Phyllotaxis
the spirals are counted, not admired
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Jugate limit — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The 2-jugate forks converge on 68.7539°. Every fork sits at a rational divergence, with denominator 4(m² + mn + n²) — 10/28, 30/76, 74/196 and so on. The limit is 68.7539°, which is 137.5078 divided by 2, and it is at none of them.

    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.

    part 4 · lattices
  2. The pairs grown stems of one, two, three and four organs a whorl count as their rise falls, against the folded rise. Four stems grown by the same rule over the same range of folded rise, 0.4 to 0.0012 — the rise between whorls times the jugacy, which puts every jugacy on one lattice axis — each whorl folded to one node and counted in a sliding window. With 1 organ a whorl the counted pairs run 1/2, 2/3, 3/5, 5/8, 8/13, 13/21; with 2 organs a whorl the counted pairs run 2/4, 4/6, 6/10, 10/16, 16/26, 26/42; with 3 organs a whorl the counted pairs run 3/6, 6/9, 9/15, 15/24, 24/39, 39/63; with 4 organs a whorl the counted pairs run 4/8, 8/12, 12/20, 20/32, 32/52, 52/84. Every jugacy passes the same transitions at the same folded rises, with its pairs multiplied by its jugacy.

    A grown stem halves its ladder

    A bijugate stem at 68.754° was derived to pass the ordinary transitions at half their rises, because a lattice wrapped twice round is the ordinary lattice at twice the rise. No bijugate stem had been grown through them. Grown by the same rule that grows an ordinary shoot, stems of two, three and four organs a whorl walk the ordinary ladder with every pair multiplied by the jugacy, change pair at the ordinary rise divided by the jugacy between whorls — and by its square per organ — lag behind the static ladder by the same hundredth of a rung, and settle within three hundredths of a degree of 137.5078 over the jugacy.

    part 5 · lattices
  3. Which rates keep a Lucas seed, on stems of one, two and three organs a whorl, placed by whorls a rung. Every rate each configuration was grown at, drawn at the number of whorls the stem places between one transition and the next: a dark cell keeps the Lucas ladder to the end of the rise, a light cell gives it up. 1 a whorl, 512: kept to 86.6 whorls a rung, lost from 87.6; 1 a whorl, 256: kept to 86.6 whorls a rung, lost from 87.6; 1 a whorl, 1024: kept to 86.6 whorls a rung, lost from 87.6; 2 a whorl, 1024: kept to 87.6 whorls a rung, lost from 88.1; 2 a whorl, 1024, imposed: kept to 87.6 whorls a rung, lost from 88.1; 2 a whorl, 512: kept to 87.6 whorls a rung, lost from 88.5; 2 a whorl, 2048: kept to 87.6 whorls a rung, lost from 88.5; 3 a whorl, 1536: kept to 87.9 whorls a rung, lost from 88.5; 3 a whorl, 1536, imposed: kept to 87.9 whorls a rung, lost from 88.5; 3 a whorl, 768: kept to 86.6 whorls a rung, lost from 88.2. On this axis every edge falls within about two whorls a rung of every other.

    A Lucas seed counts whorls

    An ordinary stem seeded on the Lucas lattice keeps that ladder when its rise falls fast and gives it up when it falls slowly, with the edge near ninety nodes a rung. A stem that grows two organs a whorl is an ordinary stem folded twice round, so the identity predicts its edge — once it says whether the edge counts placements or whorls. Grown across the edge, stems of one, two and three organs a whorl keep a Lucas seed to 86.6, 87.6 and 87.9 whorls a rung: one number to within a per cent in whorls, and one, two and three times as far out in organs. No grid moves it and imposing exact whorls moves it not at all, even where free trijugate whorls come apart completely as the seed is lost.

    part 6 · lattices
  4. Read as whorls, the three jugacies fall on one line. The rate at which a stem seeded on the Lucas lattice gives that ladder up, against how long the seed is, for one, two and three organs a whorl. The three lie on top of each other at every seed length: 15 whorls gives 30.80, 30.80, 30.80; 20 whorls gives 42.35, 42.35, 40.42; 30 whorls gives 63.52, 63.52, 63.52; 40 whorls gives 85.66, 87.58, 85.66. Every departure is one or two steps of the grain the edge is bisected to, which is two whorls of rate — 1.925 whorls a rung at every jugacy and every edge drawn.

    A seed measured in whorls

    A Lucas seed's length counts whorls, and the number published earlier for the rate edge counts nothing at all. Grown from seeds of fifteen to eighty whorls at one, two and three organs a whorl, stems of every jugacy lose the seed at the same rate in whorls a rung for the same seed length in whorls — 30.80 at fifteen, 63.52 at thirty, identically across the three — and at the same seed length in organs they differ by a factor of 3.24. So the seed is measured in whorls, as the edge is. The other half is worse for the earlier reading: the edge is not a constant but 2.24 times the seed less three, straight to within 2.7 whorls a rung over a fivefold range, so the eighty-seven whorls a rung reported everywhere is a property of the forty-whorl seed nobody varied.

    part 7 · lattices
  5. The rate at which a Lucas seed is lost, read through four counting windows. Eighteen stems — one, two and three organs a whorl, seeded with 15 to 80 whorls of Lucas lattice — each with the edge found four times, through counting windows of 20, 26, 39, 52 folded nodes, the four dots at each seed drawn side by side. 17 of the 18 stems give the same organ through every window. The exception is three organs a whorl with a 30-whorl seed, where the widest window finds the edge at 207 organs against 201 for the others. A window twice as wide as the one every earlier reading used moves no other edge by an organ, so the window is not what makes the ratio of edge to seed change with the seed.

    The window was not carrying it

    The ratio of a Lucas seed's rate edge to its length rose from 2.05 at fifteen whorls to 2.20 at eighty, and the suspect was the counting window, which is most of a short seed. Read through windows of 20, 26, 39 and 52 folded nodes, seventeen of the eighteen stems lose the seed at exactly the same organ of rate, so the window carries almost none of it. Part of the rise was the grain of rate the edge was found on, worth up to six hundredths of the ratio. What is left rises by a tenth below thirty whorls and has stopped by sixty, at a level an ordinary stem reaches about one and a half per cent lower than a bijugate or trijugate one — and a trijugate edge is not always a line.

    part 8 · lattices

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