Phyllotaxis
the spirals are counted, not admired
Generator

same-denominator

The half-width of the second-moment dip at four divergence angles with the same denominator, on heads of 1279 organs — 23 in each of 55 rows. 21/55 is the Fibonacci convergent, the fraction every earlier measurement of this law was made at and the one whose neighbourhood is emptiest; 12/55, 23/55, 17/55 are not. The four agree within a factor of 1.28, well inside the factor of two the law is stated to, so the width is a function of the denominator and not of how well the fraction approximates its neighbours. What is left over is ordered by the note beside each bar: the more crowded the neighbourhood, the wider the dip comes out, which is the direction a neighbour's own shoulder would push it.
Four fractions of 55, one widthThe half-width of the second-moment dip at four divergence angles with the same denominator, on heads of 1279 organs — 23 in each of 55 rows. 21/55 is the Fibonacci convergent, the fraction every earlier measurement of this law was made at and the one whose neighbourhood is emptiest; 12/55, 23/55, 17/55 are not. The four agree within a factor of 1.28, well inside the factor of two the law is stated to, so the width is a function of the denominator and not of how well the fraction approximates its neighbours. What is left over is ordered by the note beside each bar: the more crowded the neighbourhood, the wider the dip comes out, which is the direction a neighbour's own shoulder would push it.half-width of the dip, in degrees, at 1279 organsnote: how far the nearest other rational sits21/555.54e-313/34 at 0.193° · convergent12/555.24e-37/32 at 0.205°23/556.67e-318/43 at 0.152°17/556.71e-313/42 at 0.156°q = 55 · 1279 organsgenerated from a stated rule, not drawn to look right

Drawn at its defaults, in one denominator, four fractions. It carries a slider over q, so the claim in its caption strip is restated at every stop rather than at one. It takes no options at all, so every essay calling it gets this exact drawing.

Called by 2 essays

the blast radius of changing it

Packing and tiling

Four fractions with one denominator

The dip in a head's side-count disorder is as wide as 150·q/n², measured over six fractions — every one of them a Fibonacci convergent, which is the emptiest neighbourhood a denominator ever gets. So the law could be about the denominator or about how well the fraction approximates its neighbours. Four fractions of 55 at one head size settle it in one figure.

 Packing and tiling

A width read off a staircase

Two fractions of the fourteen measured return a dip width that moves by a factor of two when the head size changes, where the others hold to three per cent. The cause is not their neighbourhood. It is that the disorder statistic changes only when the tessellation changes, so the curve a half-width is read off is a staircase, and a width narrower than the tread cannot be read at all.

All 144 generators