The refutation index
This subject arrives with more confident wrong statements attached than any other. Some of them are decorative and some are in textbooks, and almost none of them comes with the number that would settle it — which is what the whole collection is a reply to: state the claim, state the test, report the number, and say plainly when the claim survives.
There are 12 of them below. Each carries the verdict, what was found instead, and — the part that matters — the quantity that settles it, computed while this page is built out of the same libraries the figures are drawn from. None of these numbers is typed in. If one of them stopped agreeing with the essay it points at, the build would stop rather than serve the page.
The verdicts are not all wrong, and two of the most interesting are not. A claim here usually fails by being right about one quantity and quoted about another — the nautilus really is a logarithmic spiral, a sunflower really does show consecutive Fibonacci numbers somewhere on it, Lewis's law really does hold of some tissue. Distinguishing the true half from the travelling half is most of the work, and it is the reason this site computes rather than cites.
Wrong
5 claims
137.5° is the angle that packs seeds most efficiently.
No packing criterion singles it out at a fixed head size, and the criteria disagree with each other. The golden angle wins on how close the nearest pair come — the criterion most people have in mind — and it is beaten badly on how evenly the cell areas come out, which is the criterion most people would accept if it were offered. The angles that win area-evenness are rational: their points lie on a few radial rays, every cell on a ray is a near-identical sliver, and the gaps between the rays are enormous. A criterion is not a measure of packing if it is maximised by a point set with holes in it.
What settles it: at 800 points the golden angle scores 0.731 on area evenness against 0.010 for a divergence of 5/13 of a turn — and that rational angle's largest gap is 4.9 times the golden angle's, at 0.13 of the mean spacing between its closest pair against 0.84. Argued in Packing, measured four ways.
The nautilus shell is a golden spiral.
It is a logarithmic spiral, which is the true half and the reason the claim survives being checked casually. It is not this logarithmic spiral. A golden spiral grows by the golden ratio every quarter turn, which is a factor of φ⁴ ≈ 6.85 per whole turn; a nautilus grows by about 3.2. The gap is a factor of more than two, which is not a measurement dispute — and the one free choice in fitting it, where the centre of the shell is assumed to be, moves the answer by under a percent for a quarter-radius error, so the disagreement cannot be blamed on the fit.
What settles it: 6.854 per turn claimed against about 3.2 measured — a factor of 2.14 — where the same fitting routine recovers a drawn golden spiral's growth as 6.854, to a residual of 4e-15. Argued in The nautilus question.
Lewis's law and the Aboav–Weaire relation are two properties of the same cellular tissue.
They are cited side by side and they want opposite tissue. Aboav–Weaire says a cell with many sides is surrounded by cells with few, and its parameter a comes out near the value the literature quotes — on the ordered head, where Lewis's law has just failed. On the random set, where Lewis's law holds well, Aboav's parameter is roughly half of it. Neither relation is wrong; what is wrong is the sentence that puts them together, and no amount of care about either one on its own would find that.
What settles it: a = 1.176 on the ordered head, the value usually quoted, against 0.697 on the random set — the ordering that breaks Lewis's law is what makes Aboav's hold. Argued in Two laws that want opposite tissue.
Raup's shell morphospace has an impossible region where W·D < 1.
The boundary is real and the label is an interpretation the geometry does not supply. W·D < 1 means successive whorls are in contact rather than free of one another, and whorls in contact is the ordinary condition of most gastropods — a snail, not an impossibility. The region is not empty; it is where a large share of living shells sit. This is the clearest case on the site of a computed diagram acquiring, in transmission, a claim that was never computed.
What settles it: 38.6% of the sampled morphospace has whorls in contact — 261 of 676 cells — and a shell at W = 2, D = 0.4 has W·D = 0.80, which the geometry calls "involute — whorls in contact" and not impossible. Argued in Raup's three numbers.
Counts that share a factor mean the plant is whorled.
The counts cannot decide it. An ordinary single-jugate lattice — one primordium at a time, at a divergence near half a turn — counts a pair sharing a factor, and a genuine bijugate stem counts the same pair. What separates them is the rotational symmetry of the point set, which is a measurement on positions rather than on counts, and it is not recoverable from a published parastichy pair. Every counter this site had before the scale phase asked how far it is from element i to element i+m, which is already a claim that the elements arrived one at a time.
What settles it: an ordinary lattice at 180.5° counts 2 and 4 at rotational symmetry order 1, and a genuine bijugate stem counts 2 and 4 at order 2. Argued in What "whorled" was hiding.
Right about one quantity, wrong about another
3 claims
Sunflower spirals are always consecutive Fibonacci numbers.
Fibonacci is not what the geometry hands over. Sweep the divergence angle at a fine rise and consecutive Fibonacci pairs are a minority of outcomes — and the share falls as the pattern gets finer, which is the opposite of the way the claim is usually told. Reading the census generously, up to jugacy, roughly triples the Fibonacci share, and every point of that increase is one of the coarsest pairs repeated k times, which describes no plant. What actually makes plants Fibonacci is continuity from a coarse start: stems grown from divergences spread over two thirds of a circle end on the same pair, because the pattern is on the ladder before the ladder has many rungs.
