Two numbers for a tissue, and which two

Two relations are quoted for cellular tilings more than any others. Lewis’s law says a cell’s area rises with its number of sides. Aboav’s relation, in Weaire’s form, says a many-sided cell is ringed by few-sided ones, with a coefficient $a$ near 1.2 in the materials it was measured on. Two laws that want opposite tissue found them pointing opposite ways on a golden seed head and a random set, and one law counts sides, the other pairs moved heads from one to the other by three routes and found why. Lewis’s law reads how varied the side counts are, and switches on when independent displacement makes them vary; Aboav’s relation reads whether the five- and seven-sided defects are paired, and a smooth field that makes no new defects lowers $a$ by pulling pairs apart while the variance of the side counts stays at the lattice’s.

That makes a tissue a point on a plane, one axis for each law, and it raises the question the essay ended on. Tissues are reported by one number far more often than by two — the variance of the side counts, $\mu_2$, is the usual single measure of how disordered a tiling is — and the question is whether any one number places a tissue on both laws at once, or whether two are always needed. The three routes answered it only along three lines. A plane needs the routes crossed.

Crossing the routes #

Each tissue here starts from the same 900-organ golden head the two earlier essays moved, is moved first by a smooth random field correlated over eight wall spacings, at a step of nought to three spacings, and is then displaced organ by organ, by nought to a whole spacing. Seven smooth fields by ten steps make sixty-nine moved tissues besides the ordered head, each read on ten seeds for four numbers: the variance of the side counts, Aboav’s $a$, Lewis’s slope, and the pairing — the share of five-sided cells that touch a seven. The laws’ criteria are the earlier essay’s: Lewis’s law is on when its slope reaches half a random set’s, 0.115, and Aboav’s relation holds while $a$ is within 0.35 of 1.2.

The sixty-nine points do not lie on a line. With no smooth field, the path is the one the independent route drew: $a$ rises from 1.18 to 1.44 at 0.15 of a spacing as the first defects arrive as bound pairs, and falls to 0.71 by a whole spacing, while Lewis’s slope climbs to 0.22. A smooth field of two spacings starts the same path from somewhere else, at $a = 0.84$ with a slope of −0.02, and it never catches up the part of the plane the unfielded path held: its highest $a$ is 1.22, at a tenth of a spacing of step. A field of three spacings starts lower still. Between them the paths sweep an area.

The rise, cut short #

The one feature of the independent route that surprised the earlier essay was the rise: Aboav’s $a$ climbing from 1.18 to 1.44 before it falls, because the first defects independent steps make are born as bound five–seven pairs, and a tiling of bound pairs is as strongly anticorrelated as a tiling can be. Under a smooth field the rise is still there and shorter. Its peak is 1.44 with no field, 1.45 and 1.44 under fields of a third and two thirds of a spacing, 1.39 under a field of one spacing, 1.32 under one and a half, 1.22 under two and 1.09 under three — and it comes a little sooner, at a tenth of a spacing of step rather than 0.15, once the field is a spacing or more.

That is the pairing seen from the side. The new pairs the steps make add to whatever pairing the head already has, and a field large enough to have pulled some of the lattice’s own pairs apart has left less to add to. The peak of $a$ is a measurement of how paired the tiling was before the steps began.

Two tissues with one variance #

The plainest way to see what a single number loses is two tissues that share it.

On the left, a head moved by a smooth field of three spacings and nothing else: the variance of its side counts is 0.43, Aboav’s $a$ is 0.75 and Lewis’s slope is −0.003, so it satisfies neither law. On the right, a head with every organ displaced by 0.15 of a spacing: its variance is 0.51, $a$ is 1.44 and the slope 0.088 — Aboav’s relation at its strongest, and Lewis’s law on its way on. The difference is visible in the windows. On the left the fives and sevens are scattered singletons and short pairs, 73 per cent of fives touching a seven; on the right they are threaded into chains, 93 per cent. The variance counts how many defects there are, and the two heads have about as many; it cannot count how they are arranged.

The variance reads Lewis’s law #

Put Lewis’s slope against the variance for all sixty-nine tissues and it nearly is a function of it.

