Phyllotaxis
the spirals are counted, not admired
Series

Jugacy — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. three stems: 1, 2, 3 primordia at a time. 1-jugate at 137.51° counts 2 and 3; 2-jugate at 68.75° counts 2 and 4; 3-jugate at 45.84° counts 3 and 6. Every count shares the factor k, and dividing it out leaves an ordinary lattice: what a k-jugate stem is, exactly, is k copies of an ordinary one wrapped k times round.

    Two at a time

    Every counter in these essays asks how far it is from element i to element i plus m, and that index is a claim that the elements arrived one at a time. For teasel, for Cephalaria and for a real minority of plants the claim is false — and what those patterns turn out to be is an ordinary lattice, wrapped twice.

    part 2 · lattices
  2. A bijugate stem answers in pairs, once the half turn is taken out. Removing one organ from a stem grown by a rule that places two at a time, at a rise of 0.0065, where the pattern counts 6 and 10 and has rotational symmetry of order 2. The open circles are the displacement of the next organ as it comes out of the arithmetic, which reaches 180° at offsets where the stem has demonstrably not been disturbed — the two organs of a whorl are interchangeable, so calling the other one "next" is a relabelling and not a movement. The filled circles are the same numbers read modulo 180°, which is the only way a 2-jugate divergence is defined. Read that way the response is a run of equal pairs — 68.0°, 68.0°, 42.9°, 42.9° — ending at 10, the larger parastichy number, with everything past it under 0.5°. Two organs of one whorl give the same answer to the last digit, which is the pattern's symmetry showing up in an experiment.

    What a cut costs a whorl

    A bijugate pattern is an ordinary lattice seen twice over, so the account that says a wrecked stem's repeating block is the repeat unit of the lattice underneath has a specific prediction here: three and five. It gets six and ten. And the thing a single missing organ does destroy on a whorled stem is the one property its counts cannot see.

    part 3 · cylinder
  3. Two patterns a counter cannot tell apart — counted 2/6 against 2/6. On the left, the top 90 organs of a spiral stem that never repaired after two organs were removed, settling at 175.01 degrees. On the right, a stem grown by a rule that places two organs at a time on every node. A counter shown the positions returns 2/6 for the first and 2/6 for the second, and a pair whose numbers share a factor is the usual signature of a whorled pattern. Rotate each pattern and the answer separates them at once: the whorled stem maps onto itself at a half turn and the wrecked stem maps onto itself at no fraction of a turn at all. Its shared factor is a fact about where its divergence landed, 4.99 degrees from 1 of 2 turns, and not about how it grew.

    The symmetry that is not there

    A wrecked stem counted 2/6 and a stem grown two organs at a time counted 2/6 are indistinguishable to a spiral counter. Rotate them by half a turn and one lands on itself and the other lands nowhere. A cut does not make a whorled pattern; it makes a pattern that counts like one.

    part 4 · lattices
  4. How far grown whorls of two to eight members miss exact symmetry, at four lattices. Whorls grown one member at a time at a fixed rise, on a grid of 1,680 azimuths that every jugacy divides, read at folded rises of 0.27, 0.18, 0.09, 0.036. With 2 members the largest miss is 0.00°, 0.00°, 0.00°, 0.00°; with 3 members the largest miss is 0.64°, 1.93°, 0.21°, 0.00°; with 4 members the largest miss is 0.00°, 0.00°, 0.00°, 0.00°; with 5 members the largest miss is 1.29°, 1.29°, 0.21°, 0.21°; with 6 members the largest miss is 0.86°, 0.86°, 0.43°, 0.21°; with 7 members the largest miss is 0.86°, 2.14°, 0.21°, 0.00°; with 8 members the largest miss is 0.00°, 0.00°, 0.00°, 0.00°. Whorls of two, four and eight members are exact at every lattice; three, five, six and seven miss.

    A whorl that misses its share

    A whorl of k organs is defined by its symmetry, and the rule that grows one places its members one after another, each against the members already there. Nothing tells it to put them a k-th of a turn apart. Grown that way, whorls of two, four and eight members sit exactly on their shares of the turn at every lattice measured, and whorls of three, five, six and seven do not — the pattern a mirror argument predicts, since only a power of two leaves every new member a position that mirrors every member already placed. A trijugate whorl misses by 6.5° at a coarse rise and not at all at a fine one, by an amount the rise sets almost everywhere, and none of it moves a single transition.

    part 5 · lattices

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