Each drawing has its own threshold

Half the slack measured what gluing costs and ended by saying what it had not established. Gluing one pair of a cell’s edges removes about half the freedom and costs almost nothing; gluing the second removes the other half and costs three orders of magnitude. The letters go linearly and the search does not.

What that measurement could not say is what the curve is. It was taken at one size, and a single point cannot tell a threshold from a slope — nor whether the cut sheet has a threshold of its own a little further out, with gluing merely bringing the same wall closer. The essay named the experiment that would settle it: a size sweep at fixed family and fixed order, on all four sheets, from one period to five, and called it affordable and unrun.

It is affordable. It takes about a second.

Everything starts at a third of a node #

The first thing the sweep establishes is a baseline, and it is the same baseline everywhere.

A cut cell of the square twist at one period has twelve free letters and the search settles it in five nodes. At two periods, forty letters and thirteen nodes. At three, eighty-four and twenty-six. At four, a hundred and forty-four and forty-five. At five, two hundred and twenty and seventy. That is between 0.31 and 0.42 of a node per free letter at every size, and the ratio falls slightly as the sheet grows rather than rising.

A backtracking search that spends a constant share of a node per variable is a search that is not backtracking: it assigns, propagates, and almost never has to undo. Every one of the four tilings begins there — 0.35 on the honeycomb, 0.35 on the triangular grid, 0.37 on the rhombille, at one period, cut. So does every gluing of them, at one period: the cheapest sheet in the whole sweep is a one-period torus at three nodes.

Nothing in the smallest cells distinguishes the four drawings or the four ways of gluing them. What the sweep is about is where each of them stops behaving like that.

Four drawings, four thresholds #

Each tiling leaves the baseline at a size of its own, and the sizes are not the same.

On the square grid the torus goes 3, 13, 53, 1169, 2927 as the cell grows — twenty-six times the cut sheet at four periods and forty-two at five. The crossing is between three periods and four, and it is not a slope: 13, 53, 1169 is not a curve anybody would fit a line to.

On the honeycomb the same thing happens a period earlier: 8, 95, 4186. On the triangular grid, earlier again — the two-period torus already costs 1668 nodes against its cut sheet’s 37, and the three-period one spends its whole budget without finishing.

That already changes the reading of the original measurement. Three orders of magnitude is not what gluing costs; it is what gluing costs past the threshold, and below it gluing is free or better than free, because it removes letters and leaves fewer decisions.

And the rhombille crosses without being glued #

The fourth tiling is the one that decides what the threshold is about.

The rhombille’s cut cell at one period costs 22 nodes against 60 free letters — 0.37, the same baseline as everything else. At two periods it costs 13,834 nodes against 216 free letters, which is 64 nodes per letter and a factor of 170 above where it started. Nothing has been glued. It is a sheet with four rims, cut from the plane in the ordinary way, and its search backtracks heavily.

So the question the earlier measurement left open has an answer, and it is not the tidy one. A cut sheet can have a threshold; three of these four do not reach theirs within five periods and the fourth reaches it at two. Gluing does not create the threshold. It moves it closer — by two periods on the square grid, by one on the honeycomb — and on a drawing whose threshold is already close, the cut sheet gets there on its own.

The rhombille being the exception is not a surprise once it is said. It is the one tiling here with two kinds of vertex, and it has been the awkward member of this set in every measurement that has touched it.

What crosses, and why a rim delays it #

A search that never backtracks is doing propagation with a fixed order, and its cost is the number of things it has to decide. A search that backtracks is searching, and what it costs is not the count of decisions but the shape of the space they live in. The threshold is where the constraints stop determining the next letter and start merely restricting it — where the propagation runs out before the assignment does, and the first guess has to be made.

A rim is where propagation starts cheaply. A crease ending on the rim is decided by one vertex instead of two, so a cut sheet has a supply of letters that cost nothing to place and a supply of places to recover to. What the rim was doing found the rim carrying that role at one size; the sweep here is the same fact read as a function of size. As a cell grows, its rim grows as a perimeter and its interior as an area, so the rim’s share falls — and the threshold arrives when it has fallen far enough.

