Phyllotaxis
the spirals are counted, not admired
Series

Survey spec — the series

10 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Every open question here needs under 34 specimens. The sample size at which each comparison reaches 90 per cent power at a 5 per cent false-positive rate, from the exact binomial rather than a normal approximation. The census question — do plants show consecutive Fibonacci pairs far more often than the geometry does — needs 4: 14.7% is the share of divergence angles giving a consecutive Fibonacci pair at a fine rise; 90% is what a grown history gives.

    The survey this site cannot do

    Four rounds of asking for a dataset, and it is still not here. What the work here can do instead is specify it — the fields, the sampling, the sizes, and which of this collection's claims each one would settle. Two of the four fields asked for turn out to be worth less than the asking implied, and one was never asked for at all.

    part 6 · wrong
  2. What the sequence sees that the scatter cannot. Each point is an ensemble at one amplitude, placed at the scatter it produces. A stem at three quarters of a degree of scatter has a lag-one correlation near zero if its noise arrived after the primordium was placed, and near 0.7 if it arrived before — and no measurement of the scatter can tell those apart. The separation closes above about a degree, because what the other two kinds preserve is the correlation of a lattice.

    What a quiet plant is worth

    Almost every measurement gets easier as the effect gets larger. This one gets harder — a stem's divergence sequence stops carrying information about its noise at precisely the scatter where the noise becomes obvious. The specimens worth measuring are the ones that look least interesting.

    part 7 · wrong
  3. What the pair costs, at a rise of 0.005. Five seeded stems at each length, read at four protractor errors. With no reading error the pair needs 250 internodes — against the sixty the single parastichy number costs. At 0.25° per organ it needs 250; At 0.5° per organ it needs 400; At 0.75° per organ it needs 1100. The pattern's own scatter here is 0.70°, so the last of those is a reading error larger than the signal being read.

    What the pair costs

    The single parastichy number cost sixty internodes. The pair costs two hundred and fifty, and a protractor error of three quarters of a degree takes it to eleven hundred. The arithmetic that predicts the second of those is right about the shape and wrong about the scale by a consistent factor, which is recorded rather than fitted away.

    part 8 · wrong
  4. Both statistics, on the same stems, at a rise of 0.005. Five seeded stems at each disturbance, held at a fixed rise. Bars are how many returned the pair the position counter finds; open portions are refusals. The pair comes out from 0.1 to 0.25, and across that whole range the lag-one correlation of the same sequences is -0.33, -0.58, -0.59 — decisive, negative and flat. There is no trade between the two: one stem supplies both. Below the window the sequence has locked onto the sampling grid and is a cycle rather than a sample; above it there is no lattice left, at 117° of scatter.

    What a refusal does not say

    The readout can decline for four different reasons — too quiet, too disturbed, too fast, or a window in the wrong place — and a stem that returns nothing does not say which. That is the third time this thread has failed to close the mixture problem, and the first time the failure has a shape.

    part 9 · wrong
  5. The ratio is a U across every rung, and its floor is the number that was reported. The ratio of the second comb to the main comb, on five stems at each of 9 rises spanning two rungs, against the ladder's own coordinate for where each rise sits inside its rung. Both rungs give the same shape: a floor of 0.71 and 0.79 about two thirds of the way up, climbing towards the transition at either end. The dashed line is a transported disturbance with no rule in it at 1.28, which does not vary with the rise at all — a kinematic lattice's angle sequence has no rise in it. Where the rule's curve crosses that line the two accounts are indistinguishable.

    The survey loses its second outcome

    The survey specification written earlier here names three results the survey could return, and the second — a ratio near or above 1.30, read as evidence against the placement rule — is the one that would have been worth publishing. It does not survive the measurements here. The ratio moves with where the plant sits between two transitions, and it moves again with the colour of the plant's own disturbance.

    part 10 · wrong
  6. On a rung the response is a run of offsets; near a transition it has a hole. One row per rise, from 0.02 at the top to 0.005 at the bottom, and one column per offset: the organ one place back at the left, 14 places back at the right. A cell is filled where removing that organ moves the next organ by more than 2.5°, and empty where it does not. On a rung the filled cells are a run from one to the larger parastichy number — 8, 12, 13 at the pairs shown on the left. Between a rise of 0.02 and 0.008 the run ends at 5 and one more cell is filled at 7, with the offsets between them quiet to under a degree. That isolated column is one place inside the larger number of the pair the stem is climbing towards.

