Fractions with the same neighbours

A seed head built at a rational divergence is more ordered than one built a hairsbreadth away. Its organs fall into a small number of exact rows, the cell areas repeat, and the statistic this collection uses for disorder — the mean squared departure of a cell’s side count from six — dips sharply. Sweep the divergence and the dips are at the rationals, one per fraction, with depths and widths that vary.

Two laws for the width have been established here. The dip’s depth is a property of the fraction, and its scale in degrees goes as q/n² — the denominator over the square of the head size. After those two are taken out, something looked left over, and it looked ordered: the widths within a single denominator appeared to follow how crowded the fraction’s neighbourhood is, meaning how far the nearest other rational sits.

That residual has been open for two rounds and was withdrawn in the last one, for a reason worth restating because it is the whole argument for what follows.

Why no instrument here can test it #

Every way this collection has of measuring the width has a free parameter set by the crowding.

The first instrument reads the width as a level crossing — how far from the rational the disorder has climbed back to half its background. Where that level lands depends on the staircase between this rational and its neighbours, and the staircase is the neighbourhood.

The second integrates the deficit instead and divides by the depth, giving an equivalent width. An integral needs a limit, and the limit is set by how far it may reach before it starts integrating the neighbour’s dip rather than this one’s background. That limit is the neighbourhood too.

So a hypothesis about the neighbourhood is being tested with an instrument the neighbourhood calibrates. Measured anyway, the ordering changed with the window at one denominator, came out with the loneliest fraction widest at another, and nearly matched the crowding order at a third — which is what an instrument with a neighbourhood-dependent bias in the same direction as the hypothesis would produce whether the hypothesis was true or false.

Matching rather than correcting #

The obvious repair is to model the bias and subtract it. That is a bad idea here and it is worth saying why: the bias would have to be modelled with the same machinery whose behaviour is in question, and a correction derived from an instrument cannot rescue that instrument’s evidence about the thing it is sensitive to.

The alternative is to make the confound constant by construction — to compare fractions on which the crowding does not vary, so there is nothing to correct for. Then whatever difference remains between them is a difference in the arithmetic of the fraction, which is the thing under test.

The question is whether such sets exist. They do, in quantity.

The search, and what it turns up #

Take every fraction in lowest terms with a denominator from 13 to 60, in the half of the circle this collection works in — five hundred and twenty-six of them. For each, find the nearest other rational with a denominator no larger than sixty and record how far away it is in degrees. Sort by that distance and look for runs of fractions whose distances agree within five per cent and whose denominators differ by a factor of at least 2.4.

There are twenty-four. The one used throughout is

6/19 · 7/20 · 8/21 · 9/23 · 16/33 · 19/45 · 15/47

whose nearest neighbours sit at 0.3158°, 0.3158°, 0.3117°, 0.3069°, 0.3117°, 0.3077° and 0.3064° — a spread of 3.1% across a factor of 2.47 in denominator.

It contains 8/21, which is a convergent of the golden angle and one of the fractions this collection has been measuring since the work that established the width laws. That is not a requirement and it is convenient: the set is anchored to a fraction whose behaviour is already known from a different direction.

Matching the instrument as well #

Fixing the crowding is half of it, and the other half is free.

The dip’s own scale in degrees is q/n², so at a common head size the members of the set have dips differing by a factor of 2.47 — and a window stated in degrees would be a different fraction of each dip. Stated in scaled units it would be the same fraction of each dip and a different distance in degrees. Neither is a common window.

The head size is a free parameter, so it is spent. Each member is measured at n = √(q/u) for one common scale u, which makes the scaled unit q/n² the same for every member — 8 × 10⁻⁵ here, to within 0.14%. That single choice makes three things common at once:

• the dip’s own scale, by construction;
• therefore a window in scaled units is the same window in degrees;
• and therefore, because the neighbours are matched, the same fraction of the way to the neighbour — 5.1% to 5.2% at the window used, against 3.8% to 6.1% for the unmatched comparison this replaces.

The heads run from 487 organs at 6/19 to 766 at 15/47. Every one of them is above the floor this collection uses for a dip to be worth measuring, which is fifteen organs in each of the fraction’s rows; the binding member is 15/47, whose floor is 705.

Why the crowding recurs across denominators #

The existence of matched sets is not an accident of the range searched, and the arithmetic behind it is worth a paragraph because it says how far the construction can be pushed.

The distance from p/q to the nearest other rational with denominator at most Q is governed by the mediant structure of the Farey sequence: the neighbours of p/q at level Q are the fractions p′/q′ with |pq′ − p′q| = 1 and q′ as large as the ceiling allows, and their distance from p/q is 1/(qq′) of a turn. So the crowding is set by the product of the two denominators, not by either alone.

