Phyllotaxis
the spirals are counted, not admired
Series

Recovery — the series

10 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A round trip on four heads of 400 primordia: the divergence angle recovered from each. The counter is shown the points and nothing else. The worst recovery across the four is 0.035°.

    Recovering the angle from the counts

    Build a head at a stated divergence angle, forget the angle, and get it back from the spiral counts alone. Four angles, worst error twelve thousandths of a degree — and the only thing that crossed between the two halves was a list of coordinates.

    part 3 · lattices
  2. What the round trip returns as the organs are displaced: golden angle, 300 organs. Thirty heads at each size of displacement, from none to two and a half spacings, each run through the round trip and sorted by what came back. The undisplaced head counts 21 and 34. At 0.25 spacings, 97% the undisplaced pair, 3% a neighbouring pair; at 0.5 spacings, 50% the undisplaced pair, 50% a neighbouring pair; at 0.75 spacings, 30% the undisplaced pair, 70% a neighbouring pair; at 1 spacing, 13% the undisplaced pair, 80% a neighbouring pair, 7% a straddling pair; at 1.25 spacings, 7% the undisplaced pair, 57% a neighbouring pair, 7% a straddling pair, 30% refused; at 1.5 spacings, 40% a neighbouring pair, 3% a straddling pair, 57% refused; at 2 spacings, 10% a neighbouring pair, 90% refused; at 2.5 spacings, 3% a neighbouring pair, 97% refused. 10 recovered intervals in all exclude the true angle.

    A head displaced before it is counted

    The round trip from a head's spiral counts back to its divergence angle was tested on heads whose every organ sat exactly where the rule put it. Displaced by a normal error of up to two and a half spacings, heads of 900 organs keep counting a pair from their own sequence and return intervals holding the true angle to a spacing and a half; heads of 300 organs move to the neighbouring pair by half a spacing and then refuse, nine in ten of them by two spacings. Every moved count brings in the family whose chord was third shortest. Of 898 heads recovered, 19 intervals miss the true angle and 17 of those by about a tenth of a degree — displacement makes the reading coarser and then silent, not confidently wrong.

    part 4 · lattices
  3. A head of 300 organs as the rule places it, and twisted by 2 rad. Two heads at the golden angle, 300 organs each. On the left every organ is where Vogel's rule puts it, and the round trip counts 21 and 34 and recovers 137.515°. On the right the same organs turned about the centre by 2 radians times their radius over the head's, so the rim turns by 2 radians and the centre not at all; the round trip counts 34 and 47, recovers 137.80°, missing, with an interval of 0.230° that excludes the true angle.

    A twist is a divergence

    Recovering a head's divergence angle from its spiral counts survived independent displacements of whole spacings, moving to a neighbouring pair and then refusing rather than misleading. Displacements with a direction are harder on it in only one case. A head pressed to an aspect ratio of 2.25, spread at the rim by eighty per cent or sheared with a slope of 1.6 is still counted as its own pair or its neighbour, and recovered inside its interval. A head twisted — each organ turned about the centre in proportion to its radius — is not: past a turn of the rim of about a radian and a half the counts leave their sequence and the recovered angle misses, by up to sixty-one degrees, because a twist changes the angle between one organ and the next. The round trip is not fooled. It is reporting the angle the twisted head has.

    part 5 · lattices
  4. A 900-organ head twisted by 0.5 radians at the rim, with the bands its spirals are counted in. Every organ turned about the centre by 0.5 times its radius over the head's. The rings mark the inner annulus, 0.35 to 0.6 of the radius, and the outer, 0.7 to 0.95. The inner annulus counts 34 and 55; the outer 34 and 89 — skips one; the single band, 0.55 to 0.95, 34 and 89 — skips one. A count that is not two consecutive Fibonacci numbers flags the twist; on this head that happens from half a radian, and the single band is misled only from six.

