A head is a set of points

Put a point at angle δ and radius 1. Put the next at 2δ and radius √2, the next at 3δ and radius √3, and keep going. Choose δ = 137.508° and the result is a sunflower head — not something resembling one, but a point set with the same spiral families, the same counts and the same even packing.

That is Vogel’s model, published in 1979, and it is two lines long.

The two parts of the rule, and what each is doing #

The angle is the divergence: how far round the meristem turns between making one primordium and the next. It is the only interesting parameter, and almost everything on this site is about it.

The square root is the part that gets left out of popular accounts, and it is doing real work. A disc of radius r has area proportional to r², so putting the nth point at radius √n gives every point the same area of disc to itself. Without it the head is crowded at the rim or empty in the middle, and it does not look like a sunflower at all.

That is also the only part of the rule with any biology in it. It stands in for a meristem that produces primordia at a steady rate on a surface that is growing, and it is a statement about rates rather than about geometry. Change the growth law and the radius law changes with it — which is exactly what happens in a pine cone, where the surface is a cylinder rather than a disc and the primordia sit on a helix.

What happens either side of the angle #

The knob is the subject. Turn it a degree and the pattern changes character completely.

At 137.508° the points are evenly spread and no two crowd each other; the nearest pair sit about 0.8 of the mean spacing apart. At 137.0° — half a degree away — visible radial rows appear, because 137.0 is very close to 960/7 and every seventh point lands almost on top of the ray of the one before. At 135° exactly, which is 3/8 of a turn, there are eight radial rows and nothing else.

That sensitivity is the first thing worth noticing, and it cuts against the usual telling. The claim is not that 137.5° is a bit better than its neighbours. It is that the rational neighbours collapse, and the closer a candidate angle sits to a simple fraction, the more it collapses.

What 0.8 of the mean spacing is worth #

The nearest-pair figure is the number the knob is judged by, and it means more once it is set against the two arrangements that bracket it.

Take the mean spacing to be the square root of the area each point has to itself, which is what the square-root rule hands every point equally. Then a triangular packing — the densest arrangement of equal discs there is — puts its nearest neighbours at 1.07 of that spacing, and a square packing at exactly 1. Points scattered at random, with no arrangement at all, have their nearest neighbours at about 0.5: in a random set some pairs always fall close together, and the smallest gap is what a nearest-pair statistic reports.

The golden angle gives 0.8.

So the divergence angle buys about three fifths of the distance from a random scatter to the best packing possible, and it does not reach the best packing. That gap is not a failure of the angle and it cannot be closed by choosing a different one. A head built this way places its points one at a time along a single spiral, and a triangular packing is not something a single spiral can lay down — the constraint is the construction, not the parameter.

Which is worth knowing before reading any claim that phyllotaxis is optimal packing. It is optimal within a family of arrangements that a growing meristem can actually produce, and that family tops out well below what a packing of discs achieves when the discs may be placed in any order. The interesting question is not why plants reach 0.8 rather than 1.07; it is why 0.8 is the most a one-at-a-time spiral can reach, and that is a question about the spiral rather than about the plant.

What a model that reproduces a pattern has established #

This is the point at which to be careful, because it is the point where most writing on the subject stops being careful.

Vogel’s rule produces a point set indistinguishable from a sunflower head. That establishes that the pattern is consistent with a meristem turning by a fixed angle between primordia. It does not establish that any sunflower does that, and it certainly does not establish that a sunflower is computing anything.

There are at least three different claims here and they are routinely run together.

The descriptive claim. Real seed heads have spiral counts that are usually consecutive Fibonacci numbers, and a lattice generated at the golden angle has the same counts. That is a statement about a match between a model’s output and an observation, and it is well supported.

The mechanistic claim. A meristem places each primordium at a fixed angle from the last. This is a claim about a plant and it is false as stated — the angle is not fixed, it drifts, it is noisier near the centre, and it is an outcome of the process rather than an input to it.

The explanatory claim. The plant does this because the golden angle is optimal. This one has to be taken apart carefully, and it is taken apart carefully, because the version that survives measurement is not the version usually told.

This site’s figures support the first claim, contribute a great deal to the third, and are silent about the second — which is a matter for auxin and for people with microscopes.

Where the model is visibly wrong #

Its failures are instructive and are worth drawing attention to rather than cropping out.

The centre is wrong. The first ten or twenty points of a Vogel head are irregular, because √n changes fastest there and because the model has no notion of a primordium having a size. A real meristem’s first primordia are large relative to it and arrange themselves quite differently. It matters more than it looks: on a golden head both spacing criteria are decided by the first four organs unless the centre is excluded, and a whole-head packing number is a statement about that irregular patch rather than about the head.

There is no size. Model primordia are points. Real ones are discs that grow, jostle, and are eventually squeezed into polygons, which is the tissue’s business rather than the lattice’s.

Nothing changes with time. In a real head the divergence angle drifts and the parastichy numbers transition as the head grows. Vogel’s rule has one angle for all n, and the transitions it shows come from the geometry rather than from any change in the plant.

Nothing is noisy. Real angles vary by a degree or two between successive primordia. A noiseless model produces a cleaner pattern than any flower, and a reader comparing a figure here with a photograph should expect the photograph to be messier.

The pattern is a lattice #

There is a structural fact here that makes the rest of the site possible.

A Vogel head is not a heap of points; it is a lattice in the mathematical sense, locally. Take any point well away from the centre and its neighbours sit at a small number of characteristic offsets — a fixed number of places earlier and later in birth order. Those offsets are the parastichy numbers, and they are what a person is tracing when they follow a spiral with a finger.

