Phyllotaxis
the spirals are counted, not admired
Generator

disorder-against-angle

μ₂ is the mean squared departure of a Voronoi cell's side count from six, measured on 300 points inside 86% of the radius. Swept across 1.60° it is a staircase: flat over stretches of a few hundredths of a degree, with sharp steps between them and narrow deep dips wherever a rational falls. At 137.144° — which is 360 × 8/21 — it is 0.078; At 137.648° — which is 360 × 13/34 — it is 0.197; At 138.460° — which is 360 × 5/13 — it is 0.060. The ticks along the top are the fractions p/q, placed from arithmetic rather than from the curve.
The disorder of a head against its divergence angle, 300 pointsμ₂ is the mean squared departure of a Voronoi cell's side count from six, measured on 300 points inside 86% of the radius. Swept across 1.60° it is a staircase: flat over stretches of a few hundredths of a degree, with sharp steps between them and narrow deep dips wherever a rational falls. At 137.144° — which is 360 × 8/21 — it is 0.078; At 137.648° — which is 360 × 13/34 — it is 0.197; At 138.460° — which is 360 × 5/13 — it is 0.060. The ticks along the top are the fractions p/q, placed from arithmetic rather than from the curve.00.2000.4000.600137138138139divergence angle, degreesμ₂, the mean squared departure of a cell's side count from six5/138/2113/34401 angles · 0.0040° apart · 300 points eachmarks are the fractions, placed from arithmetic

Drawn at its defaults, in the disorder, swept over the angle. It takes no options at all, so every essay calling it gets this exact drawing.

Called by 7 essays

the blast radius of changing it

Packing and tiling

Packing, measured four ways

The claim is that the golden angle packs best. It is measurable, and the measurement gives three different winners on three criteria — all near 137.5° and none of them it. That does not make the claim wrong; it makes the usual statement of it wrong.

 The claims, measured

The claim that survives

Of the three famous assertions about this subject, one is out by a factor of two, one is true of a branch rather than of plants, and one is right — in a sharper form than the version usually told, and about arithmetic rather than about packing.

 Packing and tiling

The second moment is the measurement

The mean number of sides in a cellular tissue is six, and Euler's formula leaves it no choice — so it takes the same value on a golden-angle head, a whorled head and a set of random points. The mean squared departure from six varies by a factor of eighty across the same three, and almost nobody reports it.

 Packing and tiling

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°.

 Packing and tiling

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 claims, measured

What a summary throws away

Four statistics this collection has relied on turn out to be incapable of varying with the thing they describe — one is invariant to shuffling, one is fixed by a theorem, one is a parameter that stopped mattering, one is a fitted number selected into being wrong. In each case the second statistic was free and nobody had taken it.

 The claims, measured

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.

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