That reading accounts for the tiling dependence without any extra assumption, because the ratio of rim to interior at a given period is a property of the drawing. It also accounts for the rhombille: a drawing whose interior is hard enough will cross while its rim is still substantial.

It does not account for which drawings are hard, and nothing here does. That the rhombille’s interior is the difficult one is a fact this sweep reports rather than explains. A region with no lettering is the nearest thing to an account of hardness on this subject, and it is about sector angles rather than about gluing or size.

What a period cell is, and why one was the default #

There is a reason this had not been swept, and it is worth naming because it is a habit rather than an oversight.

A repeating drawing has a period, and a period cell is the natural unit: it is the smallest piece that carries the whole pattern, it is what the two glued cylinders and the torus are built from, and every count this subject reports about a tessellation is reported per cell. So a measurement on a period cell is a measurement on the cell, and one is the number of them a cell has.

That makes the cell count invisible as a variable. It is not a parameter somebody chose and could have chosen differently; it is the definition of the object being measured. The sweep above is what happens when the definition is treated as a dial anyway — and a three-by-three torus of a twist tessellation is a perfectly good object that simply has no name in the vocabulary the measurements were taken in.

Nine periods glued into one torus is not nine times one period glued. It is a different sheet: its letters are not nine copies of a cell’s letters, because the gluing identifies them differently, and its search is not nine searches. The table’s free-letter column shows it — 12, 40, 84, 144, 220 rather than 12, 24, 36 — since the count goes as the area and the identifications go as the perimeter.

The cylinders are not halfway between #

The two cylinders were the subject of the previous measurement and the sweep says something about them the single size could not.

They track the disc almost exactly up to four periods — 4, 12, 24, 62, 66 for one of them against the disc’s 5, 13, 26, 45, 70 — and then one of them leaves. At five periods the cylinder glued the second way costs 209 nodes against the disc’s 70, and on the honeycomb at three periods the other cylinder costs 2853 against 76.

So a cylinder has a threshold too, and it lies between the disc’s and the torus’s. That is the ordering the freedom count would predict if freedom were what mattered, and the sizes at which each sheet crosses are not proportional to anything obvious about it.

It also confirms, at more sizes, what which pair is glued found at one: the two cylinders are different sheets. Below every threshold they are indistinguishable, and above it they part company by more than an order of magnitude — and which of the two goes first is not the same on the two tilings where it happens.

What the free letters do not predict #

The free-letter count is the obvious candidate for what a search costs, and the sweep shows exactly where it stops working.

Below every threshold it works perfectly: cost is a third of a letter count, and a sheet with fewer letters costs less. That is why the one-period torus is the cheapest object in the whole table — gluing removes a tenth of the letters and the cost falls in proportion.

Above the threshold the letter count is nearly useless. The square grid’s five-period torus has 200 free letters against the cut sheet’s 220 — ten per cent fewer — and costs forty-two times as much. The rhombille’s two-period torus has 192 against 216 and does not finish at all. A quantity that predicts the cost to within a few per cent on one side of a threshold and is wrong by a factor of forty on the other is not a measure of difficulty; it is a measure of size that happened to coincide with difficulty while nothing was hard.

Half the slack said this in its own terms — the letters go linearly and the search does not, and the last free letter is worth more than all the others. The sweep adds the missing half of that sentence: the last letter is worth more than all the others only once there is a threshold to be past, and whether there is one is a fact about the drawing and its size rather than about the gluing.

The one number that stays put #

Amid four thresholds at four sizes there is a quantity that does not move, and it is worth recording because it is the only thing the sweep found to be universal.

Every drawing, cut or glued, starts at between 0.31 and 0.42 of a node per free letter. Four tilings, four gluings, the smallest cell of each: sixteen sheets whose free-letter counts run from 8 to 60 and whose costs are all the same fraction of them. The one exception is the rhombille’s first cylinder at 1.19, and that sheet is one period of the drawing whose cut cell crosses at two.

