Lewis's law wants disorder
F. T. Lewis measured the epidermis of a cucumber in 1928, sorted the cells by how many sides they had, and found that the mean area of the -sided cells rose linearly with . Written in the form it is usually quoted,
so a five-sided cell has three quarters of the mean area and a seven-sided cell has five quarters.
It has been quoted across biology for nearly a century, generally as a property of packed cellular tissue and often with the implication that it reflects efficient packing. It is the sort of claim this site exists to point a measurement at — like the nautilus, like optimal packing, like the hexagon, and there are two questions worth asking that the usual citation does not answer.
Does it hold in a phyllotactic tiling? And — separately, and more importantly — how much of the variation in cell area does it actually account for?
The measurement
Nine hundred points in a disc, in two arrangements: a golden-angle head, and a Poisson set of the same size in the same disc. Voronoi cells for both, the outer 14% of the radius discarded, and the area of every interior cell measured.
For the Poisson set, the fit is , which crosses zero at . Lewis’s law says slope 0.25 and . Class by class the agreement is close: three-sided cells at 0.19 against a predicted 0.25, six-sided at 0.99 against 1.00, ten-sided at 1.94 against 2.00.
For the golden-angle head, the fit is . The slope is a fortieth of Lewis’s. Five-sided cells average 0.992 of the mean area, six-sided 1.000, seven-sided 1.011. Cell area is very nearly independent of side count.
So the law holds for the random tiling and fails for the ordered one. Which is worth pausing on, because the ordered tiling here is a golden-angle head — the arrangement the popular literature calls optimally packed.
How the cells are measured
The tiling is the Voronoi diagram of the point set: each cell is the region closer to its own point than to any other, built as the dual of the Delaunay triangulation by joining the circumcentres of the triangles around each point in angular order.
Three decisions in that pipeline affect the numbers and each is worth stating.
Unbounded cells are excluded, not estimated. A point on the convex hull has a cell that runs to infinity and has no area. Averaging over one silently is how a packing measurement goes wrong, so those cells are marked and dropped.
The rim is cut further in. Discussed below at length, because it is the decision that mattered most.
Areas are normalised by the mean. Every quantity reported is a ratio to the mean cell area of the population being measured, so nothing depends on the disc’s scale or on how many points were used.
None of this is novel; it is the standard treatment. It is set out because the failure mode this essay had to recover from came from getting the second one wrong, and a reader deciding whether to believe the slope of 0.009 needs to know what was excluded to obtain it.
Why that is the wrong way round
The usual reading of Lewis’s law runs something like: cells in a tissue are packed efficiently, so their areas are constrained by their neighbourhoods, so area and side count are related. Efficiency is doing the work in that sentence.
The measurement says the opposite. The efficiently packed arrangement — the golden-angle head, whose cells are as near identical as this subject gets — is the one where side count says nothing about area. The disordered arrangement, where cells have every size and shape, is the one that obeys the law.
Which makes sense once stated. Lewis’s law relates two kinds of variation, and it needs both to exist. In a near-regular tiling almost every cell has the same area and almost every cell has six sides, so the small residual variations in the two quantities are not strongly linked — the departures from six sides are topological defects, and a defect can occur without much change in area.
In a disordered tiling there is a genuine range of areas and a genuine range of side counts, and a large cell really does tend to touch more neighbours. The relation is a statement about that tendency.
So Lewis’s law is a law about disorder. It describes how variation in area and variation in topology travel together when there is variation to describe, and it has nothing to say about a tiling that has almost none.
What the law is competing against
It helps to say what the alternative hypothesis is, since “the law fails on the ordered tiling” is only interesting if something else is true there.
What is true on the ordered tiling is that every interior cell has essentially the same area. Five-sided cells average 0.992 of the mean, six-sided 1.000, seven-sided 1.011 — a spread of two per cent across the whole range of topologies. The standard deviation within each class is of the same order: 0.011 for the five-sided, 0.007 for the six-sided.
