The claim that survives
This site spends most of its length taking claims apart, which risks leaving an impression that the subject is a pile of errors. It is not. One of the three famous assertions about it is correct, and it is correct in a form sharper and more interesting than the version usually repeated.
The quantity
How well a number x is approximated by fractions is measured by how small q·|qx − p| can be made, taking the best p for each denominator q. A number with good approximations gets that quantity very small; a number that resists them keeps it large.
Hurwitz proved in 1891 that for every irrational number the quantity can be pushed below 1/√5 = 0.4472, and that the golden ratio is the number for which it cannot be pushed any lower. It is the extreme case, and the bound is attained by it and by nothing else that is not equivalent to it.
Measured here, the golden angle scores 0.4377 — the sampling floor approaching the theoretical value — against 0.3306 for the best of fifteen hundred other angles across a seventy-five degree window.
That is not a narrow win. It is a different regime.
Why the sweep cannot find it
A detail worth noticing, because it is the same difficulty the subject has in miniature.
The sweep steps in twentieths of a degree and therefore consists entirely of rational angles. Every one of them, being rational, has a denominator at which the approximation is exact and the score collapses — so the sweep can never land on the answer, and the best it reports is whichever grid point happens to be furthest from a low-denominator fraction.
The golden angle had to be evaluated exactly and inserted. Anything else would have been measuring the grid.
That is a small methodological point with a large moral: the property being measured is invisible to any finite sampling of the parameter, which is exactly why the packing measurements at fixed head size could not find it either.
Why this is the right form of the claim
Three reasons.
It is exact. Hurwitz’s theorem is a theorem. There is no head size at which it stops applying, no criterion under which it gives a different answer, and no dependence on how the measurement is made.
It explains the phyllotaxis. The parastichy numbers are the denominators of the good approximations, so an angle with no good approximations at any denominator never develops the rows that a good approximation produces. That is what keeps the gaps bounded, and it is the mechanism the packing story gestures at without stating.
It explains the extremeness. Every irrational angle eventually fills the disc; the golden angle is the one that never has a bad phase on the way. That is a statement about all scales at once, which is why no measurement at one scale can show it.
What it does not claim
It does not claim a plant computes it, or benefits from the last decimal place, or would be measurably worse at 137.3°.
The biological version is much weaker and is enough: a meristem that places each primordium away from its neighbours is pushed away from angles whose multiples pile up, and the golden angle is where that pressure has nowhere left to push. The theorem says why the attractor is where it is. It does not say that a plant needs to be there.
Confusing those two is how “the most irrational number” acquired its air of mysticism. The arithmetic is remarkable; the plant is just avoiding its own neighbours.
The three claims, together
The nautilus is a golden spiral. Out by a factor of 2.14, and the one free choice in the measurement cannot account for it.
Sunflower spirals are always Fibonacci. True of a branch, not of plants — the Lucas branch gives 47 and 76, and the same model reaches it.
137.5° is optimal. Right, once the criterion is named. Not as packing at any particular size, where three criteria give three answers, but as resistance to rational approximation, where it is the extreme case and there is a theorem.
One in three is a better record than this subject’s reputation suggests, and the one that survives is the one worth telling.