Phyllotaxis
the spirals are counted, not admired
Concept

Voronoi cells — where it appears

The division of a surface into the regions closest to each of a set of points, used here as the cells of a tissue. It is written out rather than imported, because the packing claims cannot be tested without cell areas and the hexagon claim cannot be corrected without side counts.

Named by 23 essays across 2 fields — each of them below, with the objects they name alongside it.

How many sides the cells actually have. The mean is 5.908, which Euler's formula forces. The spread around it is not noise — a tiling of only hexagons cannot close up on a finite patch.

Why the average cell has six sides

Not because hexagons are efficient. Because Euler's formula leaves a tiling no choice — count the edges two ways and the mean comes out at six, whatever the cells would prefer. The efficiency argument is a different claim about a different thing.

tissue · Sixsides 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 Cell area against side count, at 0% disorder. The dashed line is Lewis's law, (n−2)/4. The fitted slope here is 0.014 against his 0.25, and the side count accounts for 20% of the variation in area.

Lewis's law wants disorder

Cell area rises linearly with side count — measured on cucumber epidermis in 1928 and quoted ever since as a property of packed tissue. It holds beautifully on a random point set, with a fitted constant of 1.64 against Lewis's 2. On a phyllotactic head it does not hold at all: the slope is 0.009, and area and side count are almost independent.

tissue · Lewis Two laws, two tilings, and they disagree about which tiling is tissue. Lewis's law wants disorder: its slope is 0.231 on the random set and 0.009 on the golden head. Aboav's relation wants order: a = 1.18 on the head, 0.59 on the random set.

Two laws that want opposite tissue

Lewis's law and Aboav's relation are quoted side by side as properties of cellular tissue. Measured on the same two tilings they point opposite ways — the ordered head satisfies Aboav's with the textbook value of 1.18 and fails Lewis's completely; the random set does exactly the reverse.

tissue · Cell laws 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 The disorder of a head against its divergence angle, 300 points. μ₂ is the mean squared departure of a Voronoi cell's side count from six, measured on 300 points inside 86% of the radius. Swept across 1.60° it is a staircase: flat over stretches of a few hundredths of a degree, with sharp steps between them and narrow deep dips wherever a rational falls. At 137.142° — which is 360 × 8/21 — it is 0.078; At 137.646° — which is 360 × 13/34 — it is 0.197; At 138.458° — which is 360 × 5/13 — it is 0.060. The ticks along the top are the fractions p/q, placed from arithmetic rather than from the curve.

The disorder is a staircase

Sweep the second moment of a head's side-count distribution across the divergence angle and it is not a curve. It is flat in stretches with sharp steps between them and narrow deep dips wherever a rational falls — so 0.253 is not the golden angle's number, it is the number of every angle from 137.47° to 137.54°.

tissue · Second statistic The dip at 5/13 — 138.4615° — at three head sizes. Walking the divergence angle off an exact rational, at 300, 600, 1200 points. The floor falls as the head grows — 0.059 at 300, 0.029 at 600, 0.014 at 1200, halving for each doubling — and the dip narrows faster: half-widths of 0.0162°, 0.0048°, 0.0010°, a factor of four for each doubling rather than two. Half-width times the square of the head size is 1456, 1726, 1402, which is what makes the width a property of the sample rather than of the angle.

A dip belongs to the head

At an exact rational the disorder halves when the head doubles, and the dip around it narrows by a factor of four. So which angles look ordered is set by how many organs were counted, and a head of three hundred cannot tell 138.4615° from 138.48° while a head of twelve hundred tells it from 138.4625°.

tissue · Second statistic The neighbourhood of 21/55, and where its background was taken from. μ₂ across nine tenths of a degree either side of 21/55, on a head of 825 organs — 15 in each of its 55 rows. The deep notch at the centre is the dip. The shaded columns are the offsets now used as a background: those at least 0.03° from every rational with a denominator of 60 or less, of which there are 54 here. The marked sample at 0.2° is the one the earlier work used, and it lands 0.007° from 13/34 — inside that rational's own dip. It reads 0.115 against a floor of 0.148, so measured that way the dip is an inversion. The clear offsets give 0.339.

The background is not one sample

The dip in disorder at a rational angle is a comparison against a background, and the background was one measurement taken two tenths of a degree away. At 21/55 that lands seven thousandths of a degree from 13/34 — inside another rational's dip — and the comparison inverts. Fixed, the dip survives to a denominator of 89.

tissue · Second statistic The coefficient is not one number. n²·w, measured at the two larger of each denominator's three head sizes and averaged, for six Fibonacci fractions. Undivided it runs 1188, 1285, 2999, 7005, 9125, 14297 — a spread of 12.0. Divided by the denominator it runs 149, 99, 143, 206, 166, 161 — a spread of 2.1. So the law is w ≈ 150·q/n² to within a factor of two, and that earlier work's constant of 1,400 to 3,300 was three measurements of the small-denominator end of it.

The width carries the denominator

The earlier work measured three denominators, found the dip's half-width falling as the square of the head size, and could not say whether its coefficient depended on the denominator. Six denominators say it does: the coefficient runs from 1,188 at q = 8 to 14,297 at q = 89, and dividing by q flattens a factor of twelve into a factor of two.

tissue · Second statistic Four fractions of 55, one width. The half-width of the second-moment dip at four divergence angles with the same denominator, on heads of 1279 organs — 23 in each of 55 rows. 21/55 is the Fibonacci convergent, the fraction every earlier measurement of this law was made at and the one whose neighbourhood is emptiest; 12/55, 23/55, 17/55 are not. The four agree within a factor of 1.28, well inside the factor of two the law is stated to, so the width is a function of the denominator and not of how well the fraction approximates its neighbours. What is left over is ordered by the note beside each bar: the more crowded the neighbourhood, the wider the dip comes out, which is the direction a neighbour's own shoulder would push it.

