Phyllotaxis
the spirals are counted, not admired
Generator

width-by-window

The equivalent width of the disorder dip — the area of the deficit divided by the dip's own depth — for four fractions with a denominator of 34, against the window it was integrated over. The window is in scaled units, u = δ·n²/q, so a head of any size is sampled at the same places on its own dip; 200 scaled units is about 0.0109° at these head sizes. A width would show as a flat stretch. There is none: every line climbs to 512 scaled units or so and then turns over, because past that the window has reached far enough towards the next rational that it is integrating the neighbour's dip instead of this one's background. The dip has a depth and a scale; it does not have an outer edge, and an integral of something with no outer edge is a statement about where it was stopped.
The area never settles, so the number reported is the windowThe equivalent width of the disorder dip — the area of the deficit divided by the dip's own depth — for four fractions with a denominator of 34, against the window it was integrated over. The window is in scaled units, u = δ·n²/q, so a head of any size is sampled at the same places on its own dip; 200 scaled units is about 0.0109° at these head sizes. A width would show as a flat stretch. There is none: every line climbs to 512 scaled units or so and then turns over, because past that the window has reached far enough towards the next rational that it is integrating the neighbour's dip instead of this one's background. The dip has a depth and a scale; it does not have an outer edge, and an integral of something with no outer edge is a statement about where it was stopped.50100200400800window, in scaled units of δ·n²/qequivalentwidth13/349/3415/3411/34q = 34 · two head sizes agree to 8% at a window of 200generated from a stated rule, not drawn to look right

Drawn at its defaults, in the dip, integrated. 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

A dip with no outer edge

The disorder of a head dips at every rational divergence, and how wide that dip is has carried a long argument. Reading the width as a level crossing has a resolution problem, and the obvious repair is to integrate instead. The integral reproduces beautifully across head sizes and never settles on a value, because there is nothing out there for it to settle against.

 Packing and tiling

The window is the neighbour

An integral needs a limit, and this one has two conditions on it that pull opposite ways. It has to scale with the dip, so that two head sizes are comparable, and it has to stay clear of the next rational, which is a fixed distance in degrees. Between them there is no stretch where the answer holds still — and the limit that decides it is the crowding.

 The claims, measured

The order belonged to the method

A residual was left over after the two width laws, and it looked ordered: the most crowded fraction gave the widest dip, in all three families, in the direction a measurement artefact would take. Measured again with an instrument that has no level in it, the order changes with the window, disagrees between families, and in one of them comes out backwards.

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