Phyllotaxis
the spirals are counted, not admired
Packing and tiling

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°.
18 min read 10 figures State the claim, state the testThe angle is an outputCounted, not admired

Worth reading first: What a summary throws away · Why the average cell has six sides · The gap that grows.

The staircase has dips at the rationals and the dips are the arithmetic showing through. This essay asks what a dip is made of, and the answer turns out not to be about the angle.

Two quantities describe a dip: how deep it goes at the exact rational, and how far the angle can move before it comes back out. Both were measured at three head sizes — three hundred, six hundred and twelve hundred points — on three rationals.

The floor halves when the head doubles

At 5/13 of a turn, μ₂ is 0.059 at three hundred points, 0.029 at six hundred and 0.014 at twelve hundred. At 8/21 it is 0.078, 0.038 and 0.019. At 3/8 it is 0.032, 0.016 and 0.008.

Nine measurements, three rationals, and every doubling halves it.

The dip at 5/13 — 138.4615° — at three head sizesWalking 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.00.2000.4000.600-3-2.50-2-1.50-1-0.700offset from the rational, log₁₀ of degreesμ₂300 points600 points1200 points5/13 of a turn · 12 offsets · vertical rules are the half-widthsgenerated from a stated rule, not drawn to look right
Fig. 1 The dip at thirteen thirteenths of a turn — 138.4615° — walked at three head sizes, with the offset from the rational on a logarithmic axis. The floor falls by half for each doubling of the head and the dip narrows by four, which are the two measurements this essay is about.

The reason is the one the previous essay gave and it is worth putting quantitatively. At a rational p/q the arrangement is q radial rays; its cells are hexagons except along a fixed number of seams. The seam population does not grow with the head — there are still q rays and still the same number of places where a ray meets the boundary of its neighbour’s territory — while the number of interior cells grows with the head. So the share of non-hexagons falls as 1/n, and μ₂, which is a mean over cells, falls with it.

That is a prediction with an exponent in it, and it is the one the measurement confirms.

The dip at 3/8 — 135.0000° — at three head sizesWalking the divergence angle off an exact rational, at 300, 600, 1200 points. The floor falls as the head grows — 0.032 at 300, 0.016 at 600, 0.008 at 1200, halving for each doubling — and the dip narrows faster: half-widths of 0.0190°, 0.0051°, 0.0012°, a factor of four for each doubling rather than two. Half-width times the square of the head size is 1712, 1850, 1687, which is what makes the width a property of the sample rather than of the angle.00.5001-3-2.50-2-1.50-1-0.700offset from the rational, log₁₀ of degreesμ₂300 points600 points1200 points3/8 of a turn · 12 offsets · vertical rules are the half-widthsgenerated from a stated rule, not drawn to look right
Fig. 2 The same experiment at 3/8 — 135° — where the denominator is smaller and the floor is lower at every head size. Eight rays leave fewer seams than thirteen do, which is the ordering by denominator the staircase shows.

The width does not

The other quantity was predicted first and predicted wrong, and the correction is the point of the essay.

Move the angle off the rational by δ and the rays twist: organ i is at i·δ past where the rational would put it. The obvious argument is that the pattern still reads as q rays while the total twist across the head, n·δ, stays small, which makes the half-width go as 1/n.

Measured, it goes as 1/n². At 5/13 the half-widths are 0.016°, 0.005° and 0.001° — falling by four for each doubling, not two. The product n·w is 4.85, 2.88, 1.17, which is not flat. The product n²·w is 1456, 1726, 1402, which is flat within a quarter, and it is flat on 3/8 as well: 1712, 1850, 1687.

What the naive argument leaves out is that the thing the twist has to stay inside is shrinking too. A head of n points built on Vogel’s model has its outermost organs at a radius of about √n, so a drift of δ per index has accumulated an arc of order δ·n^1.5 by the time it reaches the rim — the extra half-power is the radius the drift is swept around. Meanwhile the radial gap between successive organs on the same ray, which is what the arc has to stay inside, is of order 1/√n, because the rings crowd together as the head fills.

The ratio of those two is δ·n², and that is the quantity a dip’s edge is at.

