Phyllotaxis
the spirals are counted, not admired
Series

Neighbour definition — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A cut-off would have to exceed 2.236 and not exceed 1.441, and nothing does both. Each of the 608 interior cells contributes two marks: its furthest wall, and its nearest partner that is not a wall, both in units of that cell's own shortest lag. A single cut-off would have to sit to the right of every mark of the first kind and to the left of every mark of the second, and the two clouds overlap — the extreme cases are 2.236 at 0.0 per cent of the radius and 1.441 at 60.0 per cent. So the interval is empty by a factor of 1.55, while 606 of the 608 cells have a cut-off that works for themselves.

    No cut-off makes them one

    Two different relations on a head have both been called neighbour: the shortest index lags a count keeps, and a shared Voronoi wall. A cut-off that turns the first into the second exists for almost every cell taken alone, and for no whole head at any size.

    part 1 · tissue
  2. One cell's six walls, and the two the pair 34 and 55 does not name. The cell of primordium 225, at 50 per cent of the head's radius, with each of its six walls labelled by the index difference across it. Its own counting instrument returns 34 and 55, which names four of them; the two drawn warm are 21 and 21, a family the instrument ranked and discarded. Over the whole head that is 33.80 per cent of the union in dispute, and it is the same fraction in every band.

    Two thirds of a cell

    The founding claim of this field is that the six sides Euler forces are the spiral families. Measured against the tessellation it names two thirds of a cell's walls exactly, in every band of a head and at every rise of a stem, and the missing third is the same third everywhere.

    part 2 · tissue
  3. Hops give 55, 34, 89, 21 and walls give 34, 55, 21, 89. The leading families of a 900-point head, ranked twice from the same points. On the left, by the median hop the lag makes divided by the local spacing — the measurement a parastichy count is; on the right, by the share of the 1,732 interior walls the lag carries. The lists hold the same four numbers and four of them change place. The lines between are the permutation, and it is why a family read off a wall count is not a family read off a hop length.

    Two rankings, one list

    An essay in this collection claimed that the four shortest index hops on a seed head and the four largest shares of its cell walls are the same four numbers in the same order, and called the correspondence exact. Measured again from the same points, the two lists hold the same four families and order them differently, and they order them differently in five of the six bands the head can be read in.

    part 3 · tissue
  4. A stem is a thread at six rises of the sweep, a ribbon at 23 and a surface at 111. The 140 rises of the sweep, split by how many index families the tessellation carries. On 6 rises above 0.5010 a node has 2 walls and the strip is a thread; on 23 between 0.2187 and 0.4833 it has 4 and is a ribbon; on 111 below 0.2109 it has 6 and is a surface. The comparison means something different in each: two families leave 33.333 per cent in dispute on the surface and 0.00 on the ribbon, and three families leave 0.000 and 33.333.

    The third family

    On a seed head no threshold makes the counted contacts and the shared cell walls the same relation. On a stem they are the same relation exactly, at every one of a hundred and eleven rises and to three decimal places of nothing, provided the contact cut keeps three families where a count keeps two.

    part 4 · tissue
  5. How often a cell on a 2,400-organ golden head has a wall the contact cut gets wrong, by its distance from the nearest flip ring. The 1,680 interior cells of a 2,400-organ golden head, grouped by distance from the nearest flip ring in twentieths of the head's wall spacing, 1.92. The bars are the share of each group with at least one wall a three-family contact cut gets wrong; the dots are the share with five or seven sides. 353 cells have such a wall and the farthest is 0.638 spacings from a ring. Of the 1,292 cells 0.66 spacings or more from every ring, none has.

    The empty interval is the rings

    No single cut-off on hop ratio turns the contacts a count keeps into the walls a tessellation draws, on any whole head at any size. Read cell by cell against the flip rings the divergence angle puts in closed form, every disputed cell lies within two thirds of a wall spacing of a ring, and with one spacing either side set aside a single cut-off between 1.430 and 1.444 serves every golden head from 900 organs to 9,000.

    part 5 · tissue
  6. The centre of a 900-organ Lucas head, with the band of two thirds of a spacing drawn round every flip ring. The organs of a 900-organ Lucas head out to a radius of 12.5, with a pale annulus 0.66 of a wall spacing either side of each flip ring, the rings keeping 1, 3, 4, 7, 11, 18. Warm: cells with five or seven sides; dark: six-sided cells a three-family contact cut gets wrong. Ringed: the one cell well inside a band that the cut reads exactly, organ 17 at radius 4.12, 0.23 of a spacing from its ring. Where two annuli overlap, a cell can sit well away from its nearest ring and still inside the next one's band.

    The blur was at the centre

    On a Lucas head the band of disputed cells round each flip ring looked blurred at its inner edge — exact cells as close as 0.23 of a wall spacing, disputed hexagons out to 0.59 where a golden head's stop at 0.43. Read a ring at a time, the two heads carry the same band on every resolved ring, to a hundredth: disputed hexagons within 0.16, exact cells from 0.64, a ring's own number of fives and of sevens and the number before it of hexagons. Every difference is inside a radius of six, where the Lucas rings of 4, 7 and 11 sit closer together than the band is wide, and the one exact cell is organ 17, which has no organ eighteen behind it.

    part 6 · tissue
  7. A 900-organ golden head displaced by 0.02 of a wall spacing, with its flip rings and the cells in dispute. Every organ of a 900-organ golden head inside the rim cut, each moved by a seeded gaussian displacement of 0.02 of a wall spacing, with the flip rings the divergence angle puts at radii 11.3, 18.2. Warm: the 150 cells with five or seven sides, 13 of them more than 0.66 of a spacing from every ring. Dark: the 59 six-sided cells with a wall a three-family contact cut gets wrong. Of the cells a spacing or more from every ring, 0 are disputed.

    A hundredth of a spacing

    Off the flip rings one hop-ratio cut-off turns a seed head's counted contacts into its cell walls, on every head from 900 organs to 9,000. Displace the organs and it is the first thing to go: shut by a fiftieth of a wall spacing on 900 organs and a two-hundredth on 9,000, because it is decided by the worst of thousands of cells. The three-family count survives two to four times further, because each cell only has to beat its own margin, and the rings keep their fives and sevens in between. All three fail from the rim inward, since the margin one spacing from a ring is 9.7 divided by the ring's family number.

    part 7 · tissue
  8. The disputed cells round the ring of 55 on a 2,400-organ head twisted by 0.2 radians at the rim. Every organ turned about the centre by 0.2 times its radius over the head's. Bars: the six-sided cells a three-family contact cut disputes (grey-blue) and the five- and seven-sided cells (warm) within two spacings of the ring the untwisted divergence puts at 55, by signed distance from it. There are 34 disputed hexagons, with a median at 0.54 of a spacing, and 110 fives and sevens. The dashed line is where the twisted divergence puts the ring in closed form, 0.54 of a spacing out; the band keeps its cells and its width and sits on it.

    The band moves, it does not blur

    Displaced organ by organ, a seed head loses its single contact cut-off first, its rings' hold on their fives and sevens next and its three-family count last. Displaced by a smooth field that moves neighbours together, the same head keeps its census — the same 353 disputed cells and 264 fives and sevens at every step up to a third of a spacing — and moves the band instead. A twist moves each flip ring exactly to where the twisted divergence puts its tie, the ring of 55 by 0.53 of a spacing, the ring of 34 the other way. Read against strain, correlation helps the cut-off and not the count, and on a 900-organ head the two fail at the same step: the order was an order of blurring.

    part 8 · tissue

All series