Phyllotaxis
the spirals are counted, not admired
Generator

order-by-window

The four fractions with a denominator of 34, ordered four ways. The left column puts them in order of how close the nearest other rational is — the crowding — with the most crowded at the top. The other three order them by the area of their dip, at windows of 50, 100, 200 scaled units. The residual claim this thread carried was that the most crowded fraction gives the widest dip, which would make all four columns the same order. They are not: the order changes between the first two windows and settles, from a window of 100 outwards, into 11/34 > 15/34 > 13/34 > 9/34 — which is not the crowding order either. A quantity that reverses when the measurement is stopped somewhere else is a property of the stopping.
The order follows the window, so it was never the fractions'The four fractions with a denominator of 34, ordered four ways. The left column puts them in order of how close the nearest other rational is — the crowding — with the most crowded at the top. The other three order them by the area of their dip, at windows of 50, 100, 200 scaled units. The residual claim this thread carried was that the most crowded fraction gives the widest dip, which would make all four columns the same order. They are not: the order changes between the first two windows and settles, from a window of 100 outwards, into 11/34 > 15/34 > 13/34 > 9/34 — which is not the crowding order either. A quantity that reverses when the measurement is stopped somewhere else is a property of the stopping.crowdednearest neighbour firstwindow 50widest firstwindow 100widest firstwindow 200widest first15/3413/349/3411/3415/3413/349/3411/3415/3413/349/3411/3415/3413/349/3411/34crowding: 13/34 0.193° · 9/34 0.200° · 15/34 0.179° · 11/34 0.286°q = 34 · 791 organsgenerated 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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