Phyllotaxis
the spirals are counted, not admired
Concept

Packing quality — where it appears

A number for how well a head fills its space, such as the closest pair, the largest empty circle or the evenness of its cell areas. They disagree about which divergence is best, so a claim that one angle packs best has to say by which number and at what scale.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Closest pair across 120–155° at 400 organs, read both ways. On the interior's scale the golden angle reads 0.9027 and ranks 1st of 72, against 0.9026 for the best grid angle at 137.5°, with the window running from 0.0668 to 0.9026; counting the rim's cells the golden angle reads 0.7076 and ranks 2nd of 72, against 0.7129 for the best grid angle at 137.5°, with the window running from 0.0331 to 0.7129. The dashed upright is the golden angle, which a grid of decimal degrees never lands on and which is therefore read separately.

Packing, measured four ways

The claim is that the golden angle packs best, and it is measurable. Read on the interior of a head, the two criteria about distance put the golden angle first among the angles near it and the two about cells are won by rational angles — which makes the claim half right, and makes the right half a statement about a class of angles. An earlier reading of the same four criteria, divided by the cells at the head's edge, said the opposite.

tissue · Packing A head growing outward beside a uniform head shaken until its cells vary as much. Left: an exact golden head of 500 organs with radius growing as the organ's index to the power 0.6; its cell areas have a coefficient of variation of 0.164, all of it a trend from centre to rim. Right: a head with an exponent of a half, whose cells would all be equal, every organ moved by 0.2 of a spacing; its coefficient of variation is 0.176, none of it a trend. Cells larger than the mean are shaded. A reading of the spread gives the two heads about the same exponent.

A spread of cell areas is not a growth law

An earlier essay proposed that a head's spread of cell areas is a measured growth law: a spread of 0.10 means a radial exponent of 0.455 or 0.555. On an exact head that holds to two figures. A uniform head with every organ moved by a tenth of a spacing also has a spread of 0.094, and reads as 0.457 with no gradient in it at all. By a fifth of a spacing it reads 0.42. Fitting cell area against radius instead is untouched on a golden head and collapses on a rational one, and ranking the organs by radius is untouched on both. The spread is law and disorder added in quadrature, which makes it a good reading of disorder once a trend has told it how much law there is.

tissue · Packing

Named alongside it

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

Voronoi cell areaAsymptotic claimCell areaClaim testingCriterion dependenceCylinderDisorderDisplacementDivergence angleHonest limitsLargest gapNearest neighbour

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