The most irrational is not the most disordered

This site has put seven criteria to the golden angle and seven of them have returned nothing. Nearest-neighbour evenness picks an angle nearby. Area-evenness is won by rational angles, whose sliver cells are near-identical. Gap-filling picks a third. Hexagon share is won by a whorl at 144°. The second moment of the side-count distribution gives 0.253 at the golden angle and 0.255 at a rational a hundredth of a degree away.

One claim survives, and it is arithmetic rather than geometric: the golden angle is the hardest number to approximate by rationals, at 0.4377 against 0.3306 for the best of fifteen hundred sampled, approaching Hurwitz’s bound of 1/√5.

The staircase invites an eighth criterion, and it is a good one — good enough that it would be embarrassing not to test it.

The repair #

If a rational divergence makes ordered tissue, and if the depth of the order falls away with the denominator, then disorder should increase as an angle becomes harder to approximate by a rational with a small denominator. And the angle that is hardest to approximate is the golden angle.

So: the golden angle should be the local maximum of μ₂.

That is not the claim anybody makes in print. It is the claim the previous two essays invite, it would give the golden angle a geometric criterion it wins, and it is exactly the shape of inference this site exists to check — a plausible step from a measured fact to a conclusion nobody has measured.

Where to test it #

The interval has to be chosen carefully or the test means nothing.

8/21 and 5/13 are Farey neighbours: 8 × 13 − 5 × 21 = −1. That is the condition for there being no fraction with a denominator below 34 anywhere between them, so the interval is bounded by the two simplest rationals in its neighbourhood and contains no simpler one. The number in it that is hardest to approximate — the noble number between two Farey neighbours — is the golden angle, at 137.5078°.

So the interval runs from 137.1429° to 138.4615°, the two ends are the two deepest dips in the region, and the hypothesis says μ₂ climbs from each end to a maximum at the golden angle.

The result #

μ₂ across the interval runs from 0.019 at the ends to 0.508 at its largest, and the largest is at 138.42° — nine tenths of a degree from the golden angle, and within a twentieth of a degree of the 5/13 dip at the far end.

The golden angle is at 0.253. Thirty-six of the sixty-seven samples are above it.

Refuted. The most badly approximable angle in the interval is not where the disorder is largest, is not near where it is largest, and is not distinguishable from the general run of angles on the same plateau.

Three things make the refutation clean rather than a matter of interpretation. The interval was chosen from arithmetic before the sweep was run. The quantity was not re-defined after the answer came out. And the failure is not marginal — the value at the predicted maximum is 0.253 and the actual maximum is 0.508, which is a factor of two on a statistic whose whole range here is a factor of twenty-seven.

Why the repair fails #

It fails because it treats “disorder” as though it were a single quantity that increases smoothly with distance from order, and the staircase says it is not.

Between the dips, μ₂ is not measuring how far the angle is from a rational. It is measuring which arrangement the head has — which cells touch which — and that is a discrete object that changes at particular angles for particular reasons. The height of a plateau depends on the mixture of five-, six- and seven-sided cells the arrangement happens to produce, and there is no monotone relation between that mixture and any measure of irrationality.

The largest plateau in the interval, at 138.42°, sits just inside the 5/13 dip’s shoulder, where the arrangement is nearly thirteen rays and the near-miss produces a tessellation full of fives and sevens. That is the geometry of a frustrated lattice, and a frustrated lattice is more disordered than a generic one — but “nearly rational” is the opposite of “badly approximable”, so the mechanism that produces the maximum runs the wrong way for the hypothesis.

The eighth negative, and what it is worth #

Seven criteria had already returned nothing and an eighth returning nothing might look like a formality. It is not, for two reasons.

The first is that this one was invited by the site’s own new result rather than inherited from the literature. The other seven are claims other people have made about packing and evenness. This one is the natural next thought after the staircase, it would have been reasonable to assert without testing, and the essay’s existence is the site’s own habit applied to its own inference.

The second is that it closes a route rather than merely failing to open one. If μ₂ had peaked at the golden angle, the site would have had a geometric criterion the golden angle wins, and the temptation to read it as “so the plant is optimising disorder” would have been considerable — a claim with no mechanism attached and a great deal of appeal. The measurement says there is nothing there to explain.

What survives, restated #

The claim about the golden angle that this site has tested and upheld is not about packing, evenness, hexagons or disorder. It is that among all angles it is the one whose multiples avoid the rationals most persistently — and the reason that matters biologically is not aesthetic. A pattern at a rational angle puts its organs on a small number of rays, which is a poor use of a disc; a pattern near a rational spends a long time nearly on those rays as the head fills. The golden angle is the angle that spends the least time nearly-rational at every scale, and that is a statement about the whole history of a growing head rather than about its finished tissue.

The staircase is consistent with that and does not demonstrate it. A frustrated lattice at 138.42° is disordered now, at nine hundred points; a head at 137.51° has no particular scale at which it is frustrated. Whether that difference matters to a plant is a question about what a plant is optimising, and this site has been careful for four phases not to answer it.

The three ways the test could have been rigged #

A refutation is only as good as the chances the claim was given, so it is worth listing what would have made this test unfair and saying that none of them was done.

