and each row has an entering and a leaving crease. Ratio four to one.

On the elongated triangular tiling’s twist tessellation, the repeat rectangle is one unit by two plus root three — a tall cell — so a horizontal loop crosses much more of the drawing than a vertical one. Ratio five to one in letters.

On the square twist the rectangle is one by one and the drawing is symmetric under exchanging the axes, so the ratio is one.

Predicting the counts is therefore reading two numbers off the construction: how many crease pieces a horizontal line through the cell crosses, and how many a vertical one crosses. That is the same pair of numbers the parity conditions are computed from, which is a small piece of tidiness: one measurement of the drawing settles both the parities and the letter savings.

The vertices, which do not move #

Amid all of this it is worth restating the control, since it is what makes any of the comparisons legitimate.

Neither gluing changes the interior vertex count, at any size, on any family. The cell’s edges are placed to miss every vertex, so identifying them joins crease pieces at ordinary points of the paper and never joins anything at a vertex.

That means the two cylinders ask the same vertices the same conditions, and so does the cut sheet, and so does the torus. Whatever differs between the four, it is not what is being asked of the pattern.

It is checked rather than assumed. Every figure that reports these counts asserts that the four gluings agree on the vertex count, and refuses to draw when they do not — which has fired once, on a cell whose corner had a crease through it, and that was a defect in the identification rather than in the claim.

The naming, corrected #

The collection has been calling these objects a cylinder and the two cylinders, and it is worth fixing the vocabulary since the essays will go on using it.

The two sheets are distinguished by which pair of edges is glued, and the pairs are distinguished by the drawing rather than by the rectangle: gluing across the pleats and gluing along them are the useful names on a corrugation, and gluing across the courses and along them on a Yoshimura.

Naming them by the drawing rather than by the rectangle’s orientation is the right convention because the rectangle’s orientation is a choice made when the cell was cut, and the pleats are a fact about the pattern.

That is a small point and it has caused at least one confusion already, in a table where two columns headed cylinder held numbers that had no reason to be equal.