Phyllotaxis
the spirals are counted, not admired
Generator

background-neighbourhood

μ₂ across nine tenths of a degree either side of 21/55, on a head of 825 organs — 15 in each of its 55 rows. The deep notch at the centre is the dip. The shaded columns are the offsets this site now takes a background from: those at least 0.03° from every rational with a denominator of 60 or less, of which there are 54 here. The marked sample at 0.2° is the one the previous phase used, and it lands 0.007° from 13/34 — inside that rational's own dip. It reads 0.115 against a floor of 0.148, so measured that way the dip is an inversion. The clear offsets give 0.339.
The neighbourhood of 21/55, and where its background was taken fromμ₂ across nine tenths of a degree either side of 21/55, on a head of 825 organs — 15 in each of its 55 rows. The deep notch at the centre is the dip. The shaded columns are the offsets this site now takes a background from: those at least 0.03° from every rational with a denominator of 60 or less, of which there are 54 here. The marked sample at 0.2° is the one the previous phase used, and it lands 0.007° from 13/34 — inside that rational's own dip. It reads 0.115 against a floor of 0.148, so measured that way the dip is an inversion. The clear offsets give 0.339.00.2000.400-0.400-0.300-0.200-0.10000.1000.2000.3000.400degrees from 21/55, at 137.4545°μ₂, the second moment of the side-count distribution13/34the background from the clear offsets: 0.339the old single sample: 0.11521/55 · head of 825generated from a stated rule, not drawn to look right

Drawn at its defaults, in the dip at a large denominator. It takes no options at all, so every essay calling it gets this exact drawing.

Called by 4 essays

the blast radius of changing it

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°.

 Packing and tiling

The background is not one sample

The dip in disorder at a rational angle is a comparison against a background, and the background was one measurement taken two tenths of a degree away. At 21/55 that lands seven thousandths of a degree from 13/34 — inside another rational's dip — and the comparison inverts. Fixed, the dip survives to a denominator of 89.

 Packing and tiling

The width carries the denominator

The previous phase measured three denominators, found the dip's half-width falling as the square of the head size, and could not say whether its coefficient depended on the denominator. Six denominators say it does: the coefficient runs from 1,188 at q = 8 to 14,297 at q = 89, and dividing by q flattens a factor of twelve into a factor of two.

 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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