Phyllotaxis
the spirals are counted, not admired
Concept

Lewis's law — where it appears

The claim that a cell's area rises linearly with how many sides it has. It wants a tissue disordered enough for side counts to vary, which is the opposite of what the neighbouring law about side counts wants.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

Cell area against side count, at 0% disorder. The dashed line is Lewis's law, (n−2)/4. The fitted slope here is 0.014 against his 0.25, and the side count accounts for 20% of the variation in area.

Lewis's law wants disorder

Cell area rises linearly with side count — measured on cucumber epidermis in 1928 and quoted ever since as a property of packed tissue. It holds beautifully on a random point set, with a fitted constant of 1.64 against Lewis's 2. On a phyllotactic head it does not hold at all: the slope is 0.009, and area and side count are almost independent.

tissue · Lewis Two laws, two tilings, and they disagree about which tiling is tissue. Lewis's law wants disorder: its slope is 0.231 on the random set and 0.009 on the golden head. Aboav's relation wants order: a = 1.18 on the head, 0.59 on the random set.

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.

tissue · Cell laws Cell area against side count on a 900-organ head, every organ displaced by 0.2 of a spacing. The joint distribution of cell area, as a multiple of the mean, and side count, over the 639 interior cells of a golden head with every organ displaced by 0.2 of a wall spacing, seed one. The dashed line is Lewis's law, a quarter of the mean area for each side; the solid line is the fit, at a slope of 0.113, and the side count explains 30 per cent of the variation in area. Classes: 4 sides, 16 cells, mean 0.75; 5 sides, 159 cells, mean 0.89; 6 sides, 295 cells, mean 1.00; 7 sides, 146 cells, mean 1.11; 8 sides, 22 cells, mean 1.22; 9 sides, 1 cells, mean 1.26.

Lewis's law needs the sides to vary

Lewis's law holds on a random set of points and fails on a golden-angle head. Walked from one to the other by displacing every organ independently, the head's Lewis slope reaches half a random set's at a fifth of a wall spacing and nine tenths by seven tenths, and in between it explains up to 41 per cent of the variation in cell area — more than the 31 per cent it explains in the random set. Moved instead by a smooth field correlated over eight spacings, the head's cell areas become nearly as varied as a random set's and its slope stays at nought, because its side counts stay the lattice's. The law is not about how varied the cells are. It is about how varied their sides are.

tissue · Lewis The cells of a golden head with every organ displaced by 0.15 of a spacing, five- and seven-sided neighbours joined. A window eleven wall spacings square, about halfway out on a 900-organ golden head with every organ displaced by 0.15 of a wall spacing, seed one. Cells are keyed by side count; every five-sided cell is joined to each seven-sided cell it touches. In the window: 26 five-sided, 75 six-sided, 26 seven-sided and 5 of other counts, with 40 five–seven contacts. Over the whole head, averaged over five seeds: Aboav's a = 1.45, and 91 per cent of five-sided cells touch a seven.

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.

tissue · Cell laws

Named alongside it

The objects these essays reach for when they reach for this one.

Order and disorderVoronoi cellsAboav–WeaireDisorderDisplacementExplained varianceHonest limitsJoint distributionSummary statisticTopological correlationTopological defect

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