That is why it recurs. A fraction with a large denominator whose best neighbour has a small one can be as lonely as a fraction with a small denominator whose best neighbour has a large one — the products are what match. In the set used here, 6/19’s neighbour is 19/60 and 15/47’s is 8/25, products of 1,140 and 1,175, and the two crowdings agree to three per cent because the products do.

It also says where the construction runs out. Holding the product fixed while spreading q means the neighbour’s denominator has to shrink as the fraction’s grows, and the ceiling on denominators bounds how far that can go. A wider span of q needs a higher ceiling, which admits more neighbours and moves everybody’s crowding down; that is a different search and it has not been run.

What is left varying #

After all of that, the members of the set differ in exactly one thing: the arithmetic of the fraction. Denominator, numerator, continued fraction expansion, how well it approximates, whether it is a convergent of anything — all of that varies. The neighbourhood does not, the dip’s scale does not, and the instrument’s reach does not.

That is a comparison that could not be made before, and it needs the crowding matched first. It is what the next essay measures.

The two sets, and why there are two #

A single set measured once is a result with no replication in it, so a second matched set is used throughout beside the first:

5/17 · 9/20 · 9/25 · 15/43 · 21/44

with neighbours at 0.3651°, 0.3529°, 0.3692°, 0.3640° and 0.3557° — a spread of 4.6% across a factor of 2.59 in denominator, at a crowding a fifth larger than the first set’s.

The two share no members and sit at different neighbourhood distances, so an agreement between them is a genuine repetition rather than the same measurement with different labels.

One fraction per denominator #

A small design decision with a reason behind it: each set takes at most one fraction from each denominator.

Two fractions sharing a denominator are exactly the comparison the residual was first read off — same q, different crowding — and including both would put that comparison back inside a set built to avoid it. Taking one per denominator makes the set’s variation entirely between denominators, which is what the arithmetic half of the hypothesis is about.

It also costs nothing, because the search has plenty of candidates. Of the twenty-four sets found, several have six or seven members after the restriction.

What the set looks like from the disorder sweep #

It is worth seeing the members where they actually live, because a list of fractions is abstract and a sweep is not.

What this does not say #

It does not say the sets are unique. Twenty-four is what a particular search returns at a particular tolerance over a particular range, and loosening any of the three would return more. The two used here were chosen for having the widest span of denominators, and the search is re-run whenever the figures are drawn so that the sets remain sets the search finds.

It does not say the neighbourhood is fully characterised by one distance. The nearest other rational is the quantity that sets both instruments’ free parameters, so it is the quantity that has to be matched. Second and third neighbours also vary, less, and the matching does not control them directly.

It does not say a matched set is better data. It is the same measurement on different fractions. What it is better at is answering one question, because the thing that made the question unanswerable is held still.

It does not say the crowding is the only confound worth matching. It is the one both instruments’ free parameters are set by, which is why it is the one that makes the hypothesis untestable. Other properties of a fraction’s neighbourhood may matter to the width in ways nothing here has looked for.

And it does not measure anything yet. This essay is the construction. What comes out of it is the next one’s business, and the answer is short.

What it would take to break the construction #

Worth stating what an objection to this design would have to look like, since the whole essay is a design rather than a measurement.

An objection would have to name a property of the instrument that varies across the set and matters. Three candidates are already controlled: the neighbour’s distance, the dip’s scale in degrees, and the window’s reach towards the neighbour. A fourth — the background the deficit is measured against — is controlled indirectly, because the background is taken at offsets clear of every rational in the range and “clear” is defined by the same neighbourhood.

The candidate that is not controlled is the second neighbour and beyond. If the shape of the profile a few times further out than the first neighbour differs systematically between small and large denominators, a wide window would pick that up and it would look like a property of the fraction. Measured, the second neighbours of this set sit between 0.343° and 0.421°, which is a wider spread than the first neighbours’ but still narrow; and the effect it could produce is a prediction about wide windows specifically, which is testable and is tested in the next essay.

The other candidate is the head size itself. The members are measured at different heads — 487 to 766 organs — and if the disorder statistic had a head-size dependence beyond the q/n² scaling, it would enter here. That is why every comparison is repeated at a second common scale, with every head a quarter larger, and the two have to agree member by member.

The check that would refuse it #

Three assertions run whenever the set is drawn.

Every member’s nearest other rational has to sit at the same distance to within five per cent, and the denominators have to span a factor of at least 2.4. Those two together are what “matched set” means, and a set failing either is refused rather than measured — including, deliberately, a hand-made pair of 8/21 and 3/8, whose crowdings differ by a factor of fourteen.

Every member’s dip has to be measured on the same scale in degrees, to within one per cent, and one window has to reach the same fraction of the way to the neighbour for all of them, to within five per cent. That is the instrument half of the matching, and it is what the head sizes are chosen for.

And every head has to have at least fifteen organs in each of its fraction’s rows. A scale that asked for a smaller head would be refused: below that floor the disorder statistic is measuring a head without rows, and every number taken from it would be about the head rather than about the fraction.