A file has to close

Three of the six places a many-organ cut can send a stem are counted at pairs whose numbers share a factor, and none of them has any rotational symmetry. So the shared factor is not jugacy. It has to be a fact about where the divergence landed, and the three divergences are 175.01°, 189.96° and 235.00°.

Half a turn is 180°. Two thirds of a turn is 240°. The three are 4.99°, 9.96° and 5.00° from those two fractions. That looks like the answer, and it is nearly the answer, and the version of it that “looks like the answer” is wrong in a way that would embarrass the whole subject.

What makes a file #

A file is a near-vertical row of organs. It exists when a small number of consecutive steps brings the pattern back to nearly where it started: at a divergence of 175.01°, two steps carry an organ 350.03° round the stem, which is 9.97° short of a full turn, so organ i and organ i + 2 sit almost above one another and the pattern has two rows running up it.

That is what a counter finds. It follows chains of near neighbours; the shortest chains are the files; and if there are q of them, the pair it returns has q in it, and the other member is a multiple of q as well because the second family has to be commensurate with the first.

So the question “what do the three destinations have in common” becomes “how nearly does a small number of steps close” — and the measurement of that is where the care is needed.

The obvious measure, and why it is wrong #

The obvious measure is the distance from the fraction. The destinations are 4.99°, 9.96° and 5.00° from a half, a half and two thirds; call anything within ten degrees “near a fraction” and the three are in and something else has to be out.

Apply it to the golden angle. 137.5077° is 6.49° from two fifths of a turn, which is 144°. That is nearer than two of the three destinations are to anything. By the obvious measure, the golden angle has five files.

It does not, and the fact that it does not is the single property this whole subject rests on. A head at the golden angle has no files at any denominator; that is why its organs pack without leaving radial gaps, and it is the content of every claim about why the angle is where it is.

So a measure that ranks the golden angle as nearly-five-file is not measuring the thing that makes files. It is measuring the wrong quantity, and the correction is one factor.

The trap has a name and this collection has walked into it before #

The failure above is not specific to files. It is the reason continued fractions exist, and it is the reason the golden angle is the golden angle rather than merely an angle near 137.5°.

Every real number has rationals arbitrarily close to it. What separates numbers is how cheaply they can be approximated: how small a denominator buys a given accuracy. A number is well approximable when a small denominator gets very close, and badly approximable when it does not — and the golden ratio is the worst approximable number there is, which is precisely the statement that no low denominator gets close for its size.

This collection has a whole thread on what a small denominator does to a head — the dip in packing quality at a rational divergence, and how wide the dip is — and every result in it is stated in terms that carry the denominator. Reverting to raw distance here would have contradicted work already done.

The right measure is q times the distance #

What makes a file is not how close the divergence is to p/q of a turn. It is how close q steps come to closing, and q steps accumulate the error q times.

At two fifths of a turn plus 6.49°, five steps overshoot by 32.46°. That is not a file; it is a third of the way round the stem, and organ i + 5 sits nowhere near above organ i. At half a turn less 4.99°, two steps fall short by 9.97°, and the two organs are visibly stacked.

Measured that way — the drift of the q-hop, over every fraction with a denominator up to six — the six destinations separate cleanly:

Three under twenty degrees and four over thirty-two. The threshold in the machinery is twenty-five, and it sits in a gap of twelve and a half degrees rather than between two adjacent measurements — which is the only state in which a threshold on a quantity that decides a table is worth having.

And the number of steps in the file is, at all three, exactly the factor the counted pair shares. Two files and a pair with a factor of two; three files and a factor of three. That is not a fit; it is the same fact counted twice.

The golden angle’s number is the theoretical floor #

The table caps denominators at six, which makes its verdict on the golden angle a statement about six fractions rather than about every fraction. That gap can be closed, and closing it turns a census into a theorem.

The quantity being tabulated — q times the distance from the nearest p/q — is the classical measure of how cheaply a number can be approximated, and it has a known floor. For any irrational divergence there are infinitely many fractions whose distance from it is smaller than one over √5 times the square of the denominator, and the √5 in that statement cannot be improved to any larger constant. Multiplying through by q — which is what turning a distance into a drift does — the drift of the best q-file is bounded below by 1/(√5 · q) of a turn, which in degrees is 161/q. The bound is attained, in the limiting sense, exactly by the golden ratio and the numbers equivalent to it — that is what “worst approximable” means, stated as a number rather than as a reputation.

Put q = 5 into it. 161 divided by five is 32.2°, and the measured drift of the golden angle’s five-file is 32.46°. The measurement is sitting on the bound.

That is worth pausing on, because it is not a fit and there was no freedom in it. The 32.46° came out of subtracting 137.5077° from 144° and multiplying by five; the 32.2° came out of a theorem about irrational numbers with no phyllotaxis in it at all. The agreement is 0.8%, and the residual is the ordinary one — the bound is approached from above along the convergents rather than met exactly at each one.

Run it up the ladder and the same thing happens at every rung. At eight steps the golden angle drifts 20.1° against a floor of 20.1°; at thirteen, 12.4° against 12.4°. The drift falls as 1/q, it never falls below 161/q, and it is the smallest such quantity any divergence has.

So the census’s verdict generalises. The golden angle has no file at any denominator, and the reason is not that six was far enough to look — it is that the floor exists and the golden angle sits on it. A destination with a two-step file drifting 9.97° is more than three times closer to closing than the best the golden angle can manage at any q whatever, and the comparison does not depend on where the table stops.

What capping q at six is really doing #

The cap is still doing work, and it is honest to say which.

