Phyllotaxis
the spirals are counted, not admired
Concept

Rational approximation — where it appears

A fraction close to a given number, whose denominator is small for how close it is. The parastichy numbers of a lattice are the denominators of the best approximations to its divergence, which is why irrationality is the property that matters.

Named by 19 essays across 4 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 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 How the largest gap behaves as the head fills, on the interior's scale, from 150 to 2000 organs. On the interior's scale, the rational angle's gap grows by a factor of 3.8 over this range, from 2.48 to 9.38 spacings; the golden angle's runs from 0.841 to 0.844, decided at radius 0.871 at every size, and 137.3° reaches 0.863. A rational angle's gap is unbounded and an irrational one's is not, which is a claim about growth rather than about a value at any one head.

The gap that grows

A rational divergence angle develops an empty wedge that grows without bound as the head fills — a factor of 2.9 between two hundred primordia and sixteen hundred, and 3.8 from a hundred and fifty to two thousand. An irrational one does not. That is the division between rational and irrational angles that survives measurement.

tissue · Gapgrowth The plane of stems: divergence across, rise up. Each shade is one parastichy pair. The marked points are the lattices where three families are equally short — the forks — and the Fibonacci ones run up the middle towards 137.51°.

The forks are exact

Where a stem's pattern has to choose between two futures, three spiral families are equally short and the lattice is exactly equilateral. A numerical solver found those points; the numbers it returned turned out to be rational, and chasing that gave a closed form — including the fact that every fork sits at a rational divergence, and the golden angle at none of them.

cylinder · Forks 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 The area never settles, so the number reported is the window. The equivalent width of the disorder dip — the area of the deficit divided by the dip's own depth — for four fractions with a denominator of 34, against the window it was integrated over. The window is in scaled units, u = δ·n²/q, so a head of any size is sampled at the same places on its own dip; 200 scaled units is about 0.0109° at these head sizes. A width would show as a flat stretch. There is none: every line climbs to 512 scaled units or so and then turns over, because past that the window has reached far enough towards the next rational that it is integrating the neighbour's dip instead of this one's background. The dip has a depth and a scale; it does not have an outer edge, and an integral of something with no outer edge is a statement about where it was stopped.

A dip with no outer edge

The disorder of a head dips at every rational divergence, and how wide that dip is has carried a long argument. Reading the width as a level crossing has a resolution problem, and the obvious repair is to integrate instead. The integral reproduces beautifully across head sizes and never settles on a value, because there is nothing out there for it to settle against.

tissue · Second statistic The area never settles, so the number reported is the window. The equivalent width of the disorder dip — the area of the deficit divided by the dip's own depth — for four fractions with a denominator of 55, against the window it was integrated over. The window is in scaled units, u = δ·n²/q, so a head of any size is sampled at the same places on its own dip; 200 scaled units is about 0.0067° at these head sizes. A width would show as a flat stretch. There is none: every line climbs to 595 scaled units or so and then turns over, because past that the window has reached far enough towards the next rational that it is integrating the neighbour's dip instead of this one's background. The dip has a depth and a scale; it does not have an outer edge, and an integral of something with no outer edge is a statement about where it was stopped.

The window is the neighbour

An integral needs a limit, and this one has two conditions on it that pull opposite ways. It has to scale with the dip, so that two head sizes are comparable, and it has to stay clear of the next rational, which is a fixed distance in degrees. Between them there is no stretch where the answer holds still — and the limit that decides it is the crowding.

tissue · Second statistic Seven fractions with one neighbour distance and every denominator. Each member of a matched set drawn on its own stretch of the divergence axis, 0.5° either side of itself, with the nearest other rational marked. The distances are 0.3158°, 0.3158°, 0.3117°, 0.3069°, 0.3117°, 0.3077°, 0.3064° — a spread of 3.1% — while the denominators run 19, 20, 21, 23, 33, 45, 47, a factor of 2.47. That is the construction this thread needed. Every instrument for the width of a disorder dip has a free parameter set by how close the neighbour is, so a hypothesis about the neighbourhood cannot be tested by varying the neighbourhood; on this set the neighbourhood is held fixed and the arithmetic of the fraction is what varies.

Fractions with the same neighbours

Every instrument this collection has for the width of a disorder dip has a free parameter set by how close the next rational sits — which makes a hypothesis about the neighbourhood untestable with any of them. The repair is not a better instrument. It is a set of fractions whose neighbourhoods are identical and whose denominators are not, and the arithmetic supplies twenty-four of them.

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 The order follows the window, so it was never the fractions'. The four fractions with a denominator of 34, ordered four ways. The left column puts them in order of how close the nearest other rational is — the crowding — with the most crowded at the top. The other three order them by the area of their dip, at windows of 50, 100, 200 scaled units. The residual claim this thread carried was that the most crowded fraction gives the widest dip, which would make all four columns the same order. They are not: the order changes between the first two windows and settles, from a window of 100 outwards, into 11/34 > 15/34 > 13/34 > 9/34 — which is not the crowding order either. A quantity that reverses when the measurement is stopped somewhere else is a property of the stopping.

The order belonged to the method

A residual was left over after the two width laws, and it looked ordered: the most crowded fraction gave the widest dip, in all three families, in the direction a measurement artefact would take. Measured again with an instrument that has no level in it, the order changes with the window, disagrees between families, and in one of them comes out backwards.

wrong · Second statistic Hold the neighbourhood and the denominator stops mattering. The equivalent width of the disorder dip — the area of the deficit divided by its own depth — for seven fractions whose nearest neighbours sit at the same distance and whose denominators run from 19 to 47. Each is measured at a head size chosen so that all of them share one scaled unit, which makes a window in scaled units the same window in degrees and the same fraction of the way to the neighbour for every member. At a window of 25 the seven widths are 38.8, 38.8, 39.0, 39.0, 38.7, 39.0, 39.0 — a spread of ×1.010 across a factor of 2.47 in denominator. The lines separate as the window widens, to ×1.147 at 200, and when they do they order by denominator rather than by crowding. So the residual this thread carried was the window: hold it and there is nothing left that belongs to the fraction.

The residual was the window

After the depth and the q over n squared scale are taken out of a disorder dip, something looked left over and looked ordered by how crowded the fraction's neighbourhood is. Measured on fractions whose neighbourhoods are identical by construction, seven widths across a factor of two and a half in denominator agree to one per cent. There is no residual; there was a comparison made at different effective windows.

wrong · Second statistic 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.

Divergence angleHonest limitsMeasurementContinued fractionArtefactConvergentsDisorderRational angleSummary statisticSamplingOrder and disorderVoronoi cells

All concepts