Phyllotaxis
the spirals are counted, not admired
Concept

Topological charge — where it appears

Six minus a cell's number of sides, so a hexagon carries none and every exception carries a signed unit. Summed over a tessellated head the total is fixed exactly by that head's own boundary, and on a golden head the freedom is spent on 264 exceptions among 1,631 interior cells that cancel to six.

Named by 9 essays across one field — each of them below, with the objects they name alongside it.

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.

tissue · Topological charge 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.

tissue · Topological charge 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.

tissue · Topological charge 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.

tissue · Topological charge 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.

tissue · Neighbour definition The second moment of five arrangements' side counts against the number of organs on the head. μ₂ on logarithmic axes for heads of 300 to 10,000 organs. Golden: 0.455 at 300 and 0.101 at 10,000; Lucas: 0.362 at 300 and 0.086 at 10,000; 137.5°: 0.453 at 300 and 0.045 at 8,000; whorled: 0.070 at 300 and 0.003 at 8,000; Poisson: 1.727 at 300 and 1.749 at 8,000. The Poisson set is a mean over three seeds. The dashed line is 6.83 over the square root of the organ count, the level the golden head returns to just before each defect ring enters the cut.

A second moment that goes to zero

The mean squared departure of a cell's side count from six separates a random tissue from a whorled head by a factor of eighty, on heads of 900 organs. Read at thirty-three head sizes it is exactly the share of cells on the defect rings of a spiral head and falls as one over the square root of the organ count, it falls as one over the count on a whorled head, and the factor is 25 at 300 organs and 677 at 8,000.

tissue · Sixsides The five kinds of cell across the ring where the 89 spirals begin, with every contact labelled. One cell of each kind from the ring of a 2,400-organ golden head where the family of 89 enters and the family of 21 leaves, drawn with a spoke to each neighbour labelled by the difference of their placement indices. Inside the ring, a hexagon has two contacts each of 21, 34 and 55. A seven has those six and one 89. A traded hexagon has one 21, two 34s, two 55s and one 89. A five has two 34s, two 55s and one 89. Outside, a hexagon has two each of 34, 55 and 89. Read outward, the cells give up the 21s and take on the 89s, and the seven and the five are the two ways of doing it unevenly.

A defect ring is where a family of spirals begins

Label every wall between two cells of a seed head with the difference of their placement indices and a hexagon is two contacts from each of three spiral families. The cells that are not hexagons sit on rings, and read by their labels each ring is a handover: one family of spirals ends there and the next begins. On a golden head the ring where 89 enters holds 34 sevens, 21 hexagons that have traded one contact and 34 fives — 34 + 21 + 34 = 89 — because every chain of the entering family begins at one of its cells and every chain of the leaving family ends at a traded hexagon. Joined by their walls, the fives and sevens make 21 clusters, and the clusters run round the ring in the order of the Fibonacci word.

tissue · Contact network A quarter of the ring where the 89 spirals begin, on a golden head displaced by 0.02 of a spacing. A quarter turn of a 4,000-organ golden head with every organ displaced by a gaussian of 0.02 of a spacing, unrolled so that angle runs across and radius down, between the radii the exact ring where 89 enters occupies. Each organ is marked by its cell's kind: seven, traded hexagon or five of that ring, a cell whose labels fit no kind, or an ordinary hexagon; and the walls joining a five to a seven are drawn, which is what makes the clusters. Round the whole ring on this head there are 32 sevens, 25 traded hexagons and 32 fives, 89 cells, in 23 clusters — 8 single pairs, 11 double and 4 of other sizes — whose order round the ring is not the Fibonacci word.

A ring keeps its count after it loses its order

On an exact golden head a defect ring holds exactly as many cells as the family of spirals entering it, and its fives and sevens join into clusters that run round it in the order of the Fibonacci word. Move every organ by a two-hundredth of a spacing and the order is gone from the outer rings on every head, while the count has not moved by a cell. The clusters' make-up goes next, at a hundredth or two. The count lasts longest — to a displacement of about three divided by the entering family, five hundredths of a spacing on the ring where 55 enter and one and a half on the ring of 233 — and while it lasts the ring pays for the displacement in kind, trading fives and sevens for hexagons that have swapped a contact, one pair for two.

tissue · Contact network A quarter of the ring where the 89 spirals begin, on a golden head moved by a drift of 0.1 of a spacing. A quarter turn of a 4,000-organ golden head moved by a drift along its spirals — each organ inheriting nine tenths of the mean error of its two nearest earlier organs, scaled to 0.1 of a spacing per coordinate — unrolled so that angle runs across and radius down, between the radii the exact ring where 89 enters occupies. Each organ is marked by its cell's kind, and the walls joining a five to a seven are drawn. Round the whole ring on this head there are 30 sevens, 31 traded hexagons and 30 fives, 91 cells, in 27 clusters, whose order is not the Fibonacci word. The same head displaced independently by 0.1 of a spacing holds 101 ring cells.

A ring outlasts an error that runs along its spirals

A golden head's defect ring holds exactly as many cells as the spiral family entering it, and displaced organ by organ the count lasts to a displacement of about three over the family. A grown head's errors are not organ by organ; they are carried along the spirals, so neighbours move together. Laid down that way, a drift of the same size keeps every ring's count to a displacement a quarter to twice as large again — a tenth of a spacing on the ring where 55 enter, against a twentieth — and keeps the clusters' make-up and order longer too, losing them in the same sequence. Stated as how far an organ moves against its neighbours, the drift's tolerance falls back into the band the independent heads set: the count law reads relative displacement.

tissue · Contact network

Named alongside it

The objects these essays reach for when they reach for this one.

Defect ringHonest limitsDelaunayVoronoi cellsRim effectTopological defectBound pairClaim testingParastichyDislocationDisorderEuler's formula

All concepts