The dip at 8/21 — 137.1429° — at three head sizesWalking the divergence angle off an exact rational, at 300, 600, 1200 points. The floor falls as the head grows — 0.078 at 300, 0.038 at 600, 0.019 at 1200, halving for each doubling — and the dip narrows faster: half-widths of 0.0195°, 0.0088°, 0.0023°, a factor of four for each doubling rather than two. Half-width times the square of the head size is 1754, 3151, 3333, which is what makes the width a property of the sample rather than of the angle.00.2000.4000.600-3-2.50-2-1.50-1-0.700offset from the rational, log₁₀ of degreesμ₂300 points600 points1200 points8/21 of a turn · 12 offsets · vertical rules are the half-widthsgenerated from a stated rule, not drawn to look right
Fig. 3 The third rational, at 8/21, where the agreement is loosest: n²·w runs 1754, 3151, 3333, a factor of 1.9 across the range rather than a quarter. The exponent is right and the constant is not a constant — a denominator of twenty-one puts more rays into the same head and the seams interact, which the two-quantity argument above does not model.

Which rationals a head can see

Turn that around and it is a statement about resolution.

A head of n organs resolves a rational p/q only if the divergence is within about 1500/n² degrees of it. At three hundred organs that is a sixtieth of a degree; at nine hundred, two thousandths; at twelve hundred, one thousandth.

So the question “is this head’s divergence rational?” does not have an answer that belongs to the head’s divergence. It has an answer that belongs to the head’s divergence and its size, and the two are not separable. A sunflower with three hundred florets and a divergence of 138.475° looks like a thirteen-ray whorl. The same divergence with twelve hundred florets does not.

The disorder of a head against its divergence angle, 300 pointsμ₂ 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 137.144° — which is 360 × 8/21 — it is 0.078; At 137.648° — which is 360 × 13/34 — it is 0.197; At 138.460° — 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.00.2000.4000.600137138138139divergence angle, degreesμ₂, the mean squared departure of a cell's side count from six5/138/2113/34401 angles · 0.0040° apart · 300 points eachmarks are the fractions, placed from arithmetic
Fig. 4 The same sweep as the previous essay, at three hundred points instead of nine hundred. The dips are fifty times wider and the staircase between them is coarser, because a smaller head resolves fewer of the steps. Nothing about the angles has changed.
The disorder of a head against its divergence angle, 900 pointsμ₂ 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 1.60° 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 137.140° — which is 360 × 8/21 — it is 0.051; At 137.455° — which is 360 × 21/55 — it is 0.135; At 137.645° — which is 360 × 13/34 — it is 0.063. The ticks along the top are the fractions p/q, placed from arithmetic rather than from the curve.00.2000.4000.600137138138139divergence angle, degreesμ₂, the mean squared departure of a cell's side count from six5/138/2113/34321 angles · 0.0050° apart · 900 points eachmarks are the fractions, placed from arithmetic
Fig. 5 And at nine hundred, where the dips have become too narrow for a sweep at this spacing to land in them. What is drawn is the staircase alone: the same function, sampled at a resolution that steps over its own deepest features.

That last pair of figures is worth dwelling on, because it is a trap this site has already recorded in another form. The nine-hundred-point sweep looks like a different curve — no dips, more steps — and it is the same curve measured with a grid too coarse for it. A person who ran only that sweep would conclude the dips were an artefact of small heads. A person who ran only the three-hundred-point one would conclude the dips are wide.

The same shape as the counting band

This site’s foundation phase found that every published spiral count is a statement about an annulus: one head gives 21 and 34 near the middle, 34 and 55 further out, 55 and 89 at the rim. The count is not a property of the head; it is a property of the head and the band it was counted in.

The dip width is the same statement made from the tissue side. The disorder is not a property of the angle; it is a property of the angle and the number of organs. And in both cases the dependence is not a nuisance term to be minimised by measuring more carefully — it is the leading behaviour, and reporting the sample size is not good practice but a condition on the number meaning anything.

The spiral counts, band by band, in one headThe same flower gives 21/34, 34/55, 55/89 at different radii. A caption saying "34 and 55 spirals" is a statement about one annulus.0255075406080fraction of the head's radiusparastichy numbers found in a band there1200 primordia at the golden angle3 different pairs
Fig. 6 The counting version, from the foundation phase: the pair the blind counter returns changes with the radius it counts at, on one head, with computable transitions. The dip width is this fact reached from the other side — how much of the head was included decides what the arrangement looks like.

What it does to the previous phase’s numbers

The moment table the previous phase built is at nine hundred points throughout, which was the right choice for what it was comparing — six arrangements at one size, so that a difference between rows is a difference between arrangements.

What the size dependence adds is that the rows are not comparable to anything measured elsewhere. A μ₂ of 0.023 for a whorled lattice at nine hundred points would be 0.046 at four hundred and fifty and 0.012 at eighteen hundred; a μ₂ of 0.253 for the golden angle would be roughly 0.253 at all of them, because a plateau is not a dip and does not dilute the same way.

