Phyllotaxis
the spirals are counted, not admired
Concept

Golden angle — where it appears

The divergence of 137.508 degrees, one full turn divided by the square of the golden ratio. It arrives here as the settled output of a rule that never mentions it, over a range of one parameter, rather than as an assumption put in at the start.

Named by 11 essays across 4 fields — each of them below, with the objects they name alongside it.

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 Every family but two is the sum of two others. Four heads and the contact families each actually has. An arc arrives at every family that is the sum of two smaller ones; the two with no arc arriving are the generators, and on every head they are the two smallest. whorled, 144°: 2, 3, 5 — golden, 137.508°: 8, 13, 21, 34, 55, 89 — Lucas, 99.502°: 11, 18, 29, 47, 76 — rational, 137.5°: 8, 13, 21, 34, 55, 89 — 137.0°: 8, 13, 21, 29, 50, 71, 92, 113. So once a pair is counted the rest is arithmetic, and a third counted family is a prediction rather than a second measurement.

Every family but two is a sum

A seed head has six spiral families and everybody reports two. That looks like a convention hiding information and it is the opposite — every family but the two smallest is the sum of two others, so a third count is a prediction rather than a measurement, and a check that catches a wrong pair.

lattices · Counting 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 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 How much a twist of the rim changes the divergence between neighbouring organs, across a 900-organ head, for six shapes of twist. A twist of 0.25 radians at the rim, shaped as a·(r/R)^p, changes the angle between one organ and the next by a·(p/2)·ρ^(p−2)/(N − 1) at a distance ρ from the centre: the six lines are exponents 0.5, 1, 1.5, 2, 3, 4. At two the change is the same everywhere — 0.0159° — which is a different divergence angle; below two it is largest at the centre, above two at the rim. The shaded ring is the outer annulus, and the band between dotted lines is the flag's resolution on an untwisted 900-organ head: grown 0.0125° off the golden angle it counts consecutive Fibonacci numbers, 0.015° off it does not. At this twist the outer annulus passes it for exponents 2, 3, 4.

The flag reads the angle, not the twist

A twisted seed head is caught first by its counts: at half a radian some band stops returning two consecutive Fibonacci numbers, which an untwisted golden head never does. Twist the head in proportion to the square of the radius instead and the organs land exactly where a head grown at the golden angle plus a/(N − 1) puts them — to a billionth of a spacing — so the two read the same pairs in every band, and the flag fires on both. The flag detects a divergence that is not golden, and it has a resolution: 0.015° on 900 organs, 0.01° on 2,400, 0.003° on 9,000. Every twist shape from a half to four is flagged exactly when its change to the divergence in the outer annulus passes that resolution. So a flagged head is not golden, and nothing in its counts says whether it was twisted or grown that way.

lattices · Recovery The divergence read from the positions ring by ring on a 900-organ head twisted half a radian at the rim, for six shapes of twist. A 900-organ golden head twisted by 0.5 radians at the rim as a·(r/R)^p, displaced by 0.1 of a spacing, read in six rings from 0.35 to 0.95 of the radius by the organs' phase sums within the counts' interval. Above the golden angle, ring by ring: p 0.5: 0.0371°, 0.0212°, 0.0179°, 0.0126°, 0.0114°, 0.0091°; p 1: 0.0451°, 0.0306°, 0.0271°, 0.0214°, 0.0192°, 0.0176°; p 1.5: 0.0426°, 0.0324°, 0.0320°, 0.0277°, 0.0264°, 0.0253°; p 2: 0.0374°, 0.0318°, 0.0334°, 0.0310°, 0.0316°, 0.0321°; p 3: 0.0241°, 0.0237°, 0.0306°, 0.0334°, 0.0385°, 0.0437°; p 4: 0.0148°, 0.0156°, 0.0249°, 0.0313°, 0.0414°, 0.0530°. The square law reads flat, 0.0310° to 0.0374°, as a head grown at another angle does; every other shape falls or rises across the head.

