Phyllotaxis
the spirals are counted, not admired
Concept

Rim effect — where it appears

The distortion in any tessellation statistic caused by cells at the boundary, whose neighbours are missing rather than few. It is large enough to account for claimed departures from Lewis's law on its own, which is why every statistic here is taken on an interior annulus.

Named by 14 essays across 2 fields — each of them below, with the objects they name alongside it.

Closest pair across 120–155° at 400 organs, read both ways. On the interior's scale the golden angle reads 0.9027 and ranks 1st of 72, against 0.9026 for the best grid angle at 137.5°, with the window running from 0.0668 to 0.9026; counting the rim's cells the golden angle reads 0.7076 and ranks 2nd of 72, against 0.7129 for the best grid angle at 137.5°, with the window running from 0.0331 to 0.7129. The dashed upright is the golden angle, which a grid of decimal degrees never lands on and which is therefore read separately.

Packing, measured four ways

The claim is that the golden angle packs best, and it is measurable. Read on the interior of a head, the two criteria about distance put the golden angle first among the angles near it and the two about cells are won by rational angles — which makes the claim half right, and makes the right half a statement about a class of angles. An earlier reading of the same four criteria, divided by the cells at the head's edge, said the opposite.

tissue · Packing The statistic everybody reports is the one that cannot vary. Six arrangements of 900 points, from a whorled lattice to a set with no rule in it. The mean number of sides per cell is 5.97–6.04 on all six, because Euler's formula forces it. The mean squared departure from six runs from 0.023 to 1.83 — a factor of 79 — and the most hexagonal tissue in the set is the whorled one, at a rational angle.

The second moment is the measurement

The mean number of sides in a cellular tissue is six, and Euler's formula leaves it no choice — so it takes the same value on a golden-angle head, a whorled head and a set of random points. On heads of nine hundred organs the mean squared departure from six varies by a factor of eighty across the same three, and almost nobody reports it.

tissue · Sixsides A cell's neighbours are its spiral families. Left: part of a 900-point golden, 137.508° head, with every contact between two cells drawn and coloured by the difference between the two nodes' placement indices. Right: the share each difference takes, across all 1903 contacts between interior cells. They are the parastichy numbers — 34, 55, 21, 89, 13, 8 — and the pair a person would count is 34 and 55. The six sides Euler forces are shared out among four of them, 5.72 edges per cell.

The six are the spirals

Label every contact between two cells in a seed head with the difference between the two nodes' placement indices. The labels are the parastichy numbers — 34, 55, 21, 89 — and the six sides Euler forces turn out to be about two from one family, one and a half from the next, and one each from two more.

tissue · Contact network 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 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 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.

tissue · Neighbour definition The cell area a 137.508° golden head's packing is divided by, at 10 sizes. Counting every bounded cell the mean area runs from 3.607 to 38.828, the largest at 150 organs, because the cells just inside the edge of the head reach out to circumcentres far beyond it. Leaving out every cell whose polygon crosses the head's own radius, it stays between 3.1425 and 3.1634 at every size drawn, which is π, the area the square-root rule gives each organ.

Packing, measured against the interior

An earlier reading of these heads reported that no packing criterion singles out the golden angle and that three criteria give three winners. Every one of those readings was divided by a mean cell area that, on a head of 150 organs, was 38.8 where the interior's is π. Divided by the interior's own, the criteria about distance put the golden angle first of 72 angles and the criteria about cells go to rational ones.

tissue · Packing The largest empty circle of a 2400-organ 137.508° golden head, ring by ring. Each dot is the largest empty circle in one ring of the head, on the interior's scale. The whole head's is 0.8437 at radius 0.871, at the centre; the rings peak at the radii where the lattice flips from one pair of neighbours to the next, and at the 4 flips beyond radius six the head reads within 0.0020 of the closed form. The dashed line is 1/√2 = 0.7071.

One over root two

On the interior's scale a golden head's largest empty circle is 0.8435 of a spacing at every size from 150 organs to 2,000, because one triangle at its centre decides it. Everywhere else it is 1/√2 — a square cell at every ring where the lattice flips — and a closed form in the angle's continued fraction says that only noble angles hold it there.

tissue · Gapgrowth 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 Four head sizes, one staircase. The same window swept at 539, 900, 1409, 3690 organs, each curve divided by its own median so that the overall fall with head size is out of the way and only the shape is left. The features line up. Across the three steps in size, 52 of the 53 features present at a smaller head are still present at the same angle at the next size up — nothing slides. What a bigger head does is resolve features between the ones already there, which is a statement about the instrument rather than about the arrangement.

Nothing in the staircase moves

Disorder swept across the divergence angle is a staircase, and every step of it had been read at one head size — which leaves open whether a step is the lattice changing or a ring of defects crossing the rim as the angle moves it. Read again at 539, 900, 1409 and 3690 organs, 52 of the 53 features present at a smaller head are still there at the same angle at the next size up. Not one slides. A bigger head adds steps between the ones already there — 4, 18, 31, 40 — so the staircase belongs to the angle and the head size decides only how much of it is resolved. The one size every other disorder figure here uses turns out to sit three per cent past a ring entry.

tissue · Sixsides A gap in the disorder staircase read at a hundredth of its grid. μ₂ on a head of 3,690 organs from 137.1975° to 137.2725°, a gap in which the staircase counts no step, read every 0.00005° — 1500 samples — against the staircase's median; the open circles are the staircase's own samples every 0.005°, which the fine sweep passes through exactly. No change between two fine samples reaches a fifth of the median. The curve is a sawtooth: 13 falls of ten cells or more, 19 at 137.2004°, 21 at 137.2009°, 18 at 137.2054°, 21 at 137.2058°, 14 at 137.2111°, 18 at 137.2115° and more, against 0 climbs that large; climbs average 2.4 cells and falls 4.0.

What lies between the steps

The disorder staircase — the spread of a head's side counts against its divergence angle — gained steps with every larger head, forty at 3,690 organs, and nothing said whether it had steps at every scale. Read again at a hundredth of its grid inside its two widest gaps, it has none: no change there reaches the size it counts as a step, and no dip hides between two of its samples. The steps stop. What the gaps hold instead is a sawtooth — μ₂ climbing a cell or two at a time and falling in teeth of ten to thirty-two cells, five of them exactly twenty-one — and a step, read at the same resolution, is not one event but two runs of flips of fifty-five cells each. How many steps a head has is a statement about where the line is drawn; the steps themselves are finite.

tissue · Sixsides Where the pattern starts, measured at two run lengths. One row per wrecked cut. The small mark is the onset a 300-organ run reports and the ring is what a 600-organ run reports; a row with only one mark reports an onset at only one length. 19 of the 24 rows that report both move by more than twenty organs, and the largest move is from 135 to 435. The range the thread has been quoting, 7 to 303 organs, becomes 28 to 473.

An onset at the end of the run

One cut reported that its pattern began 299 organs into a 300-organ run. Given twice the room it reports no pattern at all. The reading was the run stopping, not the disturbance ending, and the definition guarantees one at the last organ of every run.

mechanism · Run length

Named alongside it

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

Honest limitsDisorderRational angleVoronoi cellsDefect ringDelaunayMeasurementNegative resultSummary statisticArtefactTopological chargeClaim testing

All concepts