The three destinations counted with a shared factor sit near a half turn, a half turn and two thirds. Measuring how near is the trap: by distance from the fraction, the golden angle is closer to two fifths than two of them are to anything, and would be reported as having five files it does not have.
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.
Fig. 1 The six destinations with the fractions marked. The three above the axis sit near a low fraction of a turn; the three below sit on a ladder of the surviving family.
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.
Fig. 2 The files, drawn. Two rows of organs running up the stem on the left, which is what the counter is reporting when it says two.
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.
Fig. 3 The chains being followed, on a stem the counter was built for. It does exactly this on a stem with files in it too, which is why the files it reports are really there.Fig. 4 The same stem unrolled, so that the two files can be read directly off the positions rather than off a count.
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 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.
Fig. 5 Why that happens: with denominators allowed to grow, every angle has a good rational nearby, and the golden angle is only the worst approximable in a sense that the raw distance does not capture.
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.
Fig. 6 The golden angle, drawn. Whatever it is six and a half degrees from, it does not have five rows of organs running out of it.Fig. 7 And two fifths of a turn, which does. Six and a half degrees is the whole of the difference between these two pictures.
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.
Fig. 8 The machinery that measures approximability: the continued fraction expansion, whose terms are all ones for the golden ratio, which is what makes it the worst case.
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.
Fig. 9 The thread in question: several fractions with one denominator, and the same structural consequence at each of them.Fig. 10 And the general result it reached: how wide the effect of a rational divergence is depends on the denominator, which is the same correction applied to a different quantity.
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.
Fig. 11 Two fifths of a turn, drawn as a stem. Five steps come back to the start and five files run up it, which is what the golden angle is six and a half degrees away from doing.
Measured that way — the drift of the q-hop, over every fraction with a
denominator up to six — the six destinations separate cleanly:
destination
best file
drift of that file
175.01°
2 steps
9.97°
235.00°
3 steps
15.00°
189.96°
2 steps
19.92°
the golden angle
5 steps
32.46°
136.99°
5 steps
35.05°
208.80°
5 steps
36.00°
280.43°
5 steps
37.85°
Fig. 12 The separation on the axis. The three destinations with a file are the three whose counted pair has a factor in it, and no threshold anywhere between twenty and thirty-two degrees changes which three they are.
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.
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.
Fig. 13 The general form of the distinction: a tolerance is either separating two clouds or sitting in a hole between them, and only the second kind is worth stating.Fig. 14 And what happens when it is the first kind: tightening the tolerance moves the count, which means the count was reporting the tolerance.
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.
Fig. 15 Where the two populations live: the golden branch and its multiples on one side, the low fractions on the other, with nothing in between at these denominators.
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.
Fig. 16 The structure that puts them there: exactly one lag of the old lattice standing, and the slip closing on a whole number of its turns.Fig. 17 The two structures on one axis, over the whole sweep of cuts. A ladder of one family’s turns, and three places off it.
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.
Fig. 18 And both kinds settle. A destination here is a fixed repeating motif, whichever of the two descriptions applies to it.
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.
Fig. 19 The family the ladder is built on, in the ranking of the lattice it came from: the five, which is the second-shortest step at this rise.
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°.
Fig. 20 The angles the rule can hold, and the region the golden branch occupies. A slip moves along one family’s ladder and stays inside it.
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.
Fig. 21 The far end of that: a stem three files wide, with nothing of the lattice it was cut from left in it.Fig. 22 And the dose, which decides only whether a stem wrecks. Where it goes when it does is one of these two kinds, and the size of the cut does not choose between them.
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.
Fig. 23 Why an unbounded search says nothing: the fractions crowd everywhere as the denominator grows, and what distinguishes angles is which fractions are near them at a given size.
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.
Fig. 24 The counts as the divergence is swept, which is where the ceiling comes from: the counter’s own output at these arrangements never carries a factor above three.
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.
Fig. 25 The case the ceiling excludes and a different thread includes: three eighths of a turn, with eight files and a denominator this measurement does not reach for.
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.
Fig. 26 And what it buys, in the packing rather than in the arithmetic: a pattern with no files has no radial gaps, which is what the geometry is asked for.
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.