Choosing the interval after seeing the answer. The interval is the Farey interval between 8/21 and 5/13, fixed by the arithmetic of the two fractions: they are neighbours, so nothing simpler lies between them, and the noble number between two Farey neighbours is the golden angle. Any other interval containing the golden angle either contains a simpler fraction — in which case the deepest dip in it is not at an end and the shape of the test changes — or is a sub-interval, in which case it is a smaller version of the same experiment. There is one right interval here and it was chosen from the fractions.

Choosing the head size until the answer came out. Nine hundred points, which is the standing default for every disorder measurement here and the size every other measurement in this thread uses. At three hundred the dips are wider and the interval has less plateau in it; the maximum is still not at the golden angle.

Choosing the statistic. μ₂ about six is the disorder measure the cellular tissue literature uses and the one the previous phase measured; it was not invented for this test. If it had been — if the essay had defined a new “disorder” and then found the golden angle did not maximise it — the refutation would be about a definition rather than about a claim.

There is one thing that would strengthen the result and was not done: sweeping several Farey intervals rather than one. The golden angle is the noble number of every interval between consecutive Fibonacci fractions, so the same test could be run between 5/13 and 3/8, or between 13/34 and 8/21, and each would be an independent instance. One interval is enough to refute a claim about the golden angle in particular; it is not enough to establish the general statement that noble numbers are nowhere special on this statistic, and that general statement is not claimed here.

What a noble number is, and why the intuition is attractive #

The intuition being refuted is a good one and it is worth stating carefully, because the essay is not saying it is silly.

A number’s continued fraction is a sequence of whole numbers, and the size of those numbers controls how well the number can be approximated: a large partial quotient means a very good rational approximation just before it. The golden ratio has all its partial quotients equal to one — the smallest they can be — so it never has a good approximation, which is Hurwitz’s theorem in its concrete form. A noble number is one whose continued fraction ends in all ones, and the noble number between two Farey neighbours is the one their mediants converge to.

So the golden angle is genuinely, provably, the hardest angle in its interval to approximate. And rational angles genuinely make ordered tissue. The inference between those two facts is the step that fails, and it fails for a reason worth keeping: “far from every rational” and “disordered” are different quantities because the disorder is not a function of distance to a rational at all. It is a function of which contact graph the arrangement has, and the contact graph changes in steps.

An angle can be far from every simple rational and still produce an arrangement whose cells happen to sit in an unfortunate mixture of fives and sevens. An angle can be moderately close to a rational and produce a tidier one. The staircase is the record of which is which, and it has no monotone relation to approximability because it is not built out of approximability.

What would have to be true for the claim to hold #

A refuted claim is more useful when the conditions under which it would have held are written down, because that is what makes it a measurement rather than a verdict.

The claim needs μ₂ to be a decreasing function of how well the angle can be approximated by a rational the head can resolve. That would require three things, and the sweep says which of them fail.

Every resolvable rational produces a dip. True. Every dip found is at a rational and the ordering is by denominator.

The dips have overlapping shoulders, so that between two of them the value rises smoothly to a maximum at the point furthest from both. False. At nine hundred points a dip’s half-width is about two thousandths of a degree and the gap between the two ends of the Farey interval is one and a third degrees, so the dips are isolated spikes with a wide, structured region between them that owes nothing to either.

And the region between the dips is featureless except for that. False, and this is the substantive failure. Between the dips the curve is a staircase whose steps have heights set by which contact graph the arrangement has, and the variation across it — 0.02 to 0.51 — is larger than anything the dips’ shoulders contribute.

So the claim fails not because the arithmetic is irrelevant but because the arithmetic operates only very locally, and the space between its effects is filled by something else. A head with a hundred times more organs would have dips a hundred times narrower still, so the failure gets worse rather than better with resolution.

The index entry #

For the refutation index, which keeps every claim this site has tested and what the number returned:

Claim. The golden angle maximises the disorder of the resulting tissue, as measured by the second moment of the side-count distribution.

Test. Sweep μ₂ across the Farey interval between 8/21 and 5/13, at nine hundred points, sixty-seven samples.

Result. Maximum 0.508 at 138.42°; golden angle 0.253; thirty-six of sixty-seven samples above it. Refuted.

The claim is one this site invented in order to test it, which is worth recording as such. An index of other people’s errors is a debunking list; an index that includes the author’s own plausible inferences, tested and dropped, is a record of method.

Eight and counting #

The tally is now eight geometrical criteria and eight distinct answers, none of them the golden angle, against one arithmetic criterion the golden angle wins outright.

The pattern in that is not an accident and it is worth stating as the thread’s conclusion rather than as a running joke. Geometrical criteria are evaluated on a finished arrangement of a particular size, and a finished arrangement of a particular size is well served by a rational angle whose rays happen to fit it. The arithmetic criterion is evaluated over all scales at once, and it is the only one of the nine that has no head size in it anywhere.

So the golden angle’s claim to be special is a claim about a growing head rather than about any head, and every criterion that measures a head at one moment is measuring the wrong thing to see it. That is a hypothesis rather than a result — this site has not built a criterion that integrates over the growth — and it is the clearest thing eight negatives have to say.