What settles it: 14.7% of divergences give a consecutive Fibonacci pair at a rise of 0.008 — 50.1% read up to jugacy — while 1 of 16 stems grown down to that rise from scattered starts end on one. Argued in How often is it Fibonacci.
A sunflower has 34 and 55 spirals.
One head has several answers and the count depends on where it was taken. Run the same blind counter — one that is never shown the divergence angle — band by band up the radius of a single head, and it returns a different consecutive pair in each band. Every published count is therefore a statement about an annulus, and a count reported without the radius it was taken at is a sample from an unstated mixture. The transitions are not noise: they are computable in advance from the rise, and they are a factor of φ² apart.
What settles it: one head of 800 points gives 21/34 at 0.3 of the radius, 34/55 at 0.5 of the radius, 55/89 at 0.92 of the radius — the same rule and the same counter throughout. Argued in The counts change with radius.
Lewis's law describes the cells of a growing tissue.
It holds, and it holds on the wrong tissue. Lewis's law says a cell's area grows linearly with its number of sides, with a slope of a quarter and an intercept at two sides. On a random set of points it is close to exactly that. On an ordered golden-angle head — the pattern a phyllotaxis paper is likely to be about — the slope is a small fraction of Lewis's, because ordering the points is precisely what removes the area variation the law is a statement about. A law quoted as a property of cellular tissue is a property of disordered cellular tissue.
What settles it: slope 0.249 on a Poisson set against Lewis's 0.25, and 0.009 on a golden-angle head — a factor of 27.0 — with side count explaining 39.1% of the variance in cell area even where the law holds. Argued in Lewis's law wants disorder.
Outside the mechanism
1 claim
A reaction–diffusion mechanism explains the divergence angle.
A ring of cells running Gierer–Meinhardt does select a spacing, and the peak count is predictable from the Jacobian before any integration is run — that part works and is checked here both ways. What it does not do is define a divergence angle. Start the same ring from different small disorders and the angle between the first two peaks to appear is anything at all, while the peak count barely moves. A mechanism of this kind supplies a spacing; what supplies an order is motion, and a stationary ring has none.
What settles it: six starting disorders give first-gap angles of 176.6°, 47.2°, 109.1°, 150.7°, 47.2°, 132.7° — a spread of 129.4° — while the peak count over the same six runs is 7 or 8. Argued in A ring cannot make a spiral.
Underdetermined
1 claim
Teasel's divergence angle is 68.75°.
It is exactly half the golden angle, which is the interesting half of the claim and is not a coincidence: a k-jugate pattern is an ordinary lattice seen k times over, so every k has a limit divergence at 137.5078/k and every one of them is a noble number. But a k-jugate divergence is only defined modulo 360/k. Adding 180° to a bijugate plant's divergence relabels which member of each pair is called first and leaves the point set identical — the same coordinates, checked. So the reported number is a statement modulo 180, and quoting it without that is quoting a coordinate as though it were a measurement.
What settles it: 68.7539° is exactly half of 137.508°, and 68.75° and 248.75° are the same point set. Argued in Half the golden angle.
Survives
2 claims
The golden angle is special because it resists rational approximation.
This is the claim that holds, and it is arithmetic rather than packing. Measured as the smallest q·|qδ − p| over a range of denominators — how badly the angle can be approximated by any simple fraction of a turn — the golden angle scores higher than any other angle tried, approaching the bound Hurwitz's theorem puts on every irrational. The sharper form is the one worth carrying: a sweep over a grid of angles can never land on it, because every grid point is rational and every rational collapses the pattern onto a few rays. The famous number is the one point a search of this kind cannot reach.
What settles it: 0.438 for the golden angle against 0.422 for the Lucas angle and exactly 0 for every rational, with Hurwitz's ceiling at 0.447 — the golden angle reaching 97.9% of the best any number can do. Argued in The claim that survives.
Counting the spirals further out gives a better measurement.
It does, by a great deal, and the size of the gain is worth stating because it is the one piece of advice this collection would give anybody counting a real plant. A reported parastichy pair pins the divergence angle to a band whose width goes as one over the product of the two counts, so a count at the rim is worth a thousandfold in precision over a count near the middle. The caveat attached is the uncomfortable half: the value of a high count is entirely conditional on it being right, and a wrong 34/55 excludes the truth confidently where a wrong 2/3 excludes almost nothing.
What settles it: a report of 34 and 55 pins the divergence to 0.118° of arc against the 38.8° a report of 2 and 3 leaves — a factor of 329 — while knowing the radius it was counted at narrows the high report by a further factor of only 1.10. Argued in What a count is worth.
What this index is not
two things it is easily taken for
It is not a list of things people get wrong about plants. Every claim here is one a careful person says, and most are shorthand for something true — the trouble is that the shorthand travels and the conditions do not. A sunflower has 34 and 55 spirals is a perfectly good sentence with an annulus missing from it, and a rule of thumb with its conditions stripped off is indistinguishable, in a sentence, from a measurement.
Nor is any of this a claim about what a plant is doing. Every computation behind these verdicts is a model of form: a lattice that matches a sunflower does not show that the plant computes one, and the two mechanism models on this site are made to predict a number and then shown what they cannot do. Where a verdict rests on a model rather than on geometry the essay it points at says so, and the one verdict that turns on which model you take — whether a divergence sequence's noise arrived before or after the pattern chose — is recorded as an open question rather than as a finding.