A third of a node per letter is a specific number and it is not obviously the number a propagating search should spend. It says that on average each letter is settled after about a third of a visit — that most letters are forced the moment a neighbour is placed, and the search’s work is mostly bookkeeping. Whether the third is a property of these drawings, of the order used, or of the way the skeleton propagates is not answered here, and it is measurable by the same means.

What the sweep does not establish #

One order, one seed. Every number here is a single branch order. A backtracking search’s cost is a property of the pair — the sheet and the route through it — and nothing in this sweep separates them. That is the second experiment the earlier essay named and it is not run here.

Five periods is not asymptotic. The square grid’s cut sheet is linear to 220 free letters. Whether it stays linear to two thousand is not tested — and the rhombille is the standing warning that a threshold beyond the range measured is invisible until it is not.

One lettering, not all of them. The search stops at the first consistent assignment. The cost of enumerating them is a different quantity — one witness or forty prices that — and so is the cost of proving there is none, which the cost of a negative measures.

And the budget is a cutoff. A search that reaches it is reported with a mark rather than a number, because what it spent is a fact about the cutoff rather than about the sheet. Two rows reach it, both past their own threshold.

The pruning is the one the earlier work settled on. Pruning on proofs alone established which tests are asked in which order, and the sweep inherits that unchanged — so every number here is comparable with every other, and none is comparable with a search that prunes differently.

What the picture cannot show #

It does not say the threshold is sharp. A few sizes either side of it is what there is, and 53 to 1169 across one step is consistent with a sharp threshold and with a very steep slope. Distinguishing them needs sizes between, and the periods are whole numbers.

It does not identify what crosses. The reading above — that the propagation runs out when the rim’s share falls — is an explanation that fits, not one that is tested. The measurement that would test it is a sheet with the rim artificially preserved at a large size, and the rhombille shows it cannot be the whole story in any case.

And it says nothing about the letterings themselves. Cost is not quality: a search that settles a torus in three nodes at one period and 2927 at five has found a valid lettering in both cases, and how many there are, and whether they are alike, is not asked here.

Still open: the same sweep, many orders #

The sweep answers the first of the two questions the earlier essay left and leaves the second untouched, and the second is now the one that matters.

Every figure above is one route through one space. A backtracking search’s cost belongs to the pair, and two orders on one object can differ by more than the factor measured here — which was already known when this sweep was proposed, and is the reason the proposer expected a good deal of the factor to belong to the route.

Running it is affordable at the sizes where the threshold is, which is the part that has changed: a four-period torus under ten orders is a few seconds rather than the hours a five-period one would be. What it would settle is whether the threshold is a property of the sheet — every order dear past it — or a property of most orders, with a good one still cheap. Those two are very different claims about what a glued sheet is, and the sweep above cannot tell them apart.

The other thing now worth running is the rhombille further out, and it is the cheapest experiment of the three. Its cut sheet crosses at two periods, which means the whole interesting range sits at sizes small enough to search exhaustively. Every question this sweep asks of the square grid at four and five periods — how sharp the threshold is, what the cost does past it, whether the crossing is the same under other orders — can be asked of the rhombille at two and three, where the sheets are small enough that the answers are affordable.

Sideways from here, the linearity of the cut sheet deserves its own attention. A search that spends a third of a node per free letter is not really a search, and where that holds at every size it holds for a reason — the constraints on a cut cell must determine each letter from the ones already placed, almost always. Whether that is a theorem about three of these drawings or a happy accident of the order chosen is a question the same sweep can ask, and the rhombille is the control: whatever the explanation is, it has to fail there.

The habit worth carrying is about measurements that are missing a variable. A cost measured at one size is a cost divided by nothing, and the way to find out whether it is a property of the object is to sweep the quantity the object was arbitrarily given — here the period count, which was one because one is what a period cell means.