Phyllotaxis
the spirals are counted, not admired
Series

Counting — the series

7 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The two spiral families a counter finds between 0.68 and 0.92 of the radius. 34 spirals one way and 55 the other, found from the point positions alone — the counter is never told the divergence angle.

    Counting the spirals

    Almost every claim about phyllotaxis is a claim about how many spirals run through a pattern, and the count is almost never done. It can be done from the points alone, by a count that is never told what angle built them — and then a count of 34 is evidence rather than a restatement.

    part 2 · lattices
  2. The angles against the positions, rise by rise. three rises, five seeded stems each. A filled mark is a run whose angle readout returned the pair the position counter finds in the same stem; an open mark is a refusal. At 0.032 the counter says 3/5 and the angles agree on 0 of 5, refusing 5. At 0.013 the counter says 5/8 and the angles agree on 5 of 5. At 0.005 the counter says 8/13 and the angles agree on 5 of 5. The two instruments share no code path: one is given a list of angles, the other a list of coordinates.

    A counter that sees no positions

    This site has counted spirals two ways, and both were handed coordinates. A third counter is handed a list of angles and nothing else. It returns one number instead of two, it refuses more often, and where it refuses it would have been wrong every time.

    part 3 · lattices
  3. Every family but two is the sum of two others. Four heads and the contact families each actually has. An arc arrives at every family that is the sum of two smaller ones; the two with no arc arriving are the generators, and on every head they are the two smallest. whorled, 144°: 2, 3, 5 — golden, 137.508°: 8, 13, 21, 34, 55, 89 — Lucas, 99.502°: 11, 18, 29, 47, 76 — rational, 137.5°: 8, 13, 21, 34, 55, 89 — 137.0°: 8, 13, 21, 29, 50, 71, 92, 113. So once a pair is counted the rest is arithmetic, and a third counted family is a prediction rather than a second measurement.

    Every family but two is a sum

    A seed head has six spiral families and everybody reports two. That looks like a convention hiding information and it is the opposite — every family but the two smallest is the sum of two others, so a third count is a prediction rather than a measurement, and a check that catches a wrong pair.

    part 4 · lattices
  4. The angles against the positions, rise by rise. five rises, five seeded stems each. A filled mark is a run whose angle readout returned the pair the position counter finds in the same stem; an open mark is a refusal. At 0.032 the counter says 3/5 and the angles agree on 0 of 5, refusing 5. At 0.013 the counter says 5/8 and the angles agree on 5 of 5. At 0.01 the counter says 5/8 and the angles agree on 5 of 5. At 0.005 the counter says 8/13 and the angles agree on 5 of 5. At 0.008 the counter says 5/8 and the angles agree on 2 of 5, refusing 3. The two instruments share no code path: one is given a list of angles, the other a list of coordinates.

    The angles name the branch

    Seed the same rule at the Lucas angle and the readout returns 4 and 7, then 7 and 11 — the pairs the position counter finds, and not Fibonacci numbers. So a list of divergence angles carries not only how many spirals there are but which family of ladders the plant is on.

    part 5 · lattices
  5. One wrecked stem, lag by lag — golden, rise 0.005, organ 6 back. How much each lattice hop moves from organ to organ in a stem that never repaired after a single removal, over its last 120 organs. The lag-one hop is the divergence itself and it swings by 55 degrees. The lag-4 hop swings by 0.12 degrees and sits 0.01 degrees from where the undisturbed stem put it, so that one family of the original lattice is still standing organ by organ. Its multiples inherit the same steadiness and nothing else comes within a factor of twenty. The block of angles this stem repeats has a period of 4, which is the surviving lag and not a coincidence.

    A period that is not a count

    Eighteen wrecked stems settle into a block whose period is one of their own spiral counts, and one settles into a block of four on a lattice counted 8 and 13. The odd one is not noise. It is the case that shows what the rule is actually conserving, and it is the reason this thread is about lattice steps rather than about spirals.

    part 6 · lattices
  6. The two spiral families a counter finds between 0.68 and 0.92 of the radius. 34 spirals one way and 55 the other, found from the point positions alone — the counter is never told the divergence angle.

    What a count cannot decide

    A spiral count is the measurement this whole subject is built on, and it is deliberately blind to everything that varies inside a rung. Four results this collection now holds are results about that blindness rather than about the arrangements.

    part 7 · emergence
  7. Two readings of what a stem is, and one of them is wrong. The same runs read twice. On the left is the pair counted from the point positions by machinery that is never shown a divergence angle; on the right is the pair read from the divergence sequence, which is the column the depth thread has been quoting. Every exponent's stems are counted at 5/8 spirals from the points. Read from the angles, the deepest rule's stems come back as something else at every seed, and a pair with a 8 replaced in it is not a near miss but a family whose step is several times as long. The settled divergence moves by 0.118 degrees across all five exponents, so the rules are on one lattice and the disagreement is a defect in one of the two instruments.

    The pair read from the angles

    Checking that a jostled stem is still the lattice its panel is about turned up a disagreement. Counted from the point positions every rule's stems return five and eight spirals; read from the divergence sequence, the deepest rule's stems come back as five and seven at every seed. The points are right, and this site's founding rule is why.

    part 8 · lattices

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