Phyllotaxis
the spirals are counted, not admired
Concept

Branch point — where it appears

3 essays name this object, across 2 fields. What follows is each of them, and the objects they name alongside it.
-2-1.50-1-0.500100120140160180divergence angle (°)log₁₀ of the rise between nodes1,2,32,3,522 × 150 lattices, each solved196 runs drawn

The forks are exact

Where a stem's pattern has to choose between two futures, three spiral families are equally short and the lattice is exactly equilateral. A numerical solver found those points; the numbers it returned turned out to be rational, and chasing that gave a closed form — including the fact that every fork sits at a rational divergence, and the golden angle at none of them.

cylinder · forks 646668701234567fork number down the treedivergence at the fork (°)137.5078/2 = 68.7539°2-jugate · 7 forkslast fork 68.7365°

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.

lattices · jugate limit 024680.2000.4000.6000.8001counting only outside this fraction of the organ's length or radiustransitions inside the counted partdisc: 2 beyond 50%cone: 1 beyond 50%flare 0.12 · 4000 elements1 against 2 in the outer 50%

Why a cone can be counted once

A pineapple is described as 8 and 13 and the description holds. A sunflower is described as 34 and 55 and the description is a statement about one annulus. Both organs have the same ladder in element number — what differs is where an organ puts its elements.

cylinder · organ size

Named alongside it

The objects these essays reach for when they reach for this one.

RiseBranchFibonacciLadderRational divergenceTransitionsBijugateCensusClosed formConeCountable regionCylinder

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