That is why counting spirals is a well-defined operation rather than an impression, and why it can be done by a machine that is never told the angle. It is also why the counts change across the head: the offsets that give the shortest hops depend on the local radius, and different offsets win in different annuli.

The lattice view also makes the connection to number theory unavoidable. An offset m gives short hops exactly when m·δ is close to a whole number of turns, so which offsets win is a question about how well δ is approximated by fractions with denominator m. That question has a complete answer, it is a hundred and fifty years old, and it is why one particular angle is the extreme case.

What the model is for #

It is worth being explicit, because a model this simple invites two opposite errors.

The first is to treat it as an explanation. It is not one; it is a generator. It takes an angle and produces a pattern, and that is all it does.

The second is to dismiss it as a curve fit. It is not that either, because it has one parameter and no freedom to be adjusted. A curve fit with one parameter that reproduces the spiral counts, the packing density and the transition radii of a real seed head is not fitting; it is predicting.

What the model is genuinely for, on this site, is as a test bench. Because it produces patterns from a known angle, every claim about reading a pattern can be checked against a case where the answer is known: count the spirals and see whether the count is right, recover the angle and see whether it comes back, measure the packing and see which criterion says what. None of that is possible with photographs of flowers, where the truth is not available.

The rule in other geometries #

Vogel’s disc is one case of a more general rule, and the others matter because they are what most plants actually are.

A cylinder. Put the nth primordium at angle n·δ and height n·h and the pattern is a helical lattice — a pine cone, a pineapple, a stem with leaves. The parastichy numbers are constant up the cylinder rather than changing with radius, which makes cones a cleaner subject for counting than sunflowers.

A cone. Between the two, and the commonest real geometry.

A growing sphere. Harder, and it is where the model stops being cheap.

All of them share the divergence angle and differ only in the surface, which is a reason to be careful with the word phyllotaxis. It names a family of patterns, and the disc case that gets photographed is the one where the geometry is most complicated.

What to carry forward #

The whole of this site rests on the observation that a seed head can be produced from a rule with one parameter, and therefore that anything said about a seed head can be tested against a pattern whose parameter is known.

Everything else follows from that: the counting can be checked because the truth is available, the angle can be recovered because there is something to recover it to, and the popular claims can be measured because a claim about a pattern is a claim about a computable quantity.

The parameter itself is the one thing the model does not explain. Vogel’s rule takes 137.508° as an input and has nothing to say about where it comes from — which is a good enough reason for the next field to exist.

The model’s one virtue #

If a single thing recommends Vogel’s rule over the alternatives, it is that it has no adjustable parts.

There is no shape parameter, no smoothing, no fitted constant, no correction term. Given an angle and a count, the pattern is determined. That means a disagreement between the model and a real head cannot be fixed by tuning, which makes every agreement worth something and every disagreement informative.

Models with that character are rare and worth using where they exist. Murray’s law has it; Raup’s shell model has it; an L-system does not, which is why a convincing L-system plant establishes so much less than it appears to.

What the model leaves out #

A two-line rule that reproduces a sunflower is a strong result and an easy one to over-read, so the omissions are worth listing.

There is no time in it. Vogel’s rule places all n points at once from a closed form. A real head grows, and the point that is now at radius √n was at the boundary when there were n points in total. The model captures the finished arrangement, not the process — which is why the dynamical model is a separate thing rather than an elaboration of this one.

There is no mechanism. Nothing in the rule says why the angle is what it is. Fed 120° it produces a three-ray pattern just as confidently, and it has no opinion about which is more plausible.

There is no biology beyond the rate. The √n comes from a growth assumption; the angle is an input; and the primordia are dimensionless points that never push each other.

The head is flat and the growth is uniform. Real heads are slightly domed, real primordia are finite and jostle, and real growth rates change during development — which is one reason the counts change along a radius in ways the model gets qualitatively right and quantitatively approximate.

What the model is for is the middle of the argument: given an angle, what pattern follows. That question is worth having a clean answer to, and everything on this site that measures a pattern measures one of these.

Where the square root comes from #

The radius law is usually quoted without derivation, and it takes one line.

Primordia appear at a steady rate, so after time t there are n ∝ t of them. The head grows so that its area increases at a steady rate too — the meristem adds material at its edge — so the area A ∝ t as well. A disc of area A has radius √(A/π), so the boundary radius goes as √t, which is √n.

The nth point, placed at the boundary when there were n points, is therefore at radius proportional to √n and stays there in relative terms.

That derivation makes the assumption explicit and therefore breakable. If the head grew at a steady radial rate instead, the radius would go as n and the head would be sparse in the middle and crowded at the rim. If it grew exponentially the radius would go as an exponential of n, which is what a shell does and is why a shell is a spiral rather than a disc.

One rate assumption, three completely different biological forms. The angle is the famous parameter, but the radius law is the one that decides what kind of thing is being made.

Cylinders, cones and the shapes that are not discs #

Most phyllotaxis is not on a disc, and the model generalises in a way worth knowing because it changes what the counts mean.

A stem is a cylinder. Leaves appear at the top and the stem elongates, so the radius is constant and the “radial” coordinate is height. The lattice is the same lattice — offsets, short hops, parastichy numbers — but it lives on a cylinder, so the counts do not change along its length. A stem has one parastichy pair, and that is why divergence angles are classically measured on stems rather than on heads.

A pine cone is a cone: radius grows with height, so the counts change as they do in a head, but more slowly.

A pineapple is close to a cylinder with a slight taper, which is why its 8/13 counts are so reliably quotable.

The disc is the case with the fastest change in circumference and therefore the most dramatic transitions, which is why it is the one that gets photographed and the one whose counts are hardest to state. A caption quoting a pair for a sunflower is making a much weaker claim than the same caption for a pineapple.