So the correct description of that tiling is not “Lewis’s law with a shallow slope”. It is “constant area, with topological defects that cost almost nothing”. Those are different statements and only the second suggests the right follow-up questions — where the defects are, why they occur in pairs, and what they do to the parastichy counts.
The next essay takes those up, and the short answer is that the defect structure is exactly what Aboav’s relation measures, and that the ordered tiling scores well on it for the same reason it scores badly here.
The slider makes the point
The figure’s knob walks between the two cases: jitter each point by a fraction of the mean spacing and refit.
At zero disorder the slope is 0.009. At about half a spacing it is 0.079. By a full spacing the tiling is close to random and the slope is approaching Lewis’s value.
That continuity is the useful form of the result, because it says the two cases are not different phenomena. There is one relation, its strength is a function of how disordered the tiling is, and Lewis measured it on something disordered enough for it to be strong.
Cucumber epidermis is disordered. Anyone who has looked at one knows that; the cells are irregular polygons of visibly different sizes. The law was measured on the right material for it.
The knob also connects this to a measurement the disc essays already carry. Adding disorder to a head degrades every packing statistic on it, and the largest empty gap is the one that degrades most legibly. Lewis’s slope moves the other way, which is the whole of this essay in one sentence: the statistic that improves with disorder is the one being quoted as evidence of order.
Where the disorder comes from
The jitter is not a physical model of anything and it is worth saying so before it is over-interpreted.
Each point is displaced by an independent uniform amount, in both coordinates, up to a stated fraction of the mean spacing. That is white noise on positions, not a simulation of developmental variability, and a real tissue’s disorder has structure — cells divide, boundaries move, neighbours are correlated — that this does not reproduce.
What it does supply is a controlled interpolation between the two cases, along a single axis, which is exactly what is needed to show that they are two ends of one thing rather than two phenomena. The claim being made is about the direction of the effect and its monotonicity, and those survive the crudeness of the noise model.
The Poisson comparison set is generated the same way every build, from an explicit seed, with the radial coordinate drawn as so that density is uniform over the disc rather than over the radius. Drawing uniformly instead is the classic error and it produces a set with a dense centre, which would have given the ordered tiling an unearned advantage on every statistic here.
The number the citation leaves out
Here is the part that changes how the law should be read, and it cannot be seen from the means at all.
Take the Poisson tiling, where the law holds well. Ask what fraction of the variance in cell area is accounted for by knowing the side count.
Thirty-two per cent.
Two thirds of the variation in how big a cell is has nothing to do with how many sides it has. Within the six-sided class alone the areas run from a third of the mean to more than twice it, and the standard deviation inside a class is larger than the spread between the class means.
That is not a criticism of Lewis. It is what a relation between a mean and a variable looks like when the scatter is drawn, and Lewis drew means because in 1928 one drew means. It is a criticism of the way the law is invoked, which is almost always as though it constrained a cell rather than a class of cells.
The practical consequence: a six-sided cell in a disordered tissue is not “a mean-sized cell”. Knowing its side count moves the expected area a little and leaves most of the uncertainty in place.
Three classes are not a fit
There is a second, more mundane reason the ordered head resists the law, and it should be separated from the substantive one.
A golden-angle head’s Voronoi tiling has cells with five, six or seven sides and nothing else. That is three points to fit a two-parameter line through, with 75% of the cells in one class, and the fitted intercept is consequently meaningless — it comes out at , which is not a physical statement about anything.
So part of the failure is that the ordered case does not supply enough classes to test the law on. The jitter slider is what separates the two effects: as disorder rises, classes from four to eight appear, the fit becomes well conditioned, and the slope still has to climb from 0.009 to 0.24. The conditioning problem goes away early; the substantive gap closes slowly.
Reporting the fitted for the ordered head at all is arguably an error, and it is left in the figure’s caption deliberately, because a fitted constant that comes out at minus a hundred is a useful thing for a reader to see once. It is what a fit does when asked a question the data cannot answer.
