Phyllotaxis
the spirals are counted, not admired
Series

Topological charge — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 135 fives and 129 sevens among 1631 interior cells. The side counts of every cell strictly inside a golden, 137.508° head of 2400 organs, cut at 86% of the radius. The fives and the sevens are counted apart rather than summed into a spread, because they are opposite charges and the sum hides them. Summed over the interior, 264 cells that are not hexagons carry a charge of +6. Over the whole patch the charge is 294, which is exactly 6 + 2·144 — a number fixed by the 144 cells on the patch's own boundary and carrying nothing whatever about the interior. The defects are not scarce; they are balanced.

    An interior that is nearly neutral

    Give every cell a charge of six minus its number of sides and the total over a tessellated head is fixed by its own boundary, exactly, with nothing left over for the interior. On a golden head that freedom is spent on 264 exceptions among 1,631 cells which cancel to six.

    part 1 · tissue
  2. 264 exceptions in a 2400-organ head, and eight circles. A golden, 137.508° head of 2400 organs, tessellated inside 86% of its radius. Every interior cell is drawn, and the 135 five-sided cells and 129 seven-sided ones are marked apart from the 1367 hexagons, and the pale circles are radii computed from the divergence angle through the lattice's third-shortest vector — nothing is fitted. Every defect sits within 0.64 of a cell spacing of one of those eight circles, and between them there is not one exception in hundreds of cells.

    The defects lie on rings

    The cells in a seed head that are not hexagons are not scattered through it. Every one of 264 sits within a cell of a radius computed from the divergence angle alone, the radii are a factor of φ apart, and between two of them lie 422 consecutive cells without a single exception.

    part 2 · tissue
  3. Every five is a cell away from a seven, and the loneliest is 0.927 spacings out. How far a five-sided cell is from the nearest seven-sided one, in cell spacings, on a golden, 137.508° head of 2400 organs. The measured bar runs from the closest five to the loneliest — 0.833 to 0.927, with a median of 0.919. It stops a single cell out, so there is no unpaired tail at all rather than a small one. The nulls are seeded permutations over 200 draws: relabelling which defects are fives puts the average five 1.254 ± 0.064 spacings away, and scattering the whole multiset over the interior cells puts it 1.786 ± 0.092. 100.0% of the fives share a wall with a seven against 68.2% for the strong null, z = 7.3.

    Every five is bound to a seven

    A five-sided cell beside a seven-sided one is one object in a crystal and two exceptions in a tiling, and the phyllotaxis literature borrows the crystallographic word without measuring the binding. Measured against a seeded permutation null on a 2,400-organ head, every five in the interior shares a wall with a seven, and the loneliest one in the head is 0.927 cell spacings from the nearest.

    part 3 · tissue
  4. The defect rings are not where the counts change — they are √φ further out. A logarithmic radius axis for a golden, 137.508° head. The lower marks are the radii at which the counted parastichy pair changes, where the two shortest lattice vectors change places; the upper marks are the radii at which a cell's neighbours change, where the third-shortest does. They interleave, and the ratio of each ring to the transition inside it is 1.2715, 1.2723, 1.2723, 1.2719, 1.2719, 1.2723 against √φ = 1.27202. Consecutive rungs are a factor of φ apart in radius and √φ is their geometric midpoint, so a defect ring sits exactly halfway between two parastichy transitions. Anyone looking for the defect line at the radius where the counts change will not find it there.

    The rings are not the transitions

    A seed head has two ladders on it — the radii where the counted parastichy pair changes, and the radii where the exceptional cells sit — and the obvious guess is that they are the same ladder. They are not: the second sits a factor of the square root of phi outside the first at every rung of two different divergence ladders, which is exactly halfway between two consecutive transitions.

    part 4 · tissue
  5. A Burgers circuit round a dislocation in a crystal and round a five–seven pair on a golden head's ring. Left: a triangular lattice moved by the elastic field of one edge dislocation, its five- and seven-sided cells marked, with the tightest circuit round them, the 8 organs touching either. Rounded to whole lattice vectors the steps sum to -1, 0: the circuit fails by one lattice vector. Right: a five–seven pair on the ring at radius 29.51 of a 4,000-organ golden head and the 8 organs touching either of the two. Its steps sum to 0, 0, and the circuit closes.

    A dislocation that closes

    A defect ring on a seed head looks exactly like a low-angle grain boundary: a closed chain of bound five–seven pairs one cell thick. The defining test of a dislocation is a loop that fails to close. On an elastic crystal it fails by one lattice vector at every size. On a golden head's rings the tightest loop round a five–seven pair closes every time, 272 pairs out of 272, and one set of lattice labels holds every patch across the outer rings without a single broken edge. No lattice rotates at the ring, where a real boundary of that density would need thirty-two degrees. What a ring actually is sits in one number: its walls are within two over its own count of being replaced by the other diagonal of their cell, on one curve for all four rings.

    part 5 · tissue

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