One law counts sides, the other pairs

Two relations about cellular tilings are quoted side by side more than any others. Lewis’s law says a cell’s area rises with its number of sides. Aboav’s relation, in the form Weaire gave it, says a cell with many sides is surrounded by cells with few: the mean side count of an $n$-sided cell’s neighbours is $m(n) = 6 - a + (6a + \mu_2)/n$, with $\mu_2$ the variance of the side counts and $a$ near 1.2 in the materials it was measured on.

Two laws that want opposite tissue measured both on the same two tilings and found them pointing opposite ways. A golden-angle seed head satisfies Aboav’s relation with $a = 1.18$, the textbook value, and fails Lewis’s law completely; a random set of points fails Aboav’s, at 0.59, and satisfies Lewis’s. It then said that in between Aboav’s $a$ falls monotonically and that the two curves “cross somewhere in the middle”. Neither half was measured.

Lewis’s law needs the sides to vary then measured the Lewis half. Moving a 900-organ head towards disorder by three routes — every organ displaced independently, a share of organs replaced by random points, and a smooth field correlated over eight wall spacings — it found Lewis’s law switching on at a fifth of a wall spacing on the first route and never on the third. It ended by asking the same of Aboav’s relation: whether it switches off at the displacement Lewis’s switches on, in which case the two laws are one reading of disorder, or somewhere else.

This essay reads Aboav’s relation on exactly those heads. Every tiling below is one on which Lewis’s slope has already been measured, so each point has both laws read off the same cells.

Where the defects sit, and what happens to them #

Start with the picture, since the explanation is in it. A golden head is a set of points, and its Voronoi cells are nearly all hexagons, and the rest are five-sided and seven-sided cells bound to each other in pairs: on the 900-organ head every one of 73 seven-sided cells touches a five-sided cell, and every one of 79 five-sided cells touches a seven.

Displace every organ by 0.15 of a wall spacing and the window fills with new defects — the variance of the side counts doubles, from 0.25 to 0.54 — but they arrive the way the old ones sat: in fives and sevens that touch. Across the whole head 93 per cent of sevens still touch a five. Turn the dial to half a spacing and the chains start to include four- and eight-sided cells; at a full spacing the fives and sevens are everywhere and the lines joining them are a net rather than a set of pairs.

Aboav’s relation is a statement about exactly that. A seven-sided cell has taken a side from somewhere, and the relation measures how close to it the side was paid back. In a pair the debt is settled one cell away.

Aboav’s a rises before it falls #

The first thing the measurement shows is that the earlier essay’s “falls monotonically” is not so. Displaced by 0.05 of a wall spacing, the head’s $a$ is 1.28; by 0.1, 1.42; by 0.15, 1.45. Only then does it fall: 1.35 at 0.2, 1.03 at 0.3, 0.85 at half a spacing, 0.73 at a full one and 0.67 at three, where it reaches a random set’s.

The rise has the explanation the picture gave. The new defects are born as bound pairs, and a pair is the tightest compensation a tiling can make. The side-count variance $\mu_2$ grows with every new pair, and in Weaire’s form a steeper fall of $m(n)$ with $n$ — which is what bound pairs produce — reads as a larger $a$. So a head displaced a little is, by Aboav’s measure, more Aboav-like than the ordered head it came from. What finally lowers $a$ is not more defects but defects that are no longer paired, and that takes displacements of a third of a spacing and more.

One head’s $a$ is a noisy number. Across five seeds at one displacement it varies by up to half a unit, and its standard deviation runs from a few hundredths to 0.2. Every value here is a five-seed mean, and the crossings below are good to about a tenth of a spacing.

They switch at different places #

The question the Lewis essay asked has a direct answer. Lewis’s law is called on when its slope reaches half a random set’s, the criterion that essay used; on independent displacement that happens at 0.19 of a wall spacing. Aboav’s $a$ falls half-way from the ordered head’s 1.18 to the random set’s 0.67 at 0.44 of a spacing, and leaves the band within 0.35 of 1.2 — the criterion the two-laws essay checked the ordered head against — at 0.50. The two laws do not switch at the same place. Aboav’s switches off at a little over twice the displacement at which Lewis’s switches on.