What this does to the packing story
The site’s disc essays already found that three packing criteria pick three different angles, and that the evenness-of-areas criterion in particular is comprehensively fooled — it is won by rational angles, whose sliver cells on radial rays are near-identical while the wedges between the rays are enormous.
Lewis’s law is another single-number summary of a tiling, and it goes the same way. It is a real regularity, it can be fitted, the fit produces a number, and the number does not mean what its name suggests. Reading a high Lewis slope as evidence of good packing would be exactly backwards: the best-packed tiling here has the lowest slope there is.
That is now three separate summary statistics on this site that behave perversely on the ordered case: area evenness, Lewis’s slope, and — in the next essay — Aboav’s parameter, which behaves perversely on the disordered case instead. The common thread is that all of them were devised on disordered material and are quoted as though they were general.
What a mean hides, in one class
The variance figure is easier to feel with one class laid out.
In the Poisson tiling the six-sided cells — the largest class, 189 of them — have a mean normalised area of 0.99, sitting almost exactly on the mean of the whole population, which is what Lewis’s law says they should do.
Their standard deviation is 0.44. So a typical six-sided cell is somewhere between about half and one and a half times the mean area, and the extremes run from under a third to over twice.
Now compare classes. The five-sided cells average 0.81 and the seven-sided 1.26 — a difference between class means of 0.45, which is almost exactly one within-class standard deviation.
That is the whole of the “32% explained” figure, restated in a form that is harder to over-read: moving from a five-sided cell to a seven-sided one shifts the expected area by about as much as the ordinary scatter within either class. The relation is real and it is one standard deviation wide.
What was checked, and how it could have failed
The build asserts both halves, and both are needed.
The disordered tiling must fit Lewis’s law with a constant near his: , which it does at 1.79. Without that half, the ordered case’s failure could be a bug in the cell-area code.
The ordered tiling must fail it: slope less than a quarter of the disordered one, which it does by a factor of twenty-five. Without that half, the first is just a report that a well-known law is well known.
One implementation detail is load-bearing enough to state. The rim cut — discarding cells outside 86% of the radius — is not cosmetic. A cell one ring inside the convex hull is bounded, has a finite area, and is enormous; on the first run those cells alone put the standard deviation of the five-sided class at twice its own mean, which read as wild disorder in a tiling that has none. The cut is the same decision the Voronoi code already makes for unbounded cells, taken one ring further in.
Without it the golden head appeared to have a Lewis slope of : strongly negative, comfortably significant, and entirely an artefact of a few dozen boundary cells.
That failure is worth dwelling on for a moment, because it is the more dangerous kind. It did not look like a bug. A negative Lewis slope on an ordered tiling is a publishable-sounding result — one could write a paragraph explaining why efficient packing inverts the relation — and nothing in the figure looked wrong. The cells were drawn correctly, the fit was computed correctly, and the number was correctly derived from the data it was given.
What was wrong was the population. Sixty or so cells with areas ten to fifty times the median, all of them in the outermost ring, all of them five-sided because a boundary cell has fewer neighbours, and between them they dominated a class of eighty.
The general form: a statistic computed on a population defined by a geometric criterion inherits every artefact of that criterion, and “bounded” is a much weaker criterion than it sounds. The Voronoi code already knew this — it excludes hull cells for exactly this reason — and the fix was to apply the same reasoning one ring further in rather than to invent anything.
What holds this up
The result rests on three things and it is worth saying which one is load-bearing.
It does not rest on the Poisson set being a good model of tissue. It is not; it is a null comparison.
It does not rest on the exact value of the rim cut. Moving it between 0.80 and 0.90 of the radius changes the fitted slopes by a few per cent and changes nothing about the comparison.
It rests on the contrast — the same discipline the spiral counting is built on: the same code, the same rim cut, the same normalisation, run on two point sets that differ only in their arrangement, returning slopes that differ by a factor of twenty-five. Every criticism that applies to one half applies equally to the other, which is what makes the difference between them a measurement rather than a pair of numbers.