Lewis's law needs the sides to vary

Lewis’s law says that in a packed cellular tissue a cell’s area rises linearly with its number of sides — a five-sided cell three quarters of the mean, a seven-sided one five quarters — and it has been quoted for a century as a property of packing. Lewis’s law wants disorder pointed a measurement at it and found the opposite of the usual reading. On nine hundred random points in a disc the law holds, with a slope of 0.22 to 0.24 area-means per side against Lewis’s 0.25. On a golden-angle head of the same size, the arrangement usually called optimally packed, the slope is 0.009: area and side count are almost independent.

That essay’s slider added disorder to the head, and the slope climbed to 0.089 — about two fifths of a random set’s — at the largest setting, which moves each organ by about a sixth of a wall spacing. Where between the two the law switches on was not measured, and neither was what switches it on. This essay takes the head the rest of the way, by three routes, and finds that one of them never gets there.

Three routes away from order #

The head is the same: nine hundred organs placed by Vogel’s rule at the golden angle, its centre organ left out, its Voronoi cells measured inside 86 per cent of its radius so the rim’s swollen cells do not enter. Every fit is Lewis’s line through the side-count classes, weighted by how many cells each holds, and every number is averaged over five seeds.

The first route moves every organ by an independent gaussian step, stated in the head’s wall spacing — the median length of a wall between two interior cells, about 1.92 in the head’s units — from a twentieth of a spacing to three. The second leaves most organs where the rule put them and replaces a share of them, from one in twenty to nine in ten, by points drawn uniformly in the disc. The third moves the whole head by a smooth random field, the one that bent the band of disputed cells round each flip ring without adding to it, correlated over eight spacings, at steps from a third of a spacing to two.

The law switches on #

At a twentieth of a spacing the joint distribution looks like the ordered head’s: three side counts, five, six and seven, their mean areas within a few per cent of each other. By a fifth of a spacing it has spread into four and eight sides, and the class means have separated — fives smaller than sixes, sevens larger. The fitted slope is 0.117. By half a spacing it is 0.184, and the cloud of cells visibly tilts along Lewis’s line. By one spacing it is 0.218, a random set’s value.

Read as a curve, the switch is gradual but not slow. The slope is 0.029 at a twentieth of a spacing, 0.064 at a tenth, 0.117 at a fifth, 0.148 at three tenths, 0.184 at a half and 0.211 at three quarters, and from there to three spacings it stays between 0.196 and 0.218. It passes half of a random set’s 0.229 at 0.19 of a spacing and nine tenths of it at 0.71, so the whole switch happens over a factor of four in the size of the displacement, and the five seeds agree to within a hundredth at every step.

So the earlier slider stopped with the law about two fifths switched on, and the rest arrives within the next half spacing. Beyond that, displacing the organs further does nothing more to the slope: the head has become, as far as Lewis’s law can tell, a random set.

Lewis’s constant is never reached #

Lewis wrote his law with a second number, $n_0$, the side count at which the fitted line reaches zero area: 2 for his cucumber epidermis, the value that makes a cell’s area proportional to its sides less two. It is a harder test than the slope, because it depends on where the line sits as well as how steep it is.

On the displaced head $n_0$ has no meaning while the slope is near nought — the line crosses zero at −29 sides at a twentieth of a spacing — and it climbs through the switch: −2.5 at a fifth of a spacing, 0.1 at four tenths, 0.5 at a half, 1.2 at three quarters and 1.4 at one spacing. Then it falls back, to 1.1 at two spacings and 0.8 at three. It never reaches 2. Neither does the random set, at 1.62, and neither does the mixture, which peaks at 1.54.

So even the tissue the law was found to hold on satisfies it only in its slope. Lewis’s constant is a property of the particular tissue he measured, or of real cell walls rather than Voronoi ones, and none of the three routes from a phyllotactic head reaches it.

A partly disordered head obeys it better #

The slope says how steeply area rises with side count on average. The earlier essay argued that the number the citation leaves out is how much of the variation in area the side count accounts for, since a law that fixes the mean of each class and says nothing about the cells in it is a description of averages.