So the factor of seventy-nine between the extremes of that table is a factor at one head size, and it grows with the head, without limit, because the ordered end tends to zero and the disordered end does not.

How many sides the cells actually haveThe mean is 5.929, which Euler's formula forces. The spread around it is not noise — a tiling of only hexagons cannot close up on a finite patch.5 sides13415%6 sides65876%7 sides738%865 bounded cellsmean 5.929 sidessix is forced, not chosen
Fig. 7 The distribution the moment is taken of, on a golden-angle head: mostly hexagons with a symmetric tail of fives and sevens. On a rational head the same histogram is nearly a single bar, and it gets nearer as the head grows.

The derivation, written out

The exponent is the whole result, so the argument for it is worth having in full rather than in a sentence.

Vogel’s model puts organ i at radius √i and azimuth i·d. Take d to be a rational p/q plus a small offset δ. Then organ i sits on ray i mod q, displaced around the circle by i·δ turns.

How far has the displacement carried it? In arc length, at radius √i, the displacement is 2π·√i·i·δ, which grows as i^1.5. So the outermost organs of a head of n are displaced by an arc of order n^1.5·δ.

How far can it be displaced before the arrangement changes? The relevant distance is not the spacing along a ray — it is the gap between one ray and the next, or equivalently the size of a cell. A head of n organs on a disc of radius √n has each organ holding an area of about π, so a cell is of order one unit across, independent of n in these units.

Setting the two equal gives δ of order n^−1.5, which is not what was measured.

The missing factor is that the displacement has to be compared not with a cell but with the radial spacing along a ray, because that is the distance that decides whether the ray still reads as a row. Successive organs on the same ray are indices q apart, at radii √i and √(i+q), so the radial gap between them is q/(2√i) — which shrinks as the head grows, as one over √n.

Setting the arc against that: n^1.5·δ ~ 1/√n, so δ ~ n^−2.

That is the measured exponent, and the derivation says where it comes from: half a power from the radius the drift is swept around, and half from the rings crowding together as the head fills. Neither factor is present in the naive “total twist” argument, and they compound rather than cancel.

Where the disc's counts change, predicted from a cylinderThe dashed lines are the transition radii the cylinder's ladder gives through h = c²/4πr², with nothing fitted. The dots are what the blind counter returns from the disc: 12 of 14 bands agree, and the ones that do not sit on a transition.2040608010152025radius in the disclarger parastichy number, measured on the disc21/3434/5555/89prediction from the cylinder, counts from the disc12 of 14 bands agree
Fig. 8 The quantity that makes the second half of the argument: on a disc the local rise falls as one over the square of the radius, which is the same statement as the rings crowding together. The dip width inherits its exponent from this, which is why a tissue statistic and a counting statistic end up with related scalings.

What it predicts for a real capitulum

The derivation is about Vogel’s model, so the honest way to use it on a plant is as a scaling rather than as a number.

A sunflower head of a thousand florets resolves a rational to about a five-hundredth of a degree. A daisy of two hundred resolves one to a thirtieth. A pine cone of eighty scales resolves one to a fifth of a degree, which is wider than the gaps between the low-denominator fractions in the region — so a cone cannot be told from a rational arrangement at all by this statistic, and its μ₂ is a measurement of its size as much as of its angle.

That is a testable prediction about real material and it has an awkward consequence for the literature it touches. Reports of side-count distributions in plant epidermis and in capitula are made on samples of very different sizes, and if the arrangement is anywhere near a low-denominator rational, the reported μ₂ depends on how many cells were in the field of view. Nothing in the reporting convention records that.

The same caution applies to this site’s own numbers. The moment table is at nine hundred points because that is the default for every moment measured here; a reader comparing 0.253 to a published figure from a smaller sample is comparing two different measurements, and the difference could be a factor of two with no disagreement about the plant.

Voronoi cells of a head at 138.46°286 bounded cells, averaging 5.98 sides. Cells with six sides are shaded; the others are what a tiling has to contain to close up.286 bounded cellsmean 5.98 sides
Fig. 9 The arrangement at 138.4615° and three hundred points, where the thirteen rays are unmistakable. At twelve hundred the same angle is still thirteen rays; at twelve hundred with the angle moved by a hundredth of a degree it is not, and the two look nothing alike.

The two quantities are independent

Depth and width fall with the head size at different rates, and it is worth noticing that they are separate facts rather than two faces of one.

The depth falls as 1/n because the non-hexagonal cells at a rational angle are a fixed population — the seams — diluted by a growing one. That argument makes no reference to how far the angle is from the rational; it is a statement about the exact rational.