Replacing organs with random points tells the same story faster. Lewis’s law switches on by seven per cent of organs replaced, because a random point dropped into an ordered head makes a cell of a quite different area with a quite different side count at once. Aboav’s $a$ has no rise on this route — a random point does not make its defects in bound pairs — and falls half-way at about a fifth of the organs replaced and out of the band at about a third. The factor between the two switches is three rather than two, and the order is the same.

A tissue can satisfy both, and one route satisfies neither #

Put the two readings on one plane and the route becomes a path through four regions.

Independent displacement leaves the ordered head in Aboav’s region, rises within it, crosses into the box where both laws hold at 0.2 of a spacing, stays there to half a spacing — at 0.5, by two thousandths of the band — and leaves into Lewis’s region. Replacing organs does the same without the rise. On neither route does the tissue ever sit in the corner where neither law holds.

So the earlier essay’s guess that there is “a disorder level at which a tiling satisfies both relations moderately” is right, and the level is measured: between a fifth and a half of a wall spacing of independent displacement. A tissue reporting Aboav’s $a$ near 1.2 and a Lewis slope of 0.12 to 0.18 is not a contradiction and not a coincidence. It is a tissue in that window.

The third route is the exception, and it is the most informative. The smooth field keeps Lewis’s slope at nought throughout, as the Lewis essay found, and at its largest step of two spacings its $a$ falls to 0.84, just below the band. A smoothly deformed head, the one route of three, ends up satisfying neither law.

What the smooth field does #

Under the smooth field the side counts keep the ordered head’s variance: $\mu_2$ is 0.23 to 0.28 at every step, against a random set’s 1.64. That is why Lewis’s law never switches on — its slope needs varied sides, and the field makes none. Aboav’s $a$ holds at 1.18 to a step of 0.64 of a spacing, then falls to 1.09 at one spacing and 0.84 at two.

What falls with it is the pairing. At a step of 0.32 of a spacing every five touches a seven; at two spacings eight fives in ten do, while the sevens still nearly all touch a five, 95 per cent of them. The field stretches the head over distances of eight spacings, and where it stretches hardest it makes a few fives with no seven beside them. It makes few defects of any other kind, so the variance of the side counts barely moves, from 0.25 to 0.28; what changes is where the defects sit relative to each other.

This is the cleanest separation of the two laws the three routes allow. Lewis’s law, on this route, reads a variance that does not move and stays off. Aboav’s relation reads an arrangement that does move, and switches off. They are sensitive to different properties of the same tiling — Lewis’s to how varied the sides are, Aboav’s to whether the defects are paired — and a route that changes one without the other shows it.

It fits what the defects lie on rings and the band moves, it does not blur found from the other direction: on a golden head the defects are an organised structure, rings of bound pairs at radii the divergence sets, and a smooth field moves that structure rather than dissolving it — until its step is large enough to pull the pairs themselves apart.

Where the pairing reaches chance #

The pairing can be put on a scale. If a seven-sided cell’s seven neighbours were drawn at random from the head’s cells, the chance that at least one is five-sided would be $1 - (1 - f_5)^7$, with $f_5$ the five-sided share. That ignores that neighbours are not independent draws, so it is a scale rather than a null hypothesis; but it is the same scale at every reading, and a tiling can be placed on it.

On the ordered head, where fives are 13 per cent of cells, a random draw of neighbours would give a seven a five 62 per cent of the time, and every seven has one: the pairing is 38 points above the scale. Displaced by 0.15 of a spacing it is 12 points above; by 0.3, within 2; by half a spacing and beyond, a few points below, as on a random set. So along independent displacement the sevens’ pairing reaches the chance level at about a third of a wall spacing — close to the 0.44 at which Aboav’s $a$ is half-way off, and well past the 0.19 at which Lewis’s law switched on. The two things Aboav’s relation reads, a correlation between neighbours’ side counts and the pairing of defects, fade together.

Under the smooth field they do not fade together, and that is the exception worth recording. At two spacings the sevens still touch a five 29 points above the scale, while a fifth of the fives have lost theirs. Aboav’s $a$, fitted over all cells, follows the fives.

