Cell laws — the series
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Two laws that want opposite tissue
Lewis's law and Aboav's relation are quoted side by side as properties of cellular tissue. Measured on the same two tilings they point opposite ways — the ordered head satisfies Aboav's with the textbook value of 1.18 and fails Lewis's completely; the random set does exactly the reverse.
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One law counts sides, the other pairs
Lewis's law and Aboav's relation point opposite ways at the two ends of disorder, and the obvious guess is that they are one reading of disorder taken from two sides. Measured on the same moved heads, they are not. Displaced organ by organ, Aboav's a first rises — to 1.45 at 0.15 of a wall spacing, as the first new defects arrive as bound five–seven pairs — and falls half-way to a random set's only at 0.45 of a spacing, where Lewis's law had switched on at 0.2. Between the two a tissue satisfies both. A smooth field, which never switches Lewis's law on, lowers a by pulling the pairs apart without making any new defects. Lewis's law reads how varied the sides are; Aboav's reads whether the defects are paired.
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Two numbers for a tissue, and which two
Lewis's law and Aboav's relation read different things in a tiling — how varied the sides are, and whether the defects are paired — so a tissue has a place on a plane of both. Move a golden head by a smooth field and then displace it organ by organ, over a grid of both, and the tissues fill that plane rather than lying along a line. No single one of the four numbers a tissue is usually reported by places it on both laws: the variance of the side counts reads Lewis's slope to three times the seeds' noise and misreads Aboav's a, the pairing share reads a to two and a half times and misreads Lewis's slope. The variance with either law's own statistic places both to within one and a half times the noise; the variance with the pairing share, which is what a counter of cells records, to about twice. And the only tissues that fail both laws are heads moved by a smooth field of two spacings or more and nothing else.
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A tissue that was never shaken
Every tissue whose laws have been read here was disordered by moving its points. A growing tissue also disorders itself by dividing, and a division is a wall no set of points generates. Held as a map and divided cell by cell by three rules, a golden head's tiling switches Lewis's law on once a tenth of its cells have divided, at a variance of side counts lower than any moved tissue reaches the law at, because the commonest single division makes two half-sized fives and two full-sized sevens at once. Dividing the largest cell first reaches the corner of the plane no moved tissue reached — Lewis's law on and Aboav's a above its band, at 1.67 — because the largest cells of a golden head are its sevens. And the two numbers that placed every moved tissue on Lewis's law to one and a half times the noise misplace a divided one by fifteen times it: they were a calibration of how the tissue was disordered, not of tissue.
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A tissue that grows forgets Lewis's law
A tiling divided cell by cell switches Lewis's law on within a tenth of the cells, because the commonest division makes two half-sized fives, and nothing ever restores them. Give the walls a vertex model and each cell a target size that recovers between rounds, and the law becomes a memory that fades. A tissue regaining a tenth of its loss each round never switches it on. One regaining three per cent reaches 0.16 and is falling again by the time it has twice the cells. One regaining all of it reads 0.006 at a variance of side counts of 1.8, a place no moved tissue goes. Aboav's a hardly notices, and the variance of the side counts, as a map counts them, barely more. So the slope on a growing tissue measures how recently its cells divided, set against how fast they grow.