Every family but two is a sum

The previous essay measured the contact network of a seed head and found six spiral families where the convention reports two. Read one way that is an indictment: the practice of this whole subject throws away two thirds of what is there.

Read correctly it is a licence. This essay is the arithmetic.

The measurement #

Take the families each head actually has — the index offsets that carry more than two per cent of its cell contacts — and ask, of each one, whether it is the sum of two others in the same list.

In every case, exactly two members are not sums of others, and in every case they are the two smallest.

The set is closed under addition from the bottom. Two numbers generate it.

Why a third count is a prediction #

The consequence is the one the convention needed and had never been given.

Once two consecutive families are known, the next is settled by arithmetic. There is no head with families 34 and 55 and something other than 89 next. So a counter that reports a third family is not adding information about the divergence angle; it is either confirming the first two or contradicting them.

That makes the third count useful in a way nobody uses it. If the third counted family is not the sum of the first two, one of the three counts is wrong.

It is a check with no model in it. No divergence angle, no rise, no assumption about Fibonacci, nothing to fit. Three integers and an addition, applicable by a person with a photograph and no software.

The check catching the site’s own founding bug #

The best test of a check is an error it would have caught, and this collection has one on the record.

The site’s first spiral counter returned 21 and 34 for a head whose two nearest families are 34 and 55. It did so because it took the two smallest offsets among the local minima of the hop-length curve rather than the two shortest hops, and both numbers are real families of that head — 21 is genuinely there, carrying 17% of the contacts. Nothing about the output looked wrong: twenty-one spirals were drawn and there were twenty-one of them.

Apply the check. 21 + 34 = 55. Is the head’s next family 55?

No — the head’s families in that band run 34, 55, 21, 89, and the third strongest is 21, which is smaller than both. Read as an ordered triple the count says 21, 34, 55 and the head shows 34, 55, 89. The triple is internally consistent; what is wrong is which triple was picked out.

So the check is not a complete guard, and it is worth being exact about what it does and does not catch. It catches a count in which the third family contradicts the first two — a misread offset, a transposed digit, an offset from a different band. It does not catch a count that reports a genuine but lower window of the same sequence, because a sequence closed under addition looks the same shifted along.

The thing that caught the founding bug remains the recovery step refusing — asking which divergence angles make those two offsets the shortest, and finding none. That is a stronger check because it uses the band radius as well as the counts. This one is weaker and needs nothing but the integers, which is a different kind of useful.

The recursion is not always Fibonacci’s #

The result would be much less interesting if the closure were a fact about Fibonacci numbers, so the case that separates them is worth setting out.

Run the obvious rule — the next family is the sum of the two largest so far — from each head’s own two generators. It reproduces the measured families exactly on four of the five heads: 8, 13, 21, 34, 55, 89 from 8 and 13; 11, 18, 29, 47, 76 from 11 and 18; 2, 3, 5 from 2 and 3.

On the fifth it fails. From 8 and 13 the plain recursion predicts 21, 34, 55, 89. The head at 137.0° has 8, 13, 21, 29, 50, 71, 92, 113 — it adds 21 over and over.

Both laws are visible in that row. The closure holds: 8 + 13 = 21, 8 + 21 = 29, 21 + 29 = 50, 21 + 50 = 71, 21 + 71 = 92, 21 + 92 = 113. Every family is the sum of two others. What fails is the narrower claim that the two largest are the addends.

So “add the two previous” is a property of the golden angle’s continued fraction — all of whose partial quotients are 1 — and not of lattices. “Every family but two is a sum” is a property of lattices. The first is the famous statement and it is the weaker one.

What the families actually are #

The closure is a symptom. The underlying fact is older than this collection and the measurement recovers it, which is the strongest thing that can be said for a measurement.

Take the divergence angle as a fraction of a turn and expand it as a continued fraction. Its convergents are the best rational approximations, and their denominators are:

• golden 137.508° → 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 …
• Lucas 99.502° → 1, 3, 4, 7, 11, 18, 29, 47, 76 …
• 137.0° → 1, 2, 3, 5, 8, 21, 113 …
• 144° → 1, 2, 5

The measured families are a window on those lists — except at 137.0°, where the measured set contains 13, 29, 50, 71 and 92, none of which is a convergent denominator.

They are the intermediate fractions. Between one convergent and the next there are as many steps as the next partial quotient, each formed by adding the previous denominator again: from 8 and 21 with a partial quotient of five that gives 29, 50, 71, 92 and 113. That is precisely the measured sequence, and it is precisely the “adds 21 over and over” that broke the Fibonacci recursion.

So: the parastichy families of a head are the denominators of the convergents and intermediate fractions of its divergence angle. Checked on all five heads, against a set computed from the angle alone, with no families over and none missing.

That is the classical characterisation, and what is worth reporting is not the fact but the route: it was recovered from a tessellation that was never shown an angle. Two independent constructions — a continued fraction and a Delaunay graph — produce the same list of integers.