Four fractions with one denominator

The dip in a head's side-count disorder is as wide as 150·q/n², measured over six fractions — every one of them a Fibonacci convergent, which is the emptiest neighbourhood a denominator ever gets. So the law could be about the denominator or about how well the fraction approximates its neighbours. Four fractions of 55 at one head size settle it in one figure.

tissue · Second statistic Four fractions of 34, one width. The half-width of the second-moment dip at four divergence angles with the same denominator, on heads of 791 organs — 23 in each of 34 rows. 13/34 is the Fibonacci convergent, the fraction every earlier measurement of this law was made at and the one whose neighbourhood is emptiest; 9/34, 15/34, 11/34 are not. The four agree within a factor of 1.14, well inside the factor of two the law is stated to, so the width is a function of the denominator and not of how well the fraction approximates its neighbours. What is left over is ordered by the note beside each bar: the more crowded the neighbourhood, the wider the dip comes out, which is the direction a neighbour's own shoulder would push it.

A width read off a staircase

Two fractions of the fourteen measured return a dip width that moves by a factor of two when the head size changes, where the others hold to three per cent. The cause is not their neighbourhood. It is that the disorder statistic changes only when the tessellation changes, so the curve a half-width is read off is a staircase, and a width narrower than the tread cannot be read at all.

tissue · Second statistic The disorder of a head against its divergence angle, 900 points. μ₂ is the mean squared departure of a Voronoi cell's side count from six, measured on 900 points inside 86% of the radius. Swept across 1.32° it is a staircase: flat over stretches of a few hundredths of a degree, with sharp steps between them and narrow deep dips wherever a rational falls. At 137.143° — which is 360 × 8/21 — it is 0.025; At 137.882° — which is 360 × 18/47 — it is 0.125; At 138.002° — which is 360 × 23/60 — it is 0.089. The golden angle is marked and sits at 0.253, in the middle of a flat stretch and nowhere near the largest value on the range.

The most irrational is not the most disordered

If rational angles make ordered tissue, the most badly approximable angle should make the most disordered — which would at last give the golden angle a criterion it wins. Swept across the interval it is the most irrational point of, μ₂ peaks at 138.42° and the golden angle sits unremarkably in the middle.

wrong · Second statistic 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 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 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.

tissue · Neighbour definition 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 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.

tissue · Neighbour definition 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.

tissue · Neighbour definition 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.

tissue · Neighbour definition Cell area against side count on a 900-organ head, every organ displaced by 0.2 of a spacing. The joint distribution of cell area, as a multiple of the mean, and side count, over the 639 interior cells of a golden head with every organ displaced by 0.2 of a wall spacing, seed one. The dashed line is Lewis's law, a quarter of the mean area for each side; the solid line is the fit, at a slope of 0.113, and the side count explains 30 per cent of the variation in area. Classes: 4 sides, 16 cells, mean 0.75; 5 sides, 159 cells, mean 0.89; 6 sides, 295 cells, mean 1.00; 7 sides, 146 cells, mean 1.11; 8 sides, 22 cells, mean 1.22; 9 sides, 1 cells, mean 1.26.

Lewis's law needs the sides to vary

Lewis's law holds on a random set of points and fails on a golden-angle head. Walked from one to the other by displacing every organ independently, the head's Lewis slope reaches half a random set's at a fifth of a wall spacing and nine tenths by seven tenths, and in between it explains up to 41 per cent of the variation in cell area — more than the 31 per cent it explains in the random set. Moved instead by a smooth field correlated over eight spacings, the head's cell areas become nearly as varied as a random set's and its slope stays at nought, because its side counts stay the lattice's. The law is not about how varied the cells are. It is about how varied their sides are.

tissue · Lewis The cells of a golden head with every organ displaced by 0.15 of a spacing, five- and seven-sided neighbours joined. A window eleven wall spacings square, about halfway out on a 900-organ golden head with every organ displaced by 0.15 of a wall spacing, seed one. Cells are keyed by side count; every five-sided cell is joined to each seven-sided cell it touches. In the window: 26 five-sided, 75 six-sided, 26 seven-sided and 5 of other counts, with 40 five–seven contacts. Over the whole head, averaged over five seeds: Aboav's a = 1.45, and 91 per cent of five-sided cells touch a seven.

One law counts sides, the other pairs

Lewis's law and Aboav's relation point opposite ways at the two ends of disorder, and the obvious guess is that they are one reading of disorder taken from two sides. Measured on the same moved heads, they are not. Displaced organ by organ, Aboav's a first rises — to 1.45 at 0.15 of a wall spacing, as the first new defects arrive as bound five–seven pairs — and falls half-way to a random set's only at 0.45 of a spacing, where Lewis's law had switched on at 0.2. Between the two a tissue satisfies both. A smooth field, which never switches Lewis's law on, lowers a by pulling the pairs apart without making any new defects. Lewis's law reads how varied the sides are; Aboav's reads whether the defects are paired.

tissue · Cell laws

Named alongside it

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

Honest limitsDisorderMeasurementRational angleSummary statisticOrder and disorderEuler's formulaArtefactRational approximationRim effectContact familyDivergence angle

All concepts