On the ordered head the side count explains 16 per cent of the variation — of a variation so small, a coefficient of 1.2 per cent, that the share means little. Displaced, the share falls to 9 per cent at a twentieth of a spacing, as the displacement adds area variation the side counts have not caught up with, then climbs: 17 per cent at a tenth, 29 at a fifth, 39 at three tenths, and 41 per cent from four tenths to three quarters. Then it falls again, to 32 per cent at one and a half spacings and 13 at three.

A random set’s figure is 31 per cent. So between about three tenths and three quarters of a spacing, a head part way to disorder obeys Lewis’s law more closely than the random tissue the law was found on. The reason is visible in the joint distribution. Part way along, the head has acquired varied side counts but not yet the extra variation in area that random positions add on top; its areas are still largely set by its sides. Further along, the positions themselves become random, cells of the same side count differ widely in area, and the share the side count explains drops back to the random set’s and below it. At three spacings the displacement has become a scatter much larger than the cells, and the tiling is no longer a tissue of the head at all.

The same curve by another route #

Replacing organs switches the law on faster in terms of how many organs are touched: replacing one organ in twenty gives a slope of 0.102, about what displacing every organ by 0.15 of a spacing does, and replacing one in five gives 0.170. It reaches the random set’s slope by seven in ten.

Set against the second moment of the side count — how far the side counts spread from six — the two routes rise together. The unmoved head’s second moment is 0.25, a random set’s 1.65, and on both routes the slope climbs with it. The mixture runs a little ahead: at a second moment of 0.8 it has a slope of 0.133 where the displaced head has 0.12, and at 1.2, 0.18 against 0.15. A replaced organ drops a cell of random size into an ordered neighbourhood, and its neighbours’ sides and areas shift together, which ties area to side count a little more tightly than an even displacement does.

Varied areas are not enough #

The third route is the one that decides what the law is about.

A smooth field correlated over eight spacings stretches some regions of the head and compresses others, and the cells go with them. At a step of two spacings the coefficient of variation of cell area is 0.39 — nearly a random set’s 0.53, and more than a displaced head carries when its slope is already 0.2. And the slope is −0.012. At every step of the smooth field read, from a third of a spacing to two, it stays within 0.012 of nought, and the side count explains 2 to 4 per cent of the variation in area.

Set out against the area variation, the picture is two curves. The displaced head and the mixture lie on one line: whatever route brings the head’s cells to a given variation in area, the slope is the same function of it, rising from nought to the random set’s value. The smooth field runs along the floor.

The agreement between the first two is close enough to be read off. With every organ displaced by four tenths of a spacing, the cells’ areas vary by a coefficient of 0.325 and the slope is 0.170; with a fifth of the organs replaced, they vary by 0.321 and the slope is 0.170. At a coefficient of about 0.19 the two routes give 0.117 and 0.112. Two quite different kinds of disorder — every cell nudged a little, or a few cells replaced entirely — arrive at the same Lewis slope whenever they arrive at the same variation in area, which is what one would expect if both act through the same thing: new sides, and the areas those sides carry.

The joint distribution shows why. The smooth field’s cells range from half the mean area to more than twice it, but in every side-count class alike. Its side counts are still the lattice’s — five, six and seven for all but nine of its 741 cells, with a second moment of 0.29 against the unmoved head’s 0.25 — because a smooth field keeps the head’s census: the same fives and sevens on the same rings, moved. A large cell there is large because its region was stretched, not because it has more sides, and a small one small because its region was compressed. The class means even run the wrong way: sevens average 0.90 of the mean area and sixes 1.03, since a stretched region grows its hexagons and the fives and sevens sit on rings that the field happened to compress. Lewis’s law has nothing to fit.

What the law measures #

Put the three routes together and the law’s content is plain. It is not a statement about packed tissue, since the ordered head is packed and fails it. It is not a statement about variable tissue, since the smooth-field head is variable and fails it. It holds when a tissue’s side counts vary — when its cells have been given fives and eights and nines by disorder that breaks the lattice locally — and it then says that those side counts carry area with them.