Weaire’s second parameter #

Weaire’s form has a second number in it besides $a$: the line of $n \, m(n)$ against $n$ should cross the axis at $6a + \mu_2$, which ties the relation’s intercept to its slope and to the variance of the side counts. Fitted freely on each head, the intercept falls short of that by 0.07 on the ordered head, by 0.04 to 0.07 up to 0.3 of a spacing of displacement, and by 0.20 to 0.24 from a full spacing on; on a random set, by 0.14. Against intercepts of seven to eight, that is a one-per-cent miss where the defects are paired and a three-per-cent miss where they are not.

So the form holds best exactly where its $a$ is largest, and both are readings of the same structure. Weaire derived the form for tilings whose correlations are short-ranged, and a tiling of bound pairs is as short-ranged as a tiling can be.

Reading a tissue off the plane #

The plane of both laws can be run backwards. A tissue for which both numbers have been measured sits somewhere on it, and the three routes say what kind of departure from order could have put it there.

A Lewis slope near nought with Aboav’s $a$ near 1.2 is the ordered head, or a head moved by a smooth field of up to about a spacing; the two cannot be told apart by these numbers, and the variance of the side counts cannot separate them either, since the field leaves it at the lattice’s. A Lewis slope near nought with $a$ well below 1.2 has only one source among the three routes: a smooth deformation large enough to loosen the pairing. A slope of 0.12 to 0.18 with $a$ between 1.0 and 1.35 is independent displacement of a fifth to a half of a spacing, or a tenth to a third of the organs replaced. A slope above 0.2 with $a$ below 0.8 is a tissue indistinguishable, by these two numbers, from points thrown down at random.

Two places on the plane no route reaches: a high Lewis slope with $a$ above the band, and — on the independent routes — the corner where neither law holds. A real tissue measured there has departed from order by some route not modelled here, and that would be worth knowing. What the plane cannot do is separate the ordered head from its smooth deformations, and for that the drawing of the pairs, or where the defects lie, is needed rather than either law.

The class means behind a #

Aboav’s relation is usually drawn through class means rather than fitted over cells, and the class means show where the changes in $a$ come from.

On the unmoved head a five-sided cell’s neighbours average 6.31 sides and a seven’s 5.89. Displaced by 0.15 of a spacing the new four- and eight-sided classes appear with steep compensation — a four’s neighbours average 6.87, an eight’s 5.72 — while the fives and sevens keep theirs, and the line through the classes steepens: that is the rise in $a$. Displaced by a full spacing, every class’s neighbours average more than six except the eights and nines, the line flattens, and it lies almost on a random set’s. A six-sided cell’s neighbours average 6.29 there, where on the ordered head they average 6.02: in a random tiling a hexagon is more often beside a seven than beside a five.

What this reading does not establish #

It does not say that a Voronoi tessellation of organ centres draws the cells a tissue has, though Euler’s formula fixes its mean at six whatever draws it; that is the standing caveat of every tiling measurement here, and the second moment is the measurement set out why the side-count variance, not the mean, is the number worth reporting from one.

It does not say real disorder is either independent from cell to cell or smooth. The two routes bracket what a real epidermis might do, and a real one probably does both — which would put it on a path between the independent and smooth curves on the plane of both laws, and possibly through the corner where neither holds.

And the criteria are conventions. Half a random set’s Lewis slope and a band of 0.35 about 1.2 are the thresholds the earlier essays used; with others the crossings move, but the order of them does not, because Aboav’s $a$ is still above the ordered head’s value where Lewis’s slope has already reached half its random value.

Measurements that would undo it #

An independent displacement below 0.15 of a spacing at which Aboav’s $a$, over five seeds, is lower than the ordered head’s. A displacement at which Lewis’s law is off and Aboav’s $a$ is out of its band on either independent route. A smooth step at which $a$ falls while every five still touches a seven. Each would contradict one of the three findings rather than refine it.

Still open: the third law of the canon #

Lewis’s law and Aboav’s relation are two of three relations the cellular-tiling literature quotes; the third is the distribution of side counts itself, usually reported as its variance $\mu_2$ and compared across tissues as a single measure of disorder. The three routes here separate it from both laws: the smooth field holds $\mu_2$ fixed while Aboav’s $a$ falls, and independent displacement raises it while $a$ rises. What remains open is whether any single number — $\mu_2$, $a$, a Lewis slope, or a pairing share like the fraction of fives touching a seven — places a tissue on the plane of both laws, or whether two are always needed. The measurement is the three routes crossed with each other: heads moved by a smooth field and then displaced independently, read for all four numbers at once.