Phyllotaxis
the spirals are counted, not admired
Series

Instrument ceiling — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Every rung has a window it can be read in. For each rung of the ladder, the range of lag windows in which the pair can be read: at least three times the smaller parastichy number, so the main comb has three teeth inside the window, and less than three times the larger, so the larger number cannot itself be a candidate spacing. The band is [15, 24) at 5/8, [24, 39) at 8/13, [39, 63) at 13/21. It is non-empty at every rung, because it is empty only when the larger number is no larger than the smaller. The line at 30 is the window the earlier work used everywhere: it sits inside the 8/13 band and outside the 13/21 one, which is what was mistaken for a ceiling.

    The rung was not the instrument

    The earlier work said the pair readout has a ceiling one rung above where it works, that this is arithmetic rather than statistics, and that no amount of stem fixes it. The arithmetic is right and gives a band of lag windows that is never empty; what was actually stopping the reading was an eight-node seed and a grid of 384 azimuths.

    part 1 · cylinder
  2. What the finer grid does to the rises already published. The two rises this site has argued from and the one it published as having no answer, each read at both azimuth grids, five stems apiece. A filled mark agrees with the position counter, a half mark contradicts it, an open mark is a refusal. At 0.013 and 0.005 the readings are identical at both grids, so nothing already written depends on the sample count. At 0.008 they are not: 384 azimuths gives 5/8, 5/8 and 1152 gives 8/13, 8/13, against a counter that says 5/8. That rise was chosen in the earlier work because the three shortest lattice offsets there are within a fifth of each other, and a stem with no answer answering differently on a different grid is the object behaving as it was said to.

    What a sample grid decides

    The rule takes its minimum over 384 sampled azimuths, and that number has been a constant since this site's first commit. Tripling it changes nothing at the two rises the collection argues from — five runs of five, identical readings — and changes which answer appears at the one rise published as having no answer. A parameter of the program, measured rather than assumed.

    part 2 · emergence
  3. At 384 azimuths the ratio is 0.62; converged it is 0.82. The comb ratio and the recorded divergence scatter at a rise of 0.005, against how finely the rule samples the circle when it takes its minimum. At 384 azimuths — the grid every flat run in these essays uses, and the grid that earlier work's 0.65 was measured on — the step is 0.94°, which is larger than the 0.25° disturbance the stems carry. The quantisation is white noise, it dilutes both combs, and it does not dilute them equally. The ratio settles at 0.82 from 1152 azimuths up, and the scatter loses 0.19° that belonged to the grid rather than to the stem.

    The grid was in the number

    The rule places each organ at the least of a profile sampled at a fixed number of azimuths, and every flat run in these essays samples 384 of them — a step of 0.94°, against a disturbance of a quarter of a degree. The quantisation is the larger of the two, it is white, and it moves the discriminator from 0.79 to 0.62. The convergence study this collection had asked for and never done, in the place it turned out to matter.

    part 3 · cylinder
  4. Both edges of the front heal; the middle of it does not. The same removals, followed for 300 organs each. A cut one to three places back is undone within fifty organs and a cut ten to thirteen places back within sixty. A cut in between is never undone: the divergence sequence settles into an exactly repeating cycle of angles and holds it for the rest of the run. The rule corrects a displacement and cannot correct a deletion.

    A period the grid invented

    A wrecked stem was reported as settling into a repeating block of three angles — 219.84°, 220.31°, 220.78° — which is the smaller of its two spiral counts and would have confirmed a standing prediction. Those three numbers are three consecutive samples of the azimuth grid. There is no block; there is a constant the grid cannot write down, and the routine that found the block was working perfectly.

    part 4 · wrong
  5. Five fractions with one neighbour distance and every denominator. Each member of a matched set drawn on its own stretch of the divergence axis, 0.5° either side of itself, with the nearest other rational marked. The distances are 0.3651°, 0.3529°, 0.3692°, 0.3640°, 0.3557° — a spread of 4.6% — while the denominators run 17, 20, 25, 43, 44, a factor of 2.59. That is the construction this thread needed. Every instrument for the width of a disorder dip has a free parameter set by how close the neighbour is, so a hypothesis about the neighbourhood cannot be tested by varying the neighbourhood; on this set the neighbourhood is held fixed and the arithmetic of the fraction is what varies.

    Matching instead of correcting

    Two rounds of work failed on one question because every instrument's free parameter was set by the thing under test. The repair was not a better instrument or a model of the bias: it was choosing what to compare so that the confound could not vary. That move is available in four other places here, and three of them have already used it without anybody naming it.

    part 5 · wrong

All series