A quadratic in the variance explains 93.0 per cent of the variation in Lewis’s slope. The rest is not noise. The seeds’ own scatter in a ten-seed mean is 0.0058 in the slope, and the fit misses by 3.3 times that, and the misses sort by the smooth field: at a variance near 0.65 to 0.72, a head under a field of three spacings has a slope of 0.046 and one with no field a slope of 0.115. A smooth field raises the variance a little by itself — to 0.43 at three spacings with no step — without raising the slope at all, because the defects it makes are rearrangements of the lattice’s rather than the new, independently varied cells that make area track sides.

The same field delays Lewis’s law. Under a field of up to two thirds of a spacing the law switches on at a fifth of a spacing of independent step, as it did with no field; under a field of one spacing or more it switches on at three tenths. So the variance is the best single number for Lewis’s law, and it still needs to know where the variance came from.

The pairing reads Aboav’s relation #

For Aboav’s $a$ the best single number is the pairing share, not the variance: a quadratic in it explains 88.9 per cent, against 74.5 for the variance and 79.1 for Lewis’s slope. That is what the earlier essay’s reading of the two laws predicts, and it is good to see it hold on tissues it was not drawn from. It still misses by 2.5 times the seeds’ noise in $a$, 0.031, because $a$ is not only about whether fives touch sevens but about how the whole neighbourhood of a many-sided cell is arranged, and a head that has been both stretched and shaken has arrangements neither route alone produces.

And each law’s best number reads the other law badly. The pairing share explains 47.1 per cent of Lewis’s slope, and misses it by nine times the noise; the variance explains 74.5 per cent of $a$ and misses it by 3.9 times. No single one of the four places a tissue on both laws: fitted as a quadratic in any one of them, one law or the other is left at more than two and a half times the noise.

Which two #

Two numbers do, and it matters which two. Lewis’s slope, fitted in the variance and Aboav’s $a$, misses by 1.5 times the noise; Aboav’s $a$, fitted in the variance and Lewis’s slope, by 1.3. Nothing is left for a third number to explain in either case. The variance itself, fitted in the two laws’ statistics, misses by 1.9 times its own noise of 0.026 and is 99.3 per cent explained. So a tissue is placed on both laws by its variance and either law’s own statistic, and the three numbers $\mu_2$, $a$ and Lewis’s slope carry close to two degrees of freedom between them: any two fix the third to within one and a half to two times the scatter of ten heads.

The pair that a counter of cells would most naturally record does worse. The variance and the pairing share are both read by counting sides and looking at neighbours, with no fit made, and together they place Lewis’s slope to 2.4 times the noise and $a$ to 2.2. They are closer than any single number and not close enough to stand in for the laws.

There is a practical point hidden in the good pair. Of the four numbers only Lewis’s slope needs the cells’ areas; the variance, the pairing and Aboav’s $a$ are all read from side counts, $a$ by fitting the mean side count of each cell’s neighbours against its own. So on these tissues the variance and $a$, counted without measuring a single area, predict Lewis’s slope to 1.5 times the scatter of ten heads. A section on which areas are hard to measure — cells crushed, walls faint — still says where it sits on Lewis’s law, provided its sides can be counted and its disorder is of the kinds read here. The two numbers that are both about pairing, $a$ and the pairing share, are the worst pair of all for Lewis’s law: 8.4 times the noise, no better than either alone, because neither says anything about how varied the sides are.

A field under a spacing leaves no trace #

One more thing the crossed grid shows is what none of the four numbers can see. A smooth field of up to one wall spacing, followed by independent steps, gives the same four numbers as the steps alone, step for step, within the seeds’ noise. At a tenth of a spacing of step with no field the slope is 0.059, $a$ is 1.41, the variance 0.34 and the pairing 0.93; under a field of one spacing, 0.059, 1.39, 0.37 and 0.90. At a fifth of a spacing, 0.115, 1.32, 0.72 and 0.91 with no field and 0.115, 1.29, 0.74 and 0.90 under two thirds of a spacing.

So a tissue whose history includes a gentle smooth deformation is, by these numbers, indistinguishable from one that has none. The band moves, it does not blur found the same thing on the golden head’s own defect rings: a field correlated over eight spacings moves the rings without changing what is on them, and a tiling read cell by cell cannot tell a moved ring from one that never moved. What a smooth field changes the four numbers can see only once it is large enough to pull pairs apart, which here is a spacing and a half.

