Phyllotaxis
the spirals are counted, not admired
Ladder

Neighbour definition — the ladder

4 distinct arguments against 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.

    rung 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 on this site 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.

    rung 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.

    rung 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.

    rung 4 · tissue

All ladders