Phyllotaxis
the spirals are counted, not admired
Concept

Noble number — where it appears

A number whose continued fraction ends in all ones, so it is approximated by fractions as slowly as any number can be. The golden ratio is one, and so are others, which is why being hardest to approximate does not single it out.

Named by 10 essays across 5 fields — each of them below, with the objects they name alongside it.

The version of the claim that does survive measurement. The golden angle scores 0.4377, against 0.3462 for the best of 938 other angles sampled. The dashed line is Hurwitz's 1/√5, which no number can exceed.

The claim that survives

Of the three famous assertions about this subject, one is out by a factor of two, one is true of a branch rather than of plants, and one is right — in a sharper form than the version usually told, and about arithmetic rather than about packing.

wrong · Hurwitz Two paths down the same tree. Both start at the same first fork. Keeping the larger family every time reaches 137.473°; one different choice reaches 99.549°. Neither angle is in the arithmetic — both are limits of a path.

The tree and the attractor

The dynamical model settles on the golden angle over a range of one parameter and on the Lucas angle outside it, and it cannot reach the low-growth end at all. The lattice tree reaches everywhere, has no dynamics in it, and produces the same two angles as limits of two paths. Two routes, one pair of numbers.

emergence · Branch tree The 2-jugate forks converge on 68.7539°. Every fork sits at a rational divergence, with denominator 4(m² + mn + n²) — 10/28, 30/76, 74/196 and so on. The limit is 68.7539°, which is 137.5078 divided by 2, and it is at none of them.

Half the golden angle

The forks of the van Iterson tree converge on 137.5078° and sit at none of them. Divide the whole tree by two and the same thing happens at 68.7539° — which is where teasel is, and where a bijugate sunflower counted 42 and 68 has to be.

lattices · Jugate limit 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 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 Every angle whose counts from 34 to 144 resist approximation within a per cent of the golden angle's. Over the counts a head shows from 34 to 144, an angle scores like the golden angle when those counts add up, each the sum of the two before, from a first pair near the golden ratio. 46 angles between 20° and 180° come within one per cent of its score of 0.44718, each drawn as a stem at its angle. The five nearest are 137.51° with counts 34, 55, 89, 144 at 100.000 per cent; 99.50° with counts 47, 76, 123 at 99.989 per cent; 106.45° with counts 44, 71, 115 at 99.931 per cent; 151.14° with counts 50, 81, 131 at 99.919 per cent; 132.18° with counts 49, 79, 128 at 99.907 per cent. The golden angle is the highest, and the Lucas angle at 99.50° is a ten-thousandth of the score behind it.

What a head can mean by most irrational

Hurwitz's bound, the one famous claim about this subject that survives, is a limit over every denominator, and a head shows only the counts between its innermost spirals and its rim. Over those counts an angle resists approximation like the golden angle exactly when the counts it shows add up, each the sum of the two before, from a pair near the golden ratio — and every such pair has an angle of its own. The golden angle still scores highest over every window measured, by a ten-thousandth: over counts from 34 to 144 the Lucas angle is 99.989 per cent of it and forty-six angles are within one per cent. What separates the golden angle from them is below the counts they share, at the centre of the head.

wrong · Hurwitz The first organs of a golden head, a Lucas head and a head at 104.67°, with each one's closest pair. The organs within a radius of 7.2 of three Vogel heads — the first 51 organs after the one at the centre — at the golden angle, the Lucas angle and 104.67°, three of the forty-six angles whose counts from 34 to 144 resist approximation within a per cent of the golden angle's. The joined pair in each is its closest: organs 1 and 4, 1.602 apart, at the golden angle; 1 and 5, 1.574 apart, at the Lucas angle; 12 and 19, 1.241 apart, at 104.67°. Out here the three are already different drawings; by a radius of seventeen they are not.

The first three hundred organs

Over the counts a head shows, forty-five angles resist approximation within a per cent as well as the golden angle, and what separates them is at the centre. Grown as heads and measured there, the golden angle has the widest closest pair of all forty-six — by organs 1 and 4, the count its arithmetic names — and keeps first place only while the centre is in the reading. Its rivals stay a per cent apart from it out to a radius that tracks where their spiral counts start to add up, and every one of them is within a per cent by the 289th organ. By the largest hole it is never the best.

wrong · Hurwitz 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 The 10 sequences the settling table's destinations belong to. Every pair the counter returns is two consecutive terms of a sequence in which each term is the sum of the two before it, which is the ladder's own rule. two of these are the sequences plants are observed to follow. The rest are not, and they are not rare: the 1, 4, 5, 9, 14 sequence and the 2, 5, 7, 12, 19 sequence each supply several destinations, at every falloff exponent the table is grown at. The right-hand column is the divergences that land on each.

Ten sequences, two of them the ladder's

Every parastichy pair the settling table produces is two consecutive terms of a sequence in which each term is the sum of the two before it. Ten such sequences account for all fifteen destinations — nine, once one of the ten turns out to be a reading rather than a ladder — and the two the collection is built on are neither the largest nor the smallest.

emergence · Off-ladder

Named alongside it

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

Continued fractionDivergence angleHonest limitsRational approximationConvergentsFibonacciHurwitz's theoremClaim testingLucas numbersBranchφ, the golden ratioMeasurement

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