The positions read the twist's shape

A skipped Fibonacci count says a seed head's divergence is not golden, and cannot say whether it was twisted or grew that way; only the radial profile of the angle between neighbours can, and the counts are too coarse to see it. The organs' positions are not. The birth order is not in them — consecutive organs at the rim of a 900-organ head are a hundred and sixth of a spacing apart in radius, so a hundredth of a spacing of displacement scrambles a quarter of them — but each counted family is, and the sum of its phase angles across the organs peaks at the local divergence with no organ named. Displaced by a tenth of a spacing, a 900-organ head's outer annulus reads its divergence to 0.0001°, a hundred times finer than the flag resolves it, and the two annuli tell every twist shape from a changed angle at a quarter of a radian or less, no later than the flag fires. The square-law twist stays inside the untwisted heads' spread at every size and displacement: it is a changed angle, organ for organ.

lattices · Recovery Everywhere a cut of one to five organs can send a 5/8 stem. Every settled divergence reached by any arrangement of up to five organs removed from a stem at the 5/8 rung, on one axis. There are six of them and no more. three are slips of the lattice the stem was cut from: each keeps the lag-5 family intact and sits a whole number of turns of it from the next, which is the ladder marked below the axis with rungs 72.0 degrees apart. The other three keep no lag at all and sit near a fraction of a turn, marked above: 175.0 degrees near 1 of 2 turns, 190.0 degrees near 1 of 2 turns, 235.0 degrees near 2 of 3 turns. A stem that is cut either slides along the ladder it was on or leaves it for a lattice with files in it.

A file has to close

The three destinations counted with a shared factor sit near a half turn, a half turn and two thirds. Measuring how near is the trap: by distance from the fraction, the golden angle is closer to two fifths than two of them are to anything, and would be reported as having five files it does not have.

cylinder · Dose A seed head with half its rim missing, and three places its centre could be put. A 900-organ golden head, every organ displaced by 0.1 of a spacing, with the organs beyond seven tenths of the radius removed over half the head. The centroid of what is left lies 3.099 spacings from the centre the head grew about; the centre about which the inner annulus's two counted families are most coherent lies 0.0132 spacings from it. The inset magnifies half a spacing around the true centre.

The centre the spirals give

Reading a seed head's divergence from its organs' positions needs the head's centre, and a photograph does not give it. A misplaced centre turns every organ's angle by an amount that varies round the head and grows toward the middle, so it could read as a twist. It does not, until it is large: a 900-organ head tolerates a centre a sixth of a spacing off, a 2,400-organ head a quarter. The centroid of the organs finds a whole head's centre to a hundredth of a spacing, and read about it the positions separate every twist shape exactly as well as with the centre given. The radius law, which looks like the natural fit, does not find it at all. And when half the rim is missing the centroid moves three to five spacings and turns an untwisted head into a twisted one, while the centre about which the counted spirals are most coherent still lands within a hundredth of a spacing.

lattices · Recovery A golden seed head photographed off its axis, with the squash its second moments find. A 900-organ golden head, every organ displaced by a tenth of a spacing, photographed 20° off its axis: every coordinate along a bearing of 7° shortened by the cosine of the tilt. The head's second moments about its centroid put the short axis at 7.7° and imply a tilt of 19.75°. Read square on, the inner annulus gives a divergence of 137.5080° and the outer 137.5080°; stretched back along the short axis first, 137.5078° and 137.5077°, against the golden angle's 137.5078°. The bands count 34 and 55 spirals inside and 55 and 89 outside.

A head photographed from the side

A seed head photographed off its axis is squashed by the cosine of the tilt across one bearing, and the squash turns every organ's angle by an amount that goes round the head twice. Read square on, a 900-organ golden head tolerates twelve to fourteen degrees of tilt before its two annuli disagree by more than any untwisted head's do, and the spiral counts never notice at all. Past that, the reading is not a smooth drift but a scatter of heads misread one at a time. Stretching the head back along the short axis its own second moments find restores the square-on reading exactly to thirty degrees, even though the moments cannot see a tilt under three or four degrees. And the square-law twist stays hidden: it is a changed angle organ for organ, so no camera, at any angle, sees it differently.

lattices · Recovery The settled divergence down the golden branch. Every rise from 0.07 down to 0.00482, plotted against the divergence the rule settles on, with each rung drawn in its own stroke and the branch's limit angle marked. The curve does not slide: it turns three times in four rungs, climbing across one and falling across the next, so a value it takes on one rung it takes again on another. That is what makes a matched pair possible — two rises, different counted pairs, one angle — and it is the whole reason the design exists on this branch. The widest excursions from the limit angle, coarse rung first, are 3.195°, 3.352°, 0.961°, 0.422°.

The angle the ladder returns to

Down the golden branch the settled divergence climbs across one rung and falls across the next, turning three times in four rungs. That is why the same angle is reached at two different rises — and why the Lucas branch, which turns once, almost never offers the same thing.

cylinder · Same angle

Named alongside it

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

Divergence angleMeasurementHonest limitsParastichy pairRational angleRound tripDisorderIdentifiabilityMeasurement sensitivityResolutionVogel's modelVoronoi cells

All concepts