Phyllotaxis
the spirals are counted, not admired
Generator

how-the-rationals-crowd

For each Fibonacci fraction, the distance to the nearest other rational with a denominator of 60 or less, and the half-width of its own dip at the smallest head that resolves it. The gaps fall from 0.763° at 3/8 to 0.0735° at 34/89; the dips stay between 0.0077° and 0.0155°. The dips never touch — the closest they come is a factor of 10 — so what stops the measurement is not the dips overlapping but the background between them ceasing to be flat. The clear offsets available fall from 72 to 53.
The gaps close faster than the dips narrowFor each Fibonacci fraction, the distance to the nearest other rational with a denominator of 60 or less, and the half-width of its own dip at the smallest head that resolves it. The gaps fall from 0.763° at 3/8 to 0.0735° at 34/89; the dips stay between 0.0077° and 0.0155°. The dips never touch — the closest they come is a factor of 10 — so what stops the measurement is not the dips overlapping but the background between them ceasing to be flat. The clear offsets available fall from 72 to 53.-2-1.50-1-0.5000divergence angle, as a fraction of a turndegrees (logarithmic)3/85/138/2113/3421/5534/89to the nearest other rationalthe dip's own half-widthsix denominatorsgenerated 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 3 essays

the blast radius of changing it

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 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.

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