It is not protecting the golden angle’s row, which the bound protects. It is protecting the gap. The floor falls as 1/q, so at thirteen steps the golden angle’s own drift is 12.4° — under the twenty-five-degree threshold. Allow denominators that large and every divergence acquires a “file”, the two populations merge, and the threshold goes back to deciding the answer rather than reporting it.

It is worth being explicit that this is a genuine limitation and not a repaired one. The clean separation reported in the table is a separation among files a counter could resolve, and stated without that qualifier it would be false. Every angle has files at some denominator; what distinguishes these divergences is the denominator at which the first one appears, and a table that reports a threshold on drift is reporting that distinction in a form that hides it.

The reason the cap is legitimate anyway is that a file is something a counter finds by following chains of near neighbours, and a chain of thirteen organs on a stem carrying a hundred and forty is not a near-vertical row anybody can see. Thirteen steps at a rise of 0.013 climb a sixth of the stem’s length before returning; the organs at either end of that “file” are nowhere near each other in the picture, whatever the arithmetic says about their angles.

So the cap is a statement about what the instrument resolves rather than a convenience, and it can be stated as one: a file counts if it closes within a small number of steps and the closing organs are neighbours on the surface. The first half is the arithmetic above; the second half is what the counter is already doing. The table’s cap at six is the crude version of the second condition, and it is crude in the direction that makes the claim harder rather than easier — a finer version would exclude more candidates, not fewer.

The consequence for the census is the one that matters here: the three destinations with a shared factor have files at two and three steps, which is inside any reasonable version of that condition, and the three without have nothing until denominators no counter would resolve. The split is not an artefact of where the table stopped.

The gap, and what a threshold is doing in it #

Three measurements under twenty degrees and four over thirty-two is a gap of twelve and a half degrees, and it is worth saying what that buys.

A threshold on a quantity whose two populations overlap decides the answer. A threshold on a quantity with a wide gap in it reports the answer, and any value inside the gap gives the same table. Here the value is twenty-five, which is nearer the middle of the gap than to either edge, and moving it to twenty or to thirty-two changes nothing.

The gap is not an accident of this census either. The three destinations with files are near fractions with denominators of two and three; the three without are slips of a golden-branch lattice, and a golden-branch divergence is by construction as far from every low fraction as an angle can be. So the two populations are separated by the same property that makes the subject interesting, which is why they are separated by a lot rather than by a little.

The other three are a ladder, not a fraction #

The three destinations with no file are the three that keep a family standing, and their structure is completely different.

They sit at 136.99°, 208.80° and 280.43°. The steps between them are 71.81° and 71.63°, against 360° divided by five, which is 72.00°. Each of them keeps the five-family rigid, and each is at a different whole number of turns: zero, one and two.

So the six destinations divide into two kinds with two different descriptions. Three of them are the lattice the stem was cut from, displaced along one family by a whole number of turns. Three of them are somewhere else entirely — a divergence with a file in it, which the rule is content to hold and never arrives at unaided.

And the ladder’s own spacing is a check on the reading rather than a decoration. If a slip is the old lattice with turns threaded through one family, then consecutive rungs are exactly one turn of that family apart — 72.00° for a five-family — and the measured steps are 71.81° and 71.63°. The two-tenths of a degree of slack is the same order as the scatter of a settled divergence, which is what it should be.

Why the two kinds are exclusive #

Worth stating, because the split is asserted in both directions rather than as a description of one half.

A slip keeps a family of the old lattice standing, which means its divergence is the old one plus a whole number of turns of that family. The old divergence is 136.78° — a golden-branch value, sitting where nothing closes at any small denominator — and adding multiples of 72° to it moves it around a ring whose members are all equally far from every low fraction. So a slip cannot land on a file, and none of the three does: their best files drift by 35°, 36° and 38°.

Conversely a destination with a file has moved far enough that no lag of the old lattice is left: measured at every period up to twenty-four, none of the three has a hop within three degrees of where the control put it. There is nothing of the original lattice in them, which is exactly why there is nothing to keep them off the fractions.

The bound on the denominator, and why it is small #

The measure needs a second decision as well as the factor, and it is the ceiling on the denominator. Without one the claim is empty: allow denominators to grow and 137.5077° is four hundredths of a degree from 1237 of 3240 turns, so every angle “sits near a fraction” and the statement carries no information.

Six is the ceiling used here, and it is not chosen to make the answer come out. Six is the largest denominator that appears in any counted pair in the whole census: the pairs are 2/6, 4/6, 3/6, 5/9, 5/11 and 5/12, and the shared factors are two, two and three. A file the counter cannot report is not a file this essay has any business naming.

The bound also has to be stated before the measurement rather than fitted to it, and it is worth noticing what it costs. At a ceiling of eight, 135° — three eighths of a turn — becomes available, and there is a band of coarse rises where a stem sits exactly on it. That case is real and is a subject of its own; it is outside this census because nothing here lands there.

It is worth adding that the three fractions involved — a half, a half and two thirds — are the three smallest denominators there are after a whole turn. That is not a coincidence either: the smaller the denominator, the less exactly the steps have to close, so the fractions a wrecked stem can reach are the ones with the most room around them.

What the correction is worth outside this thread #

The measure that ranks the golden angle correctly is not a piece of local bookkeeping. It is the reason the golden angle is interesting at all, restated in the form a counter can use.

An angle is bad at making files when q consecutive steps fail to close for every small q, and that is precisely the property “badly approximable by rationals” names. The golden angle is the worst-approximable number there is, which means it is the angle at which files are hardest to make at any denominator — and a placement rule that ends up there is a rule that has minimised something files would have made worse.

So the same correction that keeps three destinations distinguishable from three others is the one that keeps 137.5° distinguishable from 144°. Getting it wrong here would have produced a tidy table; getting it wrong in general would produce a collection that could not say why its own subject exists.