Phyllotaxis
the spirals are counted, not admired
Generator

tolerance-tightens

Each dot is one configuration: a stem carried down to the stated rise, swept over the whole amplitude range, with the largest divergence scatter any surviving run showed. It ends on 5/8 at the coarse end and 8/13 at the fine one. The open marks are the width of the band of divergence angles that gives each pair at all — a quantity from a different calculation entirely, moving the same way.
The tolerance falls from 4.1° to 0.8° up the ladderEach dot is one configuration: a stem carried down to the stated rise, swept over the whole amplitude range, with the largest divergence scatter any surviving run showed. It ends on 5/8 at the coarse end and 8/13 at the fine one. The open marks are the width of the band of divergence angles that gives each pair at all — a quantity from a different calculation entirely, moving the same way.00.2000.4000.6000.80022.202.402.602.80final rise, as −log₁₀degrees, log₁₀5/88/138/13scatter survivedthe pair's band5 runs per amplitude · placement noise4.12° down to 0.83°

Drawn at its defaults, in noise, and where it gets in. It takes no options at all, so every essay calling it gets this exact drawing.

Called by 6 essays

the blast radius of changing it

Where the angle comes from

A pattern with a rate

Every lattice on this site is a static object indexed by a parameter, and a plant is not. Put the rise on a clock, place each node where the repulsion from the ones below it is least, and the object that comes out has a history — which is the first thing here that could disagree with the ladder.

 Where the angle comes from

Noise is not a slow rate

A stem seeded on the Lucas branch keeps it below ninety nodes per rung and abandons it above — which invites the objection that a real apex's fluctuations would knock it off regardless. Measured across a hundred and sixty runs of two independent kinds of noise, one escapes, at the amplitude where the pattern is already coming apart.

 Where the angle comes from

Two degrees of scatter

A lattice tolerates about two degrees of wander in its divergence angle, and two kinds of noise sharing no code agree on the number to within a third of a degree. It is not a constant: carried finer, the same stem survives 0.8°, and the tolerance tracks the band of angles that produce its pair at all.

 The claims, measured

What a quiet plant is worth

Almost every measurement gets easier as the effect gets larger. This one gets harder — a stem's divergence sequence stops carrying information about its noise at precisely the scatter where the noise becomes obvious. The specimens worth measuring are the ones that look least interesting.

 Stems and cones

What one angle says about the next

A tenth of a degree of placement noise moves a stem's divergence scatter from 0.50° to 0.62°, which nobody would report. It takes the correlation between consecutive angles from 0.54 to below zero. The other two kinds of noise, at scatters where no measurement can separate them, leave it at 0.6.

 Where the angle comes from

The boundary belongs to the pattern

Three kinds of noise, in three incommensurable units, destroy a lattice at the same place — about a degree and a half of divergence scatter. The previous phase measured that of two kinds and called it a scale rather than a constant. With a third it looks less like a coincidence and more like a property of what a lattice is.

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