Four fractions with one denominator

The second moment of a head’s side-count distribution — how far its cells depart from a common number of neighbours — has a narrow deep dip wherever the divergence angle is a rational multiple of a turn. The previous phase measured the dip’s half-width across six denominators and found

w ≈ 150 · q / n²

to within a factor of two, where q is the denominator and n the number of organs in the head.

It closed with an objection to its own result, which is the best kind to have.

The objection #

Every fraction in that measurement is a Fibonacci convergent. 3/8, 5/13, 8/21, 13/34, 21/55 and 34/89 are the convergents of the golden angle: the fractions whose continued fractions are all ones, which is the arithmetic signature of being as badly approximated by simpler fractions as a number can be.

That is not a neutral sample. A convergent sits in the emptiest neighbourhood its denominator has — the nearest other rational of comparable size is as far away as it can be — and the whole of this thread’s difficulty has been that a dip is measured against a background, and a background is a neighbourhood.

So the law could be about two different things. It could be about q — the number of rows the head is divided into, which is a structural fact about the arrangement. Or it could be about approximation quality — how nearly the angle misses being an even simpler fraction — which is a fact about the neighbourhood rather than about the head.

The two make different predictions and the test is cheap: 12/55 and 23/55 have the same denominator as 21/55 and much more crowded neighbourhoods.

The comparison needs no scaling law at all #

This is the part that makes the test easy, and it is worth spelling out because the previous phase’s machinery was built to compare across denominators and head sizes, where the law is needed.

Four fractions of 55, all measured on heads of 1,279 organs, are four measurements of one quantity. The n² is what makes different head sizes comparable; the q is what makes different denominators comparable. Hold both fixed and the law says the four widths are equal, and no fitting is involved in checking it.

The answer is the denominator #

The four widths of 55 span a factor of 1.28. The four of 34 span 1.14 and the four of 21 span 1.15.

For comparison, the same quantity across denominators — the raw n²w — spans a factor of 3.9 over these three, and it is dividing by q that brings the three families together, to 122, 191 and 180.

The width is a function of the denominator. The within-family spread is a seventh to a quarter; the across-family spread is a factor of four undivided and a factor of 1.6 divided by q. The variation the law does not capture is smaller inside a denominator than between denominators, which is what “the law is about q” means when it is stated as a measurement rather than as a preference.

And the Fibonacci convergent is not the odd one out. It is within seven per cent of its family’s mean at 21 and at 34, and eight per cent the other side at 55. Had approximation quality been what the law was about, the convergent — the extreme member of every family on that axis — would have been the extreme member on the width axis too, in the same direction each time. It is the widest at 21, the widest at 34, and the second narrowest at 55.

It is worth saying what a failure would have looked like, since the figure is easy to read as confirmation of whatever one expected. If approximation quality were the variable, the four bars of a family would not be within a quarter of each other: the convergent’s neighbourhood is emptier than 23/55’s by a factor of 1.3 in distance and its continued fraction is different in kind, and the previous phase’s own across-denominator spread — a factor of twelve undivided — is the scale on which “the neighbourhood matters” would show. A quarter is not that scale.

The other failure mode is subtler and is the reason three families were measured rather than one. A single family agreeing within a quarter could be a family whose four fractions happen to have similar neighbourhoods. Across the three families the crowding distances span 0.152° to 0.390°, a factor of 2.6, and the widths still sort by denominator rather than by crowding.

Every width is checked at a second head size #

A width read off a profile that is not a single dip can come out anywhere, and the only way this measurement can tell is that the answer changes when the head does.

So every fraction is measured twice — once at the family’s head size and once at a head 1.55 times larger — and the two are compared through the law’s own scaling, since n²w is the quantity that should be the same. Eleven of the twelve fractions agree between the two heads to within a tenth: the drifts are 0.1%, 0.4%, 1.0%, 1.2%, 1.3%, 2.0%, 2.3%, 2.7%, 3.2%, 3.3% and 21%.

The twenty-one per cent is 8/21, the convergent at the smallest denominator, which the previous phase had already flagged: the smallest head a denominator is given has fifteen organs per row, and at 21 that floor is close enough to bite. It is kept, because a threshold that excludes an inconvenient point is a threshold chosen for its result, and because at twenty-one per cent it changes nothing: the family’s spread is a seventh with it and a tenth without it.

The threshold itself is a quarter, and it is not doing delicate work. There is no fraction between three and a half per cent and twenty-one, and none between twenty-one and a hundred and fifty-five. The measurement is bimodal in a way that makes the cut obvious, which is the best position a threshold can be in and is not usually available.

The twelfth fraction disagrees between head sizes by a factor of 2.6, and is not a marginal case at all. It is refused rather than measured, and it and one other are the subject of the essay that follows.