Two laws that want opposite tissue found that Aboav’s relation, the other half of the pair Lewis’s law is quoted with, holds on the ordered head and fails on the random set. So the two laws are not two properties of tissue. Lewis’s is a readout of local disorder in the sides, Aboav’s of order in their neighbourhoods, and a tissue part way between satisfies each to the degree it has the thing each reads.

This also changes what a measured Lewis slope says about a real tissue. A slope near Lewis’s 0.25 says the side counts are as varied as a random set’s, which a photograph shows directly. A slope near nought says either that the tissue is ordered or that its variation in area is the smooth kind — growth faster in one region than another — and the two are not told apart by the slope. They are told apart by the second moment of the side count, which the slope was always standing in for.

The switch against the other thresholds #

The switch can be set against the thresholds the same head showed when displaced for its neighbour relations. There, the single contact cut-off off the flip rings was gone by a fiftieth of a spacing, the rings’ hold on their fives and sevens by four hundredths, and the three-family count’s exactness by eight hundredths. Lewis’s law is a tenth switched on at five hundredths and half switched on at a fifth.

So the order in which an ordered head gives way under independent disorder runs: its exact contact geometry first, then the confinement of its defects to the rings, then its count, and only after all of those, at displacements five to ten times larger, the relation between its cells’ sides and areas that a tissue biologist would measure. The law a century of citation attached to packing is the last thing a packed head gives up.

What the three routes leave out #

The cells here are the Voronoi cells of organ centres. A real epidermis has walls that meet at angles set by tension, not by distance, and the areas of its cells are not those of a Voronoi tessellation of their nuclei; six sides on average is forced either way, and Lewis’s slope need not be.

The routes are idealised. A real tissue’s disorder is presumably correlated over some distances and independent over others, and the arithmetic here says the two parts would act separately: the independent part switching the law on, the correlated part adding variation in area the law cannot see. A tissue with a slope well below a random set’s but areas as varied could be read that way, and nothing here measures a real one.

And the head is one size. The earlier essay found the law’s classes thin at small sizes; at nine hundred organs every class the fit uses holds at least a handful of cells, and the five seeds agree.

What a fifth of a spacing looks like #

The number at the middle of the switch is worth making concrete. A fifth of a wall spacing is a fifth of the distance between two neighbouring organ centres: on a sunflower head whose florets are three millimetres apart, six tenths of a millimetre. Every floret displaced that much, independently, is enough to give the head a Lewis slope half a random tissue’s — and, on the first seed, 159 fives, 146 sevens, 16 fours and 22 eights where the ordered head had 79 fives, 73 sevens and nothing else. A head photographed and digitised to a floret’s own width would carry that much error from the digitising alone.

The practical reading runs the other way from the usual one. A Lewis slope measured on a digitised head is not evidence about the head’s packing unless the digitising error is known to be well under a tenth of a spacing; above that, the slope measures the error as readily as the tissue.

Readings that would undo it #

A smooth field, correlated over eight spacings, that gives the head a Lewis slope above 0.05 at any step read. An independent displacement at which the slope reaches half a random set’s below a tenth of a spacing or above three tenths. A route under which the slope, set against the variation in cell area, falls off the curve the displaced head and the mixture share. Any of those would mean the law reads something other than varied sides.

Still open: Aboav’s relation along the same routes #

Aboav’s relation is the other half of the pair, and it pointed the opposite way on the two ends: held by the ordered head, failed by the random set. The same three routes would say where it switches off, and whether it switches off at the same displacement as Lewis’s switches on — in which case the two laws are one reading of disorder taken from two sides — or at a different one, in which case a tissue part way to disorder can satisfy both or neither. The smooth field is the sharpest test: if it keeps the ordered head’s Aboav relation as it keeps its side counts, then Aboav’s relation, like Lewis’s law, reads the sides and not the areas.