Phyllotaxis
the spirals are counted, not admired
Series

Dose — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. More organs removed, more stems that never come back. The share of arrangements at the 5/8 rung that never return to the divergence they were cut from, against how many organs the cut removed. One organ wrecks 2 of 8 arrangements and five wreck 63 of 64. The number of arrangements differs from bar to bar because a cut of five organs has more ways of being placed than a cut of one, and it is printed on each bar for that reason. What the dose decides is whether a stem falls off its lattice; where it lands when it does is decided by something else.

    Three organs and no mirror

    A coarse stem cut of two organs can end up as its own mirror image — the same lattice wound the other way, counts unchanged, handedness reversed. Finer stems never do it, and two accounts of why were on the table: coarseness, or the share of the neighbourhood removed. A three-organ cut at the finer arrangements settles it, and the answer is the first.

    part 1 · emergence
  2. The mirror belongs to the lattice, not to the dose. How close the closest arrangement came to the mirror of the divergence it was cut from, against the share of the front that was removed. The marked point at 40 per cent is the coarse 3/5 rung with two organs taken, which reaches the mirror exactly. Every other point is a finer rung: five sizes of cut at 5/8 running from 13 to 63 per cent, and three organs at 8/13. Taking a larger share of a larger front than the coarse rung needs gets nowhere near, so the quantity that decides it is not the fraction of the neighbourhood removed.

    The share was not the thing

    Two organs out of a front of five reverses a stem's handedness; three out of eight does not, and neither does five out of eight, which is a larger share of a larger neighbourhood. The hypothesis under test was that the dose decides the destination. It decides whether a stem falls off its lattice and nothing about where it lands.

    part 2 · emergence
  3. The short list, and what is on it. Every distinct place a wrecked stem settles, for cuts of one organ through five at one rung, with what each one turned out to be. Three of the six are the lattice the stem was cut from with one lag left rigid and a whole number of turns inserted over its period — the description the single-organ work established. The rest have no rigid lag at any period up to twenty-four, and a counter shown their positions returns a pair the original lattice does not carry. So the list is short and it is not homogeneous: a large enough cut can put a stem onto a different lattice rather than onto a slipped version of its own.

    A wreck has a short list

    Cuts of one organ through five, over two hundred and forty-six stems that never came back, land on six settled divergences between them. Removing five organs instead of one wrecks nearly everything and reaches nowhere the single cut had not already found — and half the list turns out to be the old lattice slipped by a turn, while the other half is not the old lattice at all.

    part 3 · emergence
  4. Everywhere a cut of one to five organs can send a 5/8 stem. Every settled divergence reached by any arrangement of up to five organs removed from a stem at the 5/8 rung, on one axis. There are six of them and no more. three are slips of the lattice the stem was cut from: each keeps the lag-5 family intact and sits a whole number of turns of it from the next, which is the ladder marked below the axis with rungs 72.0 degrees apart. The other three keep no lag at all and sit near a fraction of a turn, marked above: 175.0 degrees near 1 of 2 turns, 190.0 degrees near 1 of 2 turns, 235.0 degrees near 2 of 3 turns. A stem that is cut either slides along the ladder it was on or leaves it for a lattice with files in it.

    A count with a factor in it

    Cuts of several organs send a stem to six settled divergences and no more. Three of them are the old lattice with a family left standing. The other three are counted 2/6, 4/6 and 3/6 — and a pair whose numbers share a factor is the classic signature of a pattern that arrives several organs at a time.

    part 4 · lattices
  5. Everywhere a cut of one to five organs can send a 5/8 stem. Every settled divergence reached by any arrangement of up to five organs removed from a stem at the 5/8 rung, on one axis. There are six of them and no more. three are slips of the lattice the stem was cut from: each keeps the lag-5 family intact and sits a whole number of turns of it from the next, which is the ladder marked below the axis with rungs 72.0 degrees apart. The other three keep no lag at all and sit near a fraction of a turn, marked above: 175.0 degrees near 1 of 2 turns, 190.0 degrees near 1 of 2 turns, 235.0 degrees near 2 of 3 turns. A stem that is cut either slides along the ladder it was on or leaves it for a lattice with files in it.

    A file has to close

    The three destinations counted with a shared factor sit near a half turn, a half turn and two thirds. Measuring how near is the trap: by distance from the fraction, the golden angle is closer to two fifths than two of them are to anything, and would be reported as having five files it does not have.

    part 5 · cylinder

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