Phyllotaxis
the spirals are counted, not admired
Concept

Continued fraction — where it appears

An expansion of a number as a nested sequence of reciprocals, whose terms decide how well it can be approximated by fractions. The golden ratio has the slowest of all such expansions, which is the precise sense in which it is the hardest to approximate.

Named by 18 essays across 5 fields — each of them below, with the objects they name alongside it.

The spiral counts four different divergence angles produce. Fibonacci counts come from one angle. The Lucas angle — which the same dynamical model reaches on a different branch — gives 47 and 76, and neither number is a Fibonacci number.

Fibonacci is a branch, not a law

Fibonacci counts come from one branch of the model. The Lucas branch — which the same rule reaches at a different growth rate — gives 47 and 76, and neither is a Fibonacci number. The sequence is a consequence of an angle rather than a property of plants.

wrong · Fibbranch 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 The spiral counts, band by band, in one head. The same flower gives 13/21, 21/34, 34/55, 55/89 at different radii. A caption saying "34 and 55 spirals" is a statement about one annulus.

The counts change with radius

The same head gives 13 and 21 near the centre, 21 and 34 further out, 34 and 55 beyond that, and 55 and 89 at the rim. The transitions are at computable radii, and a photograph captioned with one pair is a statement about one annulus rather than about a flower.

lattices · Transitions Every transition as the rise falls. The pair climbs 1/2 → 2/3 → 3/5 → 5/8 → 8/13 → 13/21. Consecutive transitions are 0.382, 0.382, 0.383, 0.382 of the previous rise — 1/φ² is 0.3820.

The Fibonacci ladder

Lower the rise on a cylinder and the parastichy pair climbs — 1 and 2, then 2 and 3, then 3 and 5 — each rung the sum of the two before it. Nothing in the arithmetic mentions Fibonacci, the transitions sit at computable rises, and consecutive ones stand in the ratio 1/φ².

cylinder · Ladder 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 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 angles against the positions, rise by rise. five rises, five seeded stems each. A filled mark is a run whose angle readout returned the pair the position counter finds in the same stem; an open mark is a refusal. At 0.032 the counter says 3/5 and the angles agree on 0 of 5, refusing 5. At 0.013 the counter says 5/8 and the angles agree on 5 of 5. At 0.01 the counter says 5/8 and the angles agree on 5 of 5. At 0.005 the counter says 8/13 and the angles agree on 5 of 5. At 0.008 the counter says 5/8 and the angles agree on 2 of 5, refusing 3. The two instruments share no code path: one is given a list of angles, the other a list of coordinates.

The angles name the branch

Seed the same rule at the Lucas angle and the readout returns 4 and 7, then 7 and 11 — the pairs the position counter finds, and not Fibonacci numbers. So a list of divergence angles carries not only how many spirals there are but which family of ladders the plant is on.

lattices · Counting 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 Two readings of what a stem is, and one of them is wrong. The same runs read twice. On the left is the pair counted from the point positions by machinery that is never shown a divergence angle; on the right is the pair read from the divergence sequence, which is the column the depth thread has been quoting. Every exponent's stems are counted at 5/8 spirals from the points. Read from the angles, the deepest rule's stems come back as something else at every seed, and a pair with a 8 replaced in it is not a near miss but a family whose step is several times as long. The settled divergence moves by 0.118 degrees across all five exponents, so the rules are on one lattice and the disagreement is a defect in one of the two instruments.

The pair read from the angles

Checking that a jostled stem is still the lattice its panel is about turned up a disagreement. Counted from the point positions every rule's stems return five and eight spirals; read from the divergence sequence, the deepest rule's stems come back as five and seven at every seed. The points are right, and this site's founding rule is why.

lattices · Counting 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 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.

Divergence angleConvergentsMeasurementRational approximationHonest limitsFibonacciLucas numbersNoble numberRational angleArtefactClaim testingSummary statistic

All concepts