Phyllotaxis
the spirals are counted, not admired
Generator

census-up-to-jugacy

At a rise of 0.008 only 14.7 per cent of divergences give a pair of consecutive Fibonacci numbers. Allow a common factor to be divided out first and it is 50.1 per cent. The whole of the increase is pairs of the form k and 2k — k copies of the bottom rung — which is a generous reading that describes no plant.
The Fibonacci share, read two waysAt a rise of 0.008 only 14.7 per cent of divergences give a pair of consecutive Fibonacci numbers. Allow a common factor to be divided out first and it is 50.1 per cent. The whole of the increase is pairs of the form k and 2k — k copies of the bottom rung — which is a generous reading that describes no plant.rise 0.12, strictly67.5%rise 0.12, up to jugacy72.4%rise 0.03, strictly31.1%rise 0.03, up to jugacy55.7%rise 0.008, strictly14.7%rise 0.008, up to jugacy50.1%share of divergences3 rises · 3600 divergences each14.7% → 50.1% at rise 0.008

Drawn at its defaults, in whorls, and patterns with k at a time. 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

Packing and tiling

Packing, measured four ways

The claim is that the golden angle packs best. It is measurable, and the measurement gives three different winners on three criteria — all near 137.5° and none of them it. That does not make the claim wrong; it makes the usual statement of it wrong.

 The claims, measured

What "whorled" was hiding

The expansion phase's census put 35% of divergences in a bucket labelled whorled and moved on. Opened, every pair in it is k and 2k — the coarsest rung of the ladder, repeated k times — and reading the census up to jugacy takes the Fibonacci share from 14.7% to 50.1% without describing a single extra plant.

 The claims, measured

How often is it Fibonacci

The claim that plant spirals come in consecutive Fibonacci numbers is stated as a near-universal. Asked of the geometry, the answer collapses with scale — at a coarse rise 67% of divergences give Fibonacci pairs, and at a fine one 15%, with whorled and unnamed pairs taking the rest.

 The pattern itself

Half the golden angle

The forks of the van Iterson tree converge on 137.5078° and sit at none of them. Divide the whole tree by two and the same thing happens at 68.7539° — which is where teasel is, and where a bijugate sunflower counted 42 and 68 has to be.

 The claims, measured

How many plants would it take

Fourteen specimens separate the geometry's Fibonacci share from a coin weighted to a half. Four separate it from what a grown history gives. One fir cone measured at three rings settles whether its ladder is spaced as a cone's or an ogive's. The sample sizes are small, and that is the uncomfortable part.

 Where the angle comes from

Continuity from a coarse start

At a fine rise, one divergence in seven gives a Fibonacci pair. Grow a stem from a coarse start at a divergence nobody chose, down to that same rise, and sixteen runs out of sixteen end on 8/13. The expansion phase's interpretation was that continuity does the work; this is the measurement it never had.

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