The width falls as 1/n² because of how far the arrangement can be perturbed before its contact graph changes. That argument makes no reference to how many seams there are; it is a statement about the neighbourhood of the rational.

So one could imagine a statistic with one behaviour and not the other, and the measurements confirm both independently: the floor and the half-width are read off the same profiles but at different places on them, and their ratios across head sizes are 2 and 4 respectively on every rational tried.

The consequence is that a dip’s shape changes with the head as well as its size. At three hundred points it is a broad shallow bowl; at twelve hundred it is a narrow deep spike. A sweep at a fixed grid spacing therefore sees the dips get both harder to find and more dramatic when found, which is precisely the pair of effects that makes a coarse sweep at a large head size look like a curve with no dips in it at all.

The dip at 5/13 — 138.4615° — at three head sizesWalking the divergence angle off an exact rational, at 300, 1200 points. The floor falls as the head grows — 0.059 at 300, 0.014 at 1200, halving for each doubling — and the dip narrows faster: half-widths of 0.0162°, 0.0010°, a factor of four for each doubling rather than two. Half-width times the square of the head size is 1456, 1402, which is what makes the width a property of the sample rather than of the angle.00.2000.4000.600-3-2.50-2-1.50-1-0.700offset from the rational, log₁₀ of degreesμ₂300 points1200 points5/13 of a turn · 12 offsets · vertical rules are the half-widthsgenerated from a stated rule, not drawn to look right
Fig. 10 The two extremes of the same dip, with the middle removed: a broad shallow bowl at three hundred points and a narrow deep spike at twelve hundred. The two quantities that changed between them — the floor and the half-width — did so at different rates, and neither is derivable from the other.

What is not settled

The constant. n²·w is 1,400 to 3,300 across the three rationals measured, and the spread is real rather than noise — 8/21 is consistently higher than 5/13 and 3/8. A denominator of twenty-one puts twenty-one rays into a head that has room for a certain number of rings, and the seams begin to interact, which the two-quantity derivation above does not model.

So the law being asserted is the exponent and not the coefficient: the width goes as the inverse square of the head size, with a constant that is within a factor of two and a half across the rationals tried and is not claimed to be universal. Fitting a q-dependence to three denominators would be fitting a curve to three points, which this site has been caught doing once before and does not propose to do again.

The other thing not settled is what happens at large q. Every rational measured here has a denominator under twenty-five, because at larger ones the dip is too narrow to find without a sweep finer than the whole experiment is worth. Whether the same exponent holds for 55 or 89 is untested and there is no reason from the derivation to expect it to fail — which is precisely the kind of expectation this site records as untested rather than as likely.

The one-line version

Depth halves when the head doubles; width falls by four. Neither belongs to the angle, both belong to the sample, and the second is the one that decides which questions a given head can be asked.

A person with a photograph of a capitulum can now say, before measuring anything, which rationals their head could possibly resolve: those within about 1500/n² degrees of the divergence. If none is, the arrangement is on a plateau and its μ₂ is a number about the arrangement rather than about the angle. If one is, the measured value is a dip’s floor and it is a number about the head size.

Nothing in the tissue literature’s use of μ₂ requires that distinction, because a piece of epithelium has no divergence angle to be near a rational of. It is the price of importing the statistic into a subject where a one-parameter family of arrangements is the object of study.

Where this goes next

00.2000.4000.600137138138139divergence angle, degreesμ₂, the mean squared departure of a cell's side count from six5/138/2113/34261 angles · 0.0062° apart · 300 points eachmarks are the fractions, placed from arithmetic 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°. 0255075406080fraction of the head's radiusparastichy numbers found in a band there1000 primordia at the golden angle4 different pairs 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. mean sides per cell(forced to six)mean squared departure from six(not forced)whorled, 144°5.9860.023golden, 137.508°5.9900.253rational, 137.5°5.9900.255Lucas, 99.502°6.0360.255137.0°5.9900.291Poisson5.9691.830six433–637 interior cells each, inside 86% of the radiussame cells, same cut, two statistics What a summary throws away Four statistics this collection has relied on turn out to be incapable of varying with the thing they describe — one is invariant to shuffling, one is fixed by a theorem, one is a parameter that stopped mattering, one is a fitted number selected into being wrong. In each case the second statistic was free and nobody had taken it.

Shares its objects with

Essays that name at least two of the same things, and that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

AnnulusArtefactDisorderDivergence angleEuler's formulaHonest limitsLatticeMeasurementOrder and disorderRational angleRational approximationResolutionSamplingSummary statisticVoronoi cells