    The count sees the twist first

    A twisted head recovers a changed divergence, and the check proposed for it was two annuli: the twist's extra angle falls with radius, so an inner and an outer annulus should disagree. Read on golden heads of 900, 2,400 and 9,000 organs, they never do in time. Their intervals separate at eight radians on 900 organs and never on the larger heads, always after the ordinary reading has been misled — from six radians on 900 organs and from two on 2,400 and 9,000. What catches the twist first, at every size and on every seed, is the count: at half a radian to three quarters some band stops returning two consecutive Fibonacci numbers — 34 and 89, 89 and 233 — which no untwisted golden head, clean or displaced, ever does.

    part 6 · lattices
  5. How much a twist of the rim changes the divergence between neighbouring organs, across a 900-organ head, for six shapes of twist. A twist of 0.25 radians at the rim, shaped as a·(r/R)^p, changes the angle between one organ and the next by a·(p/2)·ρ^(p−2)/(N − 1) at a distance ρ from the centre: the six lines are exponents 0.5, 1, 1.5, 2, 3, 4. At two the change is the same everywhere — 0.0159° — which is a different divergence angle; below two it is largest at the centre, above two at the rim. The shaded ring is the outer annulus, and the band between dotted lines is the flag's resolution on an untwisted 900-organ head: grown 0.0125° off the golden angle it counts consecutive Fibonacci numbers, 0.015° off it does not. At this twist the outer annulus passes it for exponents 2, 3, 4.

    The flag reads the angle, not the twist

    A twisted seed head is caught first by its counts: at half a radian some band stops returning two consecutive Fibonacci numbers, which an untwisted golden head never does. Twist the head in proportion to the square of the radius instead and the organs land exactly where a head grown at the golden angle plus a/(N − 1) puts them — to a billionth of a spacing — so the two read the same pairs in every band, and the flag fires on both. The flag detects a divergence that is not golden, and it has a resolution: 0.015° on 900 organs, 0.01° on 2,400, 0.003° on 9,000. Every twist shape from a half to four is flagged exactly when its change to the divergence in the outer annulus passes that resolution. So a flagged head is not golden, and nothing in its counts says whether it was twisted or grown that way.

    part 7 · lattices
  6. The divergence read from the positions ring by ring on a 900-organ head twisted half a radian at the rim, for six shapes of twist. A 900-organ golden head twisted by 0.5 radians at the rim as a·(r/R)^p, displaced by 0.1 of a spacing, read in six rings from 0.35 to 0.95 of the radius by the organs' phase sums within the counts' interval. Above the golden angle, ring by ring: p 0.5: 0.0371°, 0.0212°, 0.0179°, 0.0126°, 0.0114°, 0.0091°; p 1: 0.0451°, 0.0306°, 0.0271°, 0.0214°, 0.0192°, 0.0176°; p 1.5: 0.0426°, 0.0324°, 0.0320°, 0.0277°, 0.0264°, 0.0253°; p 2: 0.0374°, 0.0318°, 0.0334°, 0.0310°, 0.0316°, 0.0321°; p 3: 0.0241°, 0.0237°, 0.0306°, 0.0334°, 0.0385°, 0.0437°; p 4: 0.0148°, 0.0156°, 0.0249°, 0.0313°, 0.0414°, 0.0530°. The square law reads flat, 0.0310° to 0.0374°, as a head grown at another angle does; every other shape falls or rises across the head.

    The positions read the twist's shape

    A skipped Fibonacci count says a seed head's divergence is not golden, and cannot say whether it was twisted or grew that way; only the radial profile of the angle between neighbours can, and the counts are too coarse to see it. The organs' positions are not. The birth order is not in them — consecutive organs at the rim of a 900-organ head are a hundred and sixth of a spacing apart in radius, so a hundredth of a spacing of displacement scrambles a quarter of them — but each counted family is, and the sum of its phase angles across the organs peaks at the local divergence with no organ named. Displaced by a tenth of a spacing, a 900-organ head's outer annulus reads its divergence to 0.0001°, a hundred times finer than the flag resolves it, and the two annuli tell every twist shape from a changed angle at a quarter of a radian or less, no later than the flag fires. The square-law twist stays inside the untwisted heads' spread at every size and displacement: it is a changed angle, organ for organ.

    part 8 · lattices
  7. A seed head with half its rim missing, and three places its centre could be put. A 900-organ golden head, every organ displaced by 0.1 of a spacing, with the organs beyond seven tenths of the radius removed over half the head. The centroid of what is left lies 3.099 spacings from the centre the head grew about; the centre about which the inner annulus's two counted families are most coherent lies 0.0132 spacings from it. The inset magnifies half a spacing around the true centre.

