About

This is a growing collection of illustrated essays about pattern in living things. Each takes a single claim and draws it until the argument is visible — and every pattern in every figure is generated from a stated rule rather than admired, with the claim about it given a test it could fail.

Why counting rather than admiring

Nearly everything said about phyllotaxis is a claim about a count. How many spirals run one way, how many the other, whether the two numbers are consecutive Fibonacci numbers. Those claims are checkable, and they are almost never checked — the count is quoted from a caption, and the caption is quoted from another caption.

So the arrangement in lib/lattice.js is deliberate. One function builds a head from a divergence angle. A different one is handed a set of coordinates and asked how many spiral families run through them, and it has no access to the angle, the model, or which numbers it is supposed to find. When 34 and 55 come out of it, that is evidence rather than a restatement of what went in.

Doing it turns up the first thing the captions leave out: the counts change with radius. The same head gives 21 and 34 near the middle, 34 and 55 further out, and 55 and 89 near the rim. The transitions are at computable radii. A photograph captioned “34 and 55 spirals” is a statement about one annulus.

The angle is an output

137.5° is usually presented as a constant of nature, which is the least interesting thing it could be. It is an attractor.

The rule in lib/dynamics.js is the one Douady and Couder used in 1992: each new element appears on the growing boundary at the point where the repulsion from everything already there is least. That rule contains no reference to the golden ratio, to Fibonacci, or to spirals. Run it and the divergence angle it produces settles — at 137°, from starting angles a hundred and sixty degrees apart, with the last thirty steps varying by nothing at all.

Change the one parameter — how fast the meristem grows relative to how often it makes an element — and it settles somewhere else. Sweeping it produces a diagram with a broad golden branch, a transition, and a two-whorl regime at exactly half a turn. The famous constant is one branch of that diagram.

Douady and Couder then did the same thing with magnetised ferrofluid droplets falling onto a dish of silicone oil, and got the same phenomenology out of a system with no biology in it whatsoever. That is the strongest single argument that the pattern is physics rather than a plant's arithmetic, and it is also the clearest warning about what a model like this can and cannot claim.

State the claim, state the test

This subject arrives with more confident wrong statements attached than any other on this fleet. The rule here is the same one the rest of the fleet uses: never mock a claim, measure it. Three of them, with what the measurement says:

• “The nautilus is a golden spiral.” A golden spiral grows by φ⁴ = 6.854 per turn. Measured nautilus sections give about 3.2. That is a factor of 2.14, and the two curves drawn from the same start diverge by a factor of six over two and a half turns. The one free choice in the measurement — where the centre is assumed to be — moves the answer by under half a percent for a quarter-radius error, nowhere near enough to close the gap.
• “Sunflower spirals are always Fibonacci.” Fibonacci counts come out of 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. Fibonacci is a consequence of a branch rather than a law of plants.
• “137.5° is optimal packing.” This one is the interesting case, because the popular version fails the test and a sharper version passes it. No packing criterion computed here singles out the golden angle at any particular head size: closest-pair, largest-gap and cell-area evenness pick three different winners, all near it and none of them it. What does single it out is arithmetic. The golden angle resists rational approximation better than any other angle — 0.4377 against 0.33 for the best of fifteen hundred sampled — and it approaches Hurwitz's bound of 1/√5, which no number can exceed. That is what “the most irrational number” means, and the packing story is a loose retelling of it.

Two routes, wherever there are two

• The divergence angle is used to build a head, then recovered from the spiral counts by machinery that never sees it — four angles, worst error 0.012°.
• The growth factor of a spiral is used to draw it, then recovered by fitting log r against angle on the drawn points.
• Murray's exponent is built into a tree and then fitted from the junctions rather than assumed, returning 3.000.
• The mean number of sides in a tiling is measured and compared with the six that Euler's formula forces.

What broke while this was being built

The spiral counter returned the two smallest offsets rather than the two shortest, which for a golden head means 21 and 34 when the actual nearest neighbours are 34 and 55. Nothing in the picture showed it. It was found because the angle recovery then refused: no divergence angle makes 21 and 34 the closest pair at that radius, and the refusal was correct.

The first version of the recovery scored candidate angles by how nearly their multiples landed on whole turns, and was minimised at zero degrees — where every multiple is a whole turn and the “lattice” is a single radial line. That cost function had no geometry in it.

And the shell model called W·D < 1 impossible. It is not: whorls in contact are the ordinary condition of most gastropods, where each whorl wraps over the one before. The boundary is real and the label was wrong.

Where this stops

These are models of form, not explanations of development. A lattice that matches a sunflower does not show that the plant computes it. The mechanism is auxin — a hormone moving between cells, with transporters that concentrate it where it is already concentrated — and it was worked out decades after the geometry. The geometry says what pattern a repulsion rule produces; the biology says why a meristem behaves like one.

The dynamical model has a range. Below about a growth parameter of 0.18 this discrete implementation stops converging: the elements barely move between steps, several sit on the boundary at once, and the settled angle wanders over a hundred degrees however long the run. That is a property of a sampled circle with a finite window, not a result about plants, and there is a figure that plots it rather than a range quietly chosen to avoid it.

No organism is photographed. Every pattern here is generated from its rule. That is partly a licensing rule and mostly a methodological one: a photograph would add authority the argument has not earned, and the whole point is that the argument is in the rule.

Occupancy is not geometry. Raup's model says which shells are geometrically consistent; which regions of that space real animals occupy is an empirical question that was answered by measuring shells, and this site draws the boundary rather than claiming the census.

Corrections are welcome and will be made. A pattern generated from the wrong rule looks like a pattern, which is the entire reason this site computes rather than admires, and the same courtesy applies to its own figures.