Phyllotaxis — the spirals are counted, not admired Illustrated essays on biological form with the patterns computed rather than admired: every spiral count is extracted from the point set, the divergence angle is recovered from the counts, and the famous claims are measured against a stated test. https://www.phyllotaxis.xyz/ 2026-08-05T23:21:57.837Z A head is a set of points https://www.phyllotaxis.xyz/essays/a-head-is-a-set-of-points/ 2026-08-05T23:21:57.837Z The nth primordium at n times an angle, and a radius of root n. Two lines of arithmetic produce a sunflower head, which is either remarkable or suspicious depending on how carefully the claim is stated — and stating it carefully is most of the work. The angle is an output https://www.phyllotaxis.xyz/essays/the-angle-is-an-output/ 2026-08-05T23:21:57.837Z 137.5° is not a constant of nature. It is where a rule settles — a rule that places each new element as far as it can from the ones already there, contains no reference to the golden ratio, and reaches the same answer from starting angles a hundred and sixty degrees apart. Growth as a rule https://www.phyllotaxis.xyz/essays/growth-as-a-rule/ 2026-08-05T23:21:57.837Z A logarithmic spiral is not a shape somebody admired. It is what a thing grows into when it adds material at its opening without changing shape, its one parameter is how much it grows per turn, and that parameter can be recovered from any drawn curve to the last digit. The cube law https://www.phyllotaxis.xyz/essays/the-cube-law/ 2026-08-05T23:21:57.837Z A branching network built to move fluid for the least work obeys one relation at every junction — the cube of the parent radius equals the sum of the cubes of the daughters. It is a minimisation result, it is checkable on a real tree, and it is the rare biological rule with a derivation. Why the average cell has six sides https://www.phyllotaxis.xyz/essays/why-six-sides/ 2026-08-05T23:21:57.837Z Not because hexagons are efficient. Because Euler's formula leaves a tiling no choice — count the edges two ways and the mean comes out at six, whatever the cells would prefer. The efficiency argument is a different claim about a different thing. Fibonacci is a branch, not a law https://www.phyllotaxis.xyz/essays/fibonacci-is-a-branch/ 2026-08-05T23:21:57.837Z 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. Counting the spirals https://www.phyllotaxis.xyz/essays/counting-the-spirals/ 2026-08-05T23:21:57.837Z Almost every claim about phyllotaxis is a claim about how many spirals run through a pattern, and the count is almost never done. It can be done from the points alone, by machinery that is never told what angle built them — and then a count of 34 is evidence rather than a restatement. The bifurcation diagram https://www.phyllotaxis.xyz/essays/the-bifurcation-diagram/ 2026-08-05T23:21:57.837Z Sweep the one parameter of the rule and the settled angle traces a diagram — a broad golden branch, a transition, and a two-whorl regime at exactly half a turn. The famous constant is one branch of it, which is a more useful thing to know than the constant. Raup's three numbers https://www.phyllotaxis.xyz/essays/raups-three-numbers/ 2026-08-05T23:21:57.837Z Nearly every coiled shell that has ever existed is a point in a three-dimensional space — how fast the whorl expands, how far the opening sits from the axis, how far it travels along it. That is a remarkable compression, and the most useful line in the space is the one where the whorls come apart. Fitting the exponent https://www.phyllotaxis.xyz/essays/fitting-the-exponent/ 2026-08-05T23:21:57.837Z Assuming the exponent is three and reporting the error tells you how wrong the data is. Fitting the exponent and reporting what it comes out as tells you what the network is doing — and the machinery has to be shown returning something other than three, or it is not a fit. Packing, measured four ways https://www.phyllotaxis.xyz/essays/packing-measured-four-ways/ 2026-08-05T23:21:57.837Z The claim is that the golden angle packs best. It is measurable, and the measurement gives three different winners on three criteria — all near 137.5° and none of them it. That does not make the claim wrong; it makes the usual statement of it wrong. The claim that survives https://www.phyllotaxis.xyz/essays/the-claim-that-survives/ 2026-08-05T23:21:57.837Z 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. The counts change with radius https://www.phyllotaxis.xyz/essays/the-counts-change-with-radius/ 2026-08-05T23:21:57.837Z The same head gives 21 and 34 near the middle, 34 and 55 further out, 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. Droplets with no biology in them https://www.phyllotaxis.xyz/essays/droplets-with-no-biology/ 2026-08-05T23:21:57.837Z Douady and Couder dripped magnetised ferrofluid into a dish of silicone oil, and got spiral phyllotaxis with Fibonacci parastichy numbers out of a system containing no cells, no genes and no plant. That is the strongest evidence the pattern is physics — and the clearest warning about what a model can claim. The nautilus question https://www.phyllotaxis.xyz/essays/the-nautilus-question/ 2026-08-05T23:21:57.837Z A golden spiral grows by 6.854 per turn. Measured nautilus sections give about 3.2. That is a factor of 2.14 — not a rounding error, not an artefact of where the centre is assumed to be, and not close. L-systems describe, they do not explain https://www.phyllotaxis.xyz/essays/l-systems-describe/ 2026-08-05T23:21:57.837Z Two rewriting rules and a turtle produce something indistinguishable from a plant, and there is no plant in it — no light, no water, no auxin, no mechanics. That is worth demonstrating precisely because the output is so convincing. The gap that grows https://www.phyllotaxis.xyz/essays/the-gap-that-grows/ 2026-08-05T23:21:57.837Z A rational divergence angle develops an empty wedge that grows without bound as the head fills — a factor of nearly four between two hundred primordia and sixteen hundred. An irrational one does not. That is the statement about the golden angle that survives measurement. Recovering the angle from the counts https://www.phyllotaxis.xyz/essays/recovering-the-angle/ 2026-08-05T23:21:57.837Z Build a head at a stated divergence angle, forget the angle, and get it back from the spiral counts alone. Four angles, worst error twelve thousandths of a degree — and the only thing that crossed between the two halves was a list of coordinates. Where the model stops https://www.phyllotaxis.xyz/essays/where-the-model-stops/ 2026-08-05T23:21:57.837Z Below a growth parameter of about 0.18 this implementation does not converge — the settled angle wanders over a hundred degrees however long the run. That is the range where the literature says the interesting behaviour lives, and it is worth a figure rather than a quietly chosen axis.