A dip belongs to the head

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 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 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.

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.

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 #

The founding essays 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.

Which puts the dips out of reach of a plant #

The resolution statement is arithmetic about heads and it has a consequence about material, and the consequence is large enough to be the reason this essay matters outside the sweep.

A head resolves a rational only within about 1500/n² degrees of it. At three hundred organs that is a sixtieth of a degree, which a real arrangement could plausibly sit inside. At a thousand it is fifteen ten-thousandths of a degree. At two thousand — a large sunflower — it is under four ten-thousandths.

Nothing biological holds an angle to four ten-thousandths of a degree. The scatter this collection works at is half a degree, the amplitude at which a lattice stops being one is one to two degrees, and no measurement anybody has made on a plant is good to better than a quarter. So a large head is on a plateau, always, and the question of whether its divergence is “really” rational has no observable content at that size.

That inverts how the dips should be read. They are not the ordered arrangements a large head might fall into; they are arrangements only a small head can be in, and they get harder to reach as the head grows — not because the angle changes but because the tolerance shrinks as one over the square of the count.

Which is worth setting beside the ordering by denominator. The deepest dips belong to the smallest denominators, and the smallest denominators are exactly the arrangements a young head with few organs can occupy. A shoot with twenty organs at three ray-rows is squarely in a dip; the same shoot at two thousand florets cannot be in any dip at all. The arithmetic and the growth pull the same way.

What it does to that earlier work’s numbers #

The moment table the earlier work 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.

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.

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.

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.

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.

Neither exponent is a width in the sense the level crossing meant, and the quantity that replaced it has no outer edge to be measured against — which is a later problem and not this one.

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.

A later note on the background #

Every dip on this page is a comparison against a background, and the background is one sample taken two tenths of a degree from the rational. At the denominators here — 8, 13 and 21 — that offset sits in a flat stretch and the number is sound. At 34 and above it is not: 21/55 plus two tenths of a degree lands seven thousandths of a degree from 13/34, inside that rational’s own dip, and the comparison inverts.

The method that replaces it, and what it recovers, is in The background is not one sample. The numbers on this page are unaffected; what changes is that no new measurement should use the single-sample background, and the width law’s coefficient turns out to carry a factor of the denominator that three denominators could not show.