Phyllotaxis
the spirals are counted, not admired
Concept

Normalisation — where it appears

Dividing a measurement by a reference quantity so that readings taken at different scales can be compared. The reference decides the comparison: dividing a head's cell areas by a mean inflated by its rim cells made the rim, not the divergence angle, rank the arrangements.

Named by 3 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 The cell area a 137.508° golden head's packing is divided by, at 10 sizes. Counting every bounded cell the mean area runs from 3.607 to 38.828, the largest at 150 organs, because the cells just inside the edge of the head reach out to circumcentres far beyond it. Leaving out every cell whose polygon crosses the head's own radius, it stays between 3.1425 and 3.1634 at every size drawn, which is π, the area the square-root rule gives each organ.

Packing, measured against the interior

An earlier reading of these heads reported that no packing criterion singles out the golden angle and that three criteria give three winners. Every one of those readings was divided by a mean cell area that, on a head of 150 organs, was 38.8 where the interior's is π. Divided by the interior's own, the criteria about distance put the golden angle first of 72 angles and the criteria about cells go to rational ones.

tissue · Packing A random tissue of 700 cells whose size grows outward as a Vogel head of exponent 0.8 does. Points placed at random with the density of a head of exponent 0.8. Cells above the median area are shaded, and they lie outside: 537 cells whose sizes differ far more by where they are than by how many sides they have. A cell's side count does not follow its radius at any setting.

A gradient keeps Lewis's slope and takes its share

Lewis's law is reported as two numbers, a slope and how much of the variation in cell area it explains, and they get read as one. Grow a tissue's cell size steadily across the field of view and the two part completely. Over a gradient that takes cells from one size to several times that, the slope of a random tissue stays at 0.22 and its explained share falls from 33 per cent to 15. Both are predicted by a line of arithmetic from the flat tissue's own numbers, to within a percentage point or two. Dividing the field into twelve bands of radius restores nearly all of it. So a weak Lewis correlation in a published field may be a statement about the field.

tissue · Lewis

Named alongside it

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

Cell areaHonest limitsLargest gapNearest neighbourRim effectSelf-correctionSummary statisticVoronoi cellsVoronoi cell areaArtefactAsymptotic claimCriterion dependence

All concepts