Order and disorder — where it appears
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.
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.
The disorder is a staircase
Sweep the second moment of a head's side-count distribution across the divergence angle and it is not a curve. It is flat in stretches with sharp steps between them and narrow deep dips wherever a rational falls — so 0.253 is not the golden angle's number, it is the number of every angle from 137.47° to 137.54°.
A dip belongs to the head
At an exact rational the disorder halves when the head doubles, and the dip around it narrows by a factor of four. So which angles look ordered is set by how many organs were counted, and a head of three hundred cannot tell 138.4615° from 138.48° while a head of twelve hundred tells it from 138.4625°.
The most irrational is not the most disordered
If rational angles make ordered tissue, the most badly approximable angle should make the most disordered — which would at last give the golden angle a criterion it wins. Swept across the interval it is the most irrational point of, μ₂ peaks at 138.42° and the golden angle sits unremarkably in the middle.
Named alongside it
The objects these essays reach for when they reach for this one.
Voronoi cellsDisorderDivergence angleMeasurementRational angleRational approximationSummary statisticArtefactContinued fractionEuler's formulaHonest limitsLattice