Phyllotaxis
the spirals are counted, not admired
Generator

three-profiles

Left: the meridian of a cone, an ogive of arc radius 12, and a spherical cap of radius 40, each scaled to its own length. Right: d log r / d log s along it. The cone sits at 1 the whole way; the ogive starts near 2 at its blunt tip and falls to 0.23 by the end of the 100 per cent shown; the cap starts at 1 and falls slowly.
Three organs, and the exponent each has at each placeLeft: the meridian of a cone, an ogive of arc radius 12, and a spherical cap of radius 40, each scaled to its own length. Right: d log r / d log s along it. The cone sits at 1 the whole way; the ogive starts near 2 at its blunt tip and falls to 0.23 by the end of the 100 per cent shown; the cap starts at 1 and falls slowly.the meridiana conean ogivea convex head00.50011.50200.2000.4000.6000.8001along the meridian, as a fractionlocal exponent a(s)a = 1, a conedifferenced from each profile, not looked upogive 2.00 → 0.23

Drawn at its defaults, in organs with no single exponent. 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

Stems and cones

The shape and the law

A cone's transitions are a factor of φ² apart and a disc's a factor of φ, and the temptation is to read the ratio as the shape. It is not. Five surfaces built and counted show that the ratio measures one exponent, and that the exponent is the shape multiplied by the way material arrives.

 Stems and cones

An organ has no single exponent

The previous phase measured that a surface whose circumference grows as a power of arc length puts its transitions a fixed factor apart, and checked it on five surfaces. Every one of them had a single exponent, and no organ does — a fir cone is an ogive, whose exponent runs from 2 at the tip to nearly 0 at the shoulder.

 Stems and cones

What one exponent reports

Fit a single shape exponent to an organ that has four of them and it returns a real quantity — the harmonic mean of what its individual steps report. Harmonic means sit below arithmetic ones, so the fit understates, systematically, in a known direction, and invisibly.

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