The continued fraction orders them backwards #

The sharper version of the same point is worth making, because it turns a non-result into a small positive one.

Take the largest partial quotient of each fraction’s continued fraction, which is the textbook measure of how well a number is approximated by simpler fractions — a large quotient means the fraction is very close to a simpler one. Within the family of 34 it runs 2 for the convergent, 3 for 15/34, and 11 for 11/34.

If the width followed approximation quality, 11/34 would be the extreme. Its width is 9.68 × 10⁻³, the narrowest of the three, and the convergent’s is 1.11 × 10⁻², the widest — a difference of fourteen per cent in the direction opposite to the hypothesis, over a range of partial quotients from 2 to 11.

The same reversal appears at 21: 10/21 has a partial quotient of 10 and the narrowest width in its family, and 8/21 has 2 and the widest.

So approximation quality is not merely absent from the law. What little ordering it has is backwards, and it is small enough that the honest reading is that it is not the variable.

Why the fractions were chosen this way #

Four per family, and the choice was made before any of them was measured, which matters for a test whose result is an agreement.

Each family has its Fibonacci convergent, because that is the fraction the law was fitted at. Each has one fraction with a large partial quotient — 10/21, 11/34, 17/55 — because that is the extreme of the alternative hypothesis. Each has one whose nearest neighbour is unusually close and one whose nearest neighbour is unusually far, because that is the axis the residual might follow. And every one is in lowest terms, which is not a formality: 12/34 is 6/17 and would give a head with seventeen rows rather than thirty-four, so it would be a measurement of a different denominator wearing the wrong label.

The machinery refuses a fraction that is not in lowest terms rather than silently measuring the reduced one, which is the sort of refusal this collection has learned to write after finding a counter that returned the two smallest offsets instead of the two shortest and was believed for a phase.

What does order the residual, weakly #

The widths inside a family are not identical, and the residual is not noise.

Order each family by how far away the nearest other rational is — the crowding this thread had to compute in order to find a clean background at all — and the most crowded fraction gives the widest dip and the loneliest the narrowest, in all three families.

That direction is the one a measurement artefact would take, and saying so is more useful than treating it as a finding. A half-width here is read off the climb from the dip’s floor towards the background, and a neighbour a sixth of a degree away contributes its own shoulder to that climb. A crowded fraction’s shoulder is lifted, the half-way level is reached later, and the width comes out larger.

The effect is a third of the width over a factor of 1.4 in crowding, on four points per family. It is reported as an ordering rather than as a law, and it matters mainly because it explains why the within-family spread is not zero.

The prediction that was written down #

The previous phase did not merely leave the question open. It made the prediction, in the essay that established the law, and it named the fractions.

Its reconstruction of why the width carries a factor of q goes: at denominator q the head has q rows, each holding n/q organs, so the organs within a row are q/n apart along it. What has to stay small is the twist relative to a row’s own internal spacing. Making the denominator larger makes the rows sparser, a given twist displaces an organ by a smaller fraction of its row’s spacing, and more twist is tolerated — which gives w ∝ q/n².

The essay labelled that a reconstruction rather than a derivation, on the ground that it was written after the measurement, and then stated what would make it one:

The same argument says the coefficient should be independent of which rational of a given denominator is used, so 21/55 and 34/89 — with denominators of 55 and 89 — should not be the only cases; 12/55 or 23/55 should give the same 166. Nobody has looked.

12/55 gives 156. 23/55 gives 199. 21/55 gives 165.

The prediction is confirmed to within a fifth, which is inside the factor of two the law is quoted to and inside the resolution the measurement turns out to have. The two fractions the previous phase named by hand are the two nearest and furthest from its own convergent in this family, and they bracket it.

That is a reconstruction promoted to a derivation, or as near to one as this collection’s standard allows: an argument made after a measurement, which stated a consequence nobody had checked, checked in the phase that followed, in the place it named.

It is worth being clear about what is still missing. The argument predicts that the coefficient does not depend on the fraction; it does not predict the value 150, and the residual factor of 1.6 across denominators is not accounted for by anything. What has been established is the shape of the law and the independence its account requires — not the constant.

What this settles and what it opens #

The objection is answered. The width law was fitted over a sample that could not distinguish two hypotheses, and the sample was not the reason it came out as it did. Nothing about the law needs to change.

The law is now about a structural quantity. q is the number of rows the organs fall into, which is a fact about how the head is built rather than about number theory. That makes the law’s shape less surprising: a dip whose width scales with the number of rows and inversely with the square of the head size is saying that the dip is resolved when a row’s worth of organs can be told from its neighbour’s, and both scalings follow.

And it leaves the coefficient unexplained. 150 is measured, not derived, and dividing by q leaves a residual factor of 1.6 across three denominators that nothing here accounts for. That residual is now known not to be approximation quality, which is one hypothesis fewer.