Phyllotaxis
the spirals are counted, not admired
Series

Count value — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A count of m and n pins the divergence to 221°/mn. Each dot is one reported pair, and its height is the total width of the divergence angles that could have produced it at some rise. 2/3 leaves 38.8° open; 34/55 leaves 0.118°. The line is 221°/mn, taken from the three highest pairs and drawn back through the rest.

    What a count is worth

    A reported parastichy pair pins the divergence angle to a band 221°/mn wide — so 2 and 3 says almost nothing and 34 and 55 fixes it to a tenth of a degree. Each step up the Fibonacci sequence is worth a factor of φ², and recording the radius a pair was counted at adds only ten per cent.

    part 5 · wrong
  2. For a high pair a miscount does not blur the report: it moves it. The divergence axis from 50° to 145°. For 21/34 and 34/55, the dark band is what the report allows if both counts are right and the pale bands what it allows if either is wrong by one. 21/34 allows 0.308° exactly and 0.931° in all, in 3 pieces: 20/33 at 54.37° and 22/35 at 82.13° beside the true band at 137.48°; 34/55 allows 0.118° exactly and 0.354° in all, in 3 pieces: 33/56 at 109.23° and 35/54 at 113.27° beside the true band at 137.52°. Each wrong band is as narrow as the right one and tens of degrees from it, so the reading does not widen; it becomes a short list of sharp candidates, and neither wrong candidate contains the golden angle.

    A count that can be wrong by one

    A reported parastichy pair pins the divergence angle to a band 221°/mn wide only if both counts are right. Allowing either to be off by one adds the bands of every neighbouring pair whose counts share no factor, and those bands sit where their own lattices live: for 2/3 they swallow the whole range, and for 34/55 they are two bands as narrow as the true one at 109° and 113°, twenty-five degrees away. So a high count that may be wrong is not a blurred reading but a short list of sharp candidates, costing log₂ 3 bits. And on the Fibonacci pairs, two in every six — 21/34 and 34/55 among them — cannot be miscounted silently by one count at all, because every such miscount shares a factor.

    part 6 · wrong
  3. What a report of 34 and 55 allows as a count is allowed to drift by one, two and three. The divergence axis from 20° to 180°. Dark: the band a report allows if both counts are right. Pale: the bands of every pair within the tolerance whose counts share no factor and which are the two shortest families somewhere in that range. 34/55 read with a tolerance of 1 allows 3 bands, 0.354° in all, the nearest wrong one 35/54 at 113.27°; 34/55 read with a tolerance of 2 allows 13 bands, 1.550° in all, the nearest wrong one 35/54 at 113.27°; 34/55 read with a tolerance of 3 allows 29 bands, 3.445° in all, the nearest wrong one 37/54 at 126.60°, and 2 more pairs whose bands lie below 20°; 21/34 read with a tolerance of 3 allows 28 bands, 8.767° in all, the nearest wrong one 18/31 at 139.55°, and 2 more pairs whose bands lie below 20°.

    A count that drifts by two

    A reported pair of 34 and 55 that may be wrong by one allows three sharp bands; allowed to drift by two it allows thirteen, and by three, twenty-nine — and the information lost is exactly the logarithm of that count, because every band is as narrow as the true one. The nearest wrong band stays twenty-four degrees away until a drift of three brings one to eleven. What does not survive is the protection: 21/34 and 34/55 could not be miscounted silently by one, but every Fibonacci pair can be by two, so a counter who drifts by two as readily as by one reports 34/55 silently wrong 9.5 per cent of the time rather than 0.13.

    part 7 · wrong
  4. The divergence a reading of 34/55 reports when the counter closes the circle early or late. A counter traces each family round 1 + ε of a turn, so both counts are multiplied by 1 + ε and rounded; the horizontal axis is ε in degrees of the circle, out to 54.0° either way. Where the rounded pair shares no factor it is drawn at the centre of the band it would put the divergence in; where it shares one it is a grey tick in the lane below. 34/55 is read right for closing errors under 3.27°; the first silent readings are 33/53 at −9.82° and 35/57 at +9.82°, reporting 54.4° and 82.2°. The silent band nearest the truth is 31/49's, 1.86° from it, at a closing error of 36.0°. Marked: a closing error of 9.8°, which reads 35/57 · reports 82.2°, silently wrong.

    Two counts that slip together

    A counter who closes the circle a few degrees late counts a sliver of the head twice, in both families at once, so the two counts of a reported pair drift together rather than apart. Coupled that way the count is safer than it was: fourteen Fibonacci pairs in twenty-three admit no silent equal shift of one, against seven that admit no silent single miscount, and 34/55 announces every closing error short of 9.82°. The check is what breaks. Two annuli closed at the same wrong mark pass 17.6 per cent of wrong readings of 34/55 and 76.8 per cent of 13/21's, because a linear relation survives multiplication — and what catches them instead is a protractor good to twelve degrees.

    part 8 · wrong
  5. What a reading of 34/55 becomes when each family is closed at a mark of its own. Each axis is one family's closing error in degrees of the circle, out to 24° either way; every rectangle is the set of closing errors that give one rounded pair, keyed as read right, sharing a factor (so announcing itself), or sharing none (so passing silently). The diagonal is one mark shared by both families, where the first silent readings are 33/53 and its mirror at 9.82°; off it the nearest are 33/56 and its mirror at 5.29°, with the two marks erring on opposite sides. The ellipses are one and two spreads of the marks' joint distribution at 7.2° a mark and a correlation of 0.50: 21.0% of readings right, 58.6% announced and 20.4% silent.

    Two marks chosen by one eye

    A counter traces each family of spirals from a starting organ of its own, so a reported pair carries two closing errors, correlated because one eye chose both. Letting them differ costs 34/55 its ten-degree margin — 33/56 and 35/54 share no factor, and marks that err 5.3° in opposite directions reach them — while 21/34 keeps its margin whatever the marks do. And it decides the second annulus. At a spread of 7.2° the relation passes right readings 2.8 times as readily as silent ones when the marks are independent, 1.25 times at a correlation of 0.9, and stops telling them apart at 0.98; where it does work it keeps one reading in forty-six.

    part 9 · wrong

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