    The ablation a plant would survive

    The intervention proposed earlier returns a spiral count from a yes-or-no answer, needs no protractor, and was specified at one rise. Measured across the ladder it acquires three conditions a real experiment would have to meet — and one of them is that the plant must not be too coarsely patterned, or nothing will go wrong at all.

    part 11 · wrong
  7. The disturbance with the largest wander leaves none in the sequence. How much of a divergence sequence's variance survives being averaged over blocks, on kinematic lattices. The vertical quantity is B² times the variance of the block means divided by the variance of the sequence, which is one at every block size for independent errors — the arithmetic is normalised for a differenced stream, since a divergence is the difference of two organs' errors. A line that climbs is a sequence with power at frequencies below one per block. The disturbances that remember the last error climb to 32 at a block of 128. The ones inherited between touching organs do not climb at all — 1.06 and 0.83 at the same block — although their own deviates carry ×5 and ×49 an independent stream's variance in exactly this statistic. What they do instead is dig a hole: at block sizes of 8 and 13, which are the offsets they couple at, the statistic falls to 0.22 and 0.21.

    The second statistic was the first

    The experiment this collection has been specifying was priced as two readings off one sequence, the second of them free. The two readings turn out to be one function looked at twice, so the specification loses a statistic — and gains a cheaper one, a warning about how observables get priced, and a question it could not previously ask.

    part 12 · wrong
  8. Which family survives, along the 5/8 rung. The family a wrecked stem keeps, at every offset that wrecks and every rise on one rung. A counter shown any of these stems returns 5 and 8 spirals, at all twelve rises, so nothing in the pair distinguishes the columns. The cells say otherwise: at an offset of 4 the stem keeps the 5-family at the coarse end of the rung and the other one at the fine end. The offsets that wreck at all grow from 1 to 5 as the rise falls, because the front deepens, and the extra offsets are the ones that keep the larger number. So the rule stated over the offset alone is a rule with the rise left out of it.

    One rise per rung is a sample

    Every census on this site takes one rise from each rung, because the question was always which pair. Any rule later scored on those rows inherits a variable that was never varied — and two of this collection's results turn out to be about the sampling as much as about the rule.

    part 13 · emergence
  9. The column the census never carried. One row per lattice the ablation census was grown at. The bar shows where inside its own rung that rise sat, measured in the logarithm of the rise because the ladder is geometric, with zero the coarse transition and one the fine one. The mark on each bar is that rung's own handover, the rise where the two contact steps change places. Of the ten lattices that ever wreck, eight sit past their handover and one sit before it, with one sitting so close to one that the two steps differ by parts in a thousand. The rise was recorded in every table this collection has published; this fraction was in none of them.

    The column that cost no stems

    Every table in this collection records the rise a stem was grown at. None records where inside its own rung that rise sat, and the fraction turns out to be computable from numbers already written down — which makes it the cheapest column anybody here has ever added and the one that changes the most about how the tables read.

    part 14 · emergence
  10. The column the census never carried. One row per lattice the ablation census was grown at. The bar shows where inside its own rung that rise sat, measured in the logarithm of the rise because the ladder is geometric, with zero the coarse transition and one the fine one. The mark on each bar is that rung's own handover, the rise where the two contact steps change places. Of the ten lattices that ever wreck, eight sit past their handover and one sit before it, with one sitting so close to one that the two steps differ by parts in a thousand. The rise was recorded in every table this collection has published; this fraction was in none of them.

    The side the census sat on

    Eight of the ten lattices the ablation census wrecks at were grown past their rung's handover, one before it, and one so close that the ordering it quotes differs by parts in a thousand. A reading scored over the step ordering was therefore scored against a quantity the census was nearly holding fixed.

    part 15 · emergence

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