It also explains the closure without any further work. Both convergent and intermediate denominators are built by adding an earlier denominator to a later one, so the set is closed under addition by construction, and the two smallest members of any window are the two that have their addends outside it.

What the generators are #

The two numbers that are not sums are the two smallest families, and on these heads they are 8 and 13, 11 and 18, or 2 and 3.

They are not fundamental. Measure the same head at three hundred points instead of nine hundred and the families are 5, 8, 13, 21, 34, 55 — one rung coarser, and the generators are now 5 and 8. The set is a window on an infinite sequence, and which window is visible is set by how much of the head is in the band.

That matters for what the closure claims. It does not say a head has two fundamental families and derives the rest; it says that in whatever window is visible, all but the two smallest members of it are sums of members also visible. The two smallest are generators only in the sense that they are the smallest things in view — they are themselves sums of families too fine to have survived the contact threshold.

So the honest statement is about sufficiency of a pair, not about a distinguished pair: any two consecutive families determine every larger one, and the pair a person counts is as good as any other.

The whorled case, which is the boundary #

At 144° the families are 2, 3 and 5 and the sequence stops. There is no 8.

That is not the closure failing; 2 + 3 = 5 holds. It is the sequence being short, because 144° is two fifths of a turn exactly and the pattern is five straight rows with no finer structure to have a finer family. A rational angle has a finite continued fraction, and the family sequence terminates where the fraction does.

It is a useful boundary case for two reasons. It shows the law holding where the pattern is degenerate, which is where laws about lattices usually stop. And it gives the check a distinguishable failure signature: a head whose family list is three long and stops is a whorled head, and the counted pair on it will share a factor — which is exactly the condition the site’s angle recovery refuses on.

What this does to the collection’s own practice #

Four phases of this site have reported parastichy pairs. This essay says that was enough, which is a comfortable conclusion and therefore one worth stating carefully.

What is licensed. Reporting two counts loses no information about the divergence angle, because the remaining families are determined. The site’s recovery machinery, which takes a pair, is not leaving anything on the table.

What is not. The shares — how much of the contact network each family carries — are not determined by the pair in any way this collection has established. Those are a property of the tessellation, and the previous essay had to measure them. So “two counts determine the families” is true and “two counts determine the tissue” is not.

And what changes. A third count should now be taken when it is cheap, not as a better measurement but as a check, and reported as one. That is a small addition to the survey specification and it costs nothing: anyone counting 34 and 55 on a photograph can see whether the next family is 89.

Why a lattice has short hops at those offsets #

The continued fraction explains which integers appear. It is worth one more step to say why an integer appearing there means a spiral family, because that step is what connects an arithmetic fact to a picture.

An offset $m$ is a spiral family when stepping $m$ places is a short move. Stepping $m$ places is a turn of $m\delta$, so the angular part of that step is how far $m\delta$ sits from a whole number of turns — which is small exactly when $m$ is the denominator of a good rational approximation to $\delta$. The radial part grows steadily with $m$. So the hop length is a trade-off between an angular term that is small only at special $m$ and a radial term that punishes large $m$, and the offsets that win are the approximation denominators below whatever the radial term can afford at that radius.

That is the whole of it, and it accounts for the window as well. As the head grows the radial penalty per step falls relative to the spacing, so larger denominators become affordable and the visible families move up the list. A head does not acquire new families; it becomes able to show more of the sequence it always had.

The site’s angle recovery is the same argument run backwards — given which offsets are shortest, which $\delta$ makes them so — and its habit of returning an interval rather than a value is the same fact again: a range of angles shares a window of approximation denominators.

The one-line version #

A seed head has more spiral families than anyone counts, and the ones nobody counts are sums of the ones everybody does. That is why two numbers have always been enough, and it is the first time this collection has been able to say why rather than observe that it seems to work.

What is worth taking from it #

Three things, in descending order of how much they change.

A third count is a free check. It costs one more trace on a photograph, it needs no model, and it catches a class of error that every other check on this site needs a radius and a fitted scale to catch. It should be in the survey specification and it was not.

The two-number convention is justified rather than merely traditional. That is a smaller finding and it matters for how this collection reads its own back catalogue: four phases of reporting pairs did not lose information, so nothing needs revisiting.

And “Fibonacci” is the wrong level of description. The famous statement — that the families are consecutive Fibonacci numbers — is true of the golden angle and false of a head half a degree away, which has 8, 13, 21, 29, 50, 71 and is a perfectly ordinary lattice. The statement that survives across angles is about approximation denominators, and it contains the Fibonacci case as the instance where every partial quotient is one.

That last is the phase’s habitual finding arriving in the oldest part of the subject: the famous version of a claim is a special case, the general version is duller, and the general version is the one that can be tested on a plant that turns out not to be at 137.5°.