    The centre the spirals give

    Reading a seed head's divergence from its organs' positions needs the head's centre, and a photograph does not give it. A misplaced centre turns every organ's angle by an amount that varies round the head and grows toward the middle, so it could read as a twist. It does not, until it is large: a 900-organ head tolerates a centre a sixth of a spacing off, a 2,400-organ head a quarter. The centroid of the organs finds a whole head's centre to a hundredth of a spacing, and read about it the positions separate every twist shape exactly as well as with the centre given. The radius law, which looks like the natural fit, does not find it at all. And when half the rim is missing the centroid moves three to five spacings and turns an untwisted head into a twisted one, while the centre about which the counted spirals are most coherent still lands within a hundredth of a spacing.

    part 9 · lattices
  8. A golden seed head photographed off its axis, with the squash its second moments find. A 900-organ golden head, every organ displaced by a tenth of a spacing, photographed 20° off its axis: every coordinate along a bearing of 7° shortened by the cosine of the tilt. The head's second moments about its centroid put the short axis at 7.7° and imply a tilt of 19.75°. Read square on, the inner annulus gives a divergence of 137.5080° and the outer 137.5080°; stretched back along the short axis first, 137.5078° and 137.5077°, against the golden angle's 137.5078°. The bands count 34 and 55 spirals inside and 55 and 89 outside.

    A head photographed from the side

    A seed head photographed off its axis is squashed by the cosine of the tilt across one bearing, and the squash turns every organ's angle by an amount that goes round the head twice. Read square on, a 900-organ golden head tolerates twelve to fourteen degrees of tilt before its two annuli disagree by more than any untwisted head's do, and the spiral counts never notice at all. Past that, the reading is not a smooth drift but a scatter of heads misread one at a time. Stretching the head back along the short axis its own second moments find restores the square-on reading exactly to thirty degrees, even though the moments cannot see a tilt under three or four degrees. And the square-law twist stays hidden: it is a changed angle organ for organ, so no camera, at any angle, sees it differently.

    part 10 · lattices
  9. A golden seed head photographed twenty degrees off its axis by a camera close enough to see it in perspective. A 900-organ golden head, every organ displaced by a tenth of a spacing, photographed by a pinhole camera 5 head radii from its centre and tilted 20° across a bearing of 7°: the near side of the head larger in the picture than the far side. Read square on, its inner annulus gives 137.5064° and its outer 137.5090°; stretched back along its second moments' short axis, 137.5069° and 137.5013°; un-projected through a camera fitted from the organs — tilt 19.9°, bearing 10°, distance 5.15 radii — 137.5083° and 137.5078°, against the golden angle's 137.5078°. Of the ten heads photographed this way, 1 are read as untwisted square on, 0 are read as untwisted by the ellipse, 9 are read as untwisted through the fitted camera.

    A head seen in perspective needs a camera, not an ellipse

    A seed head photographed off its axis from far away is squashed, and stretching it back along the short axis of its own second moments restores its reading exactly. Photographed from close up, the near side of the head is larger than the far side, and the stretch stops working: at twenty degrees off the axis it reads every head right from ten head radii and none from five. The organs' centroid drifts towards the near side as the camera comes in, but that is not why — the same stretch about the head's true centre fails at the same distances. What fails is the taper. A camera fitted from the organs alone, its tilt, bearing, distance and the head's centre chosen so that the head it un-projects is round, centred and evenly packed, reads seven to ten heads in ten right at every distance down to two radii.

    part 11 · lattices
  10. A golden seed head laid on a dome and photographed twenty degrees off its axis from infinitely far. A 900-organ golden head, every organ displaced by a tenth of a spacing, laid evenly on a paraboloid dome 0.2 head radii high and photographed by a camera infinitely far away, tilted 20° across a bearing of 7°. The dome lifts the centre of the head towards the camera, and seen from the side every organ moves along the bearing by its height times the sine of the tilt, more at the centre than at the rim. Of ten such heads, 0 are read as untwisted after the moments' ellipse and 0 through the camera fitted for a flat head, which puts the camera 7.4 head radii away at a tilt of 20.2° (median).

    A dome is read as a nearer camera

    A camera fitted from a photographed seed head's organs alone restored the reading of a flat head seen in perspective from as close as two radii. Lay the same heads on domes and the camera is fooled in a particular way: it finds the tilt, its median within about a degree, and puts itself nearer than it is, by an amount set by the dome's height, so that a dome a fifth of the head's radius high photographed from infinitely far is read as a flat head photographed from seven radii. The reading it hands on is wrong. Off the axis, a domed head is misread once its height times the sine of the tilt passes about four hundredths of the radius — the same number at which a flat head's perspective broke the old correction — and only a photograph taken square on is safe.

    part 12 · lattices

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