Where a tissue fails both laws #

The earlier essay found that on its two independent routes no tissue ever sat in the corner where neither law holds, and that the smooth field alone reached it, at two spacings, with $a$ just below the band. The crossed grid says how large that corner is.

Of the sixty-nine tissues, two satisfy neither law: a smooth field of two spacings and nothing else, at $a = 0.84$ and a slope of −0.023, and a smooth field of three spacings and nothing else, at 0.75 and −0.003. Add independent steps and Aboav’s relation comes back at once — a twentieth of a spacing takes the field of two to $a = 1.03$, a tenth takes the field of three to 1.05 — because the first independent steps make new defects as bound pairs, and the pairing the field had loosened is restored by pairs it did not make. The corner is reachable only by a deformation that is smooth and nothing else.

Weaire’s intercept follows the pairing #

Weaire’s form ties Aboav’s intercept to its slope and the variance: the line of $n \, m(n)$ against $n$ should cross the axis at $6a + \mu_2$. The earlier essay found the fitted intercept falling short of that by one per cent where the defects are paired and three where they are not. Across the crossed grid the shortfall runs from 0.02 to 0.34 and tracks the pairing share with a correlation of 0.68. It is 0.07 on the ordered head, 0.03 at 0.15 of a spacing of step where new pairs have just arrived, 0.18 under a smooth field of two spacings, 0.28 under three and 0.32 under three with a fifth of a spacing of step on top. Weaire’s form was derived for tilings with short-ranged correlations, and a tiling of defects bound in pairs is the shortest-ranged there is; loosen the pairs and the form loosens with them.

The empty corner #

The other empty corner of the plane stays empty too. No crossed tissue has Lewis’s law on and $a$ above the band: the highest $a$ among tissues with the law on is 1.32, a fifth of a spacing of step with no field. A tissue whose sides vary enough to make area track them has defects too many and too varied to keep the strong anticorrelation a large $a$ needs.

What a report of one number is worth #

The practical reading is a warning about the commonest way disorder is reported. A variance of the side counts places a tissue on Lewis’s law to within the scatter of a few heads, provided it was disordered in one way; it places it on Aboav’s relation hardly at all. Two tissues with variances of 0.43 and 0.51 can sit on opposite sides of both laws. A report of the variance and $a$ together, or the variance and a Lewis slope, fixes both laws and so fixes the third number; a report of any one leaves a direction on the plane unmeasured, and the direction it leaves is the one that says whether the disorder was smooth or local.

That is also what the second moment is the measurement claimed for the golden head’s own rings, from another side: the variance is the right single number for how much a lattice departs from order, and the wrong one for how the departure is arranged.

What these tissues leave out #

They are Voronoi tilings of organ centres, one head size, one correlation length of smooth field and one order of the two moves — the field first, the steps after. A real tissue’s disorder arrives in no fixed order, and a field of a different correlation length would sweep a different band of the plane. The quadratic fit is a convention; a more flexible fit would lower every error somewhat, and the ranking of the numbers and pairs is what is read, not the last decimal of any one. Every value is a ten-seed mean on 900 organs, and one head’s $a$ scatters by a tenth or more.

Readings that would undo it #

A single one of the four numbers whose quadratic places both laws within twice the seeds’ noise. A pair that includes the variance and a law’s own statistic leaving either law at more than twice the noise. A crossed tissue with Lewis’s law on and $a$ above the band. A tissue satisfying neither law that has any independent step in its history. Each would change the answer to how many numbers a tissue needs, rather than refine it.

Still open: a tissue whose defects are placed rather than shaken #

Both routes here make disorder by moving points. Growing tissues also make it by division: a cell splits, and the two daughters and their neighbours gain and lose sides in a pattern set by where the new wall goes. Division makes defects in bound pairs by construction, which is what the first independent steps did and the smooth field undid. The next measurement is a head disordered by successive divisions at a stated rule — the longest axis, a random axis — read for the same four numbers, asking where division puts a tissue on the plane of both laws, and whether the variance with $a$ still fixes Lewis’s slope for a tissue that was never shaken at all.