What a head can mean by most irrational

Of the three famous assertions about this subject, the one that survives is that the golden angle is the hardest divergence angle to approximate by fractions. In Hurwitz’s form it is a theorem: for every irrational number, $q \cdot |q\,d - p|$ can be pushed below $1/\sqrt{5}$ at infinitely many denominators $q$, and the golden ratio and the numbers equivalent to it are the ones for which it cannot be pushed any lower. The golden angle is the simplest of those, the noble numbers.

A theorem about infinitely many denominators is a statement about every scale at once, and that was part of its appeal: no head size at which it stops applying. It is also a statement no head can witness. A head is counted from its inner spirals to its rim, and the counts between those are the only denominators it shows. So this essay asks what the claim picks out over the counts a head actually shows.

A head shows a window of counts #

A spiral count is a denominator. A head showing 34 and 55 spirals at one radius and 89 and 144 at its rim has had the rows those four numbers make, and the counts change with radius because the good approximations change with scale. A denominator below the innermost count belongs to rows inside the smallest organs, and one above the outermost to rows beyond the rim.

The natural finite version of the claim therefore measures an angle’s resistance over a window of denominators from the innermost count to the outermost: the least $q \cdot |q\,d - p|$ with $q$ in the window. Over counts from 34 to 144 the golden angle scores 0.44718, within a ten-thousandth of Hurwitz’s $1/\sqrt{5} = 0.44721$.

Only the convergents set the score #

Two facts make that measure tractable, and the first is a theorem of Legendre’s that can be watched. For any angle, a fraction in lowest terms whose denominator is not one of the angle’s continued-fraction convergents scores at least a half, which is above $1/\sqrt{5}$. So only the convergents can decide how well an angle is approximated over a window.

For the Lucas angle, 99.5016°, ten of the 376 denominators up to 377 score below a half: 4, 7, 11, 18, 29, 47, 76, 123, 199 and 322. Those are its convergents, and from 4 onward each is the sum of the two before. Every other denominator’s best fraction scores a half or more, as Legendre’s theorem requires.

A convergent scores by the ratio of its counts #

The second fact is a closed form. A convergent with denominator $q_k$ scores exactly $1/(\alpha + q_{k-1}/q_k)$, where $\alpha$ is the rest of the continued fraction from the next partial quotient onward. Once the partial quotients are all one, $\alpha$ is $\varphi$, and the score is $1/(\varphi + q_{k-1}/q_k)$.

That is $1/\sqrt{5}$ exactly when consecutive counts stand in the golden ratio, and it is checked to a millionth at every convergent up to a thousand on four noble angles. A partial quotient of two instead of one pushes $\alpha$ above two, and with the counts around it in anything like the golden ratio the score falls to about 0.38, so a score near the golden angle’s needs the quotients in the window to be one.

There is one exception, and it is worth naming because it is not the head in question. An angle whose counts barely change across the window — a large partial quotient below it, so that a single count holds from 34 nearly to 144 — can score above the golden angle, because it has almost no convergents in the window to be caught out at. Such a head shows the same spiral pair across its whole face. Everything below is about heads whose counts change inside the window, as a head counted from 34 to 144 does.

So the claim becomes a claim about counts #

A partial quotient of one means the next convergent’s denominator is the sum of the two before it. So over the counts a head shows, an angle whose counts change inside the window resists approximation like the golden angle exactly when those counts add up — each the sum of the two before — starting from a pair whose ratio is near the golden ratio.

The golden angle’s convergents score 0.472, 0.438, 0.451, 0.446, 0.448 and then 0.447 from 21 onward. The Lucas angle’s score 0.422 at 4, 0.457 at 7, 0.444 at 11, 0.449 at 18 and then 0.447 from 29 onward. The angle at 151.14° dips to 0.321 at 2 and settles by 50; the one at 106.45° dips to 0.339 at 3 and settles by 71. Past their first few counts all four are within a few thousandths of each other.

Every pair of counts starts one #

Nothing about adding up requires the counts to be Fibonacci numbers. Any two coprime counts $a < b < 2a$ start a sequence $a, b, a + b, \ldots$, and there is exactly one angle whose convergents run through that sequence with every later quotient one: its earlier quotients are the continued fraction of $b/a$ read backwards, and after $b$ it is noble.

Over counts from 34 to 144 there are 550 such angles between 20° and 180°. Their scores run from 0.3860, for a first pair such as 34 and 35 whose ratio is nearly one, up to the golden angle’s 0.44718, and they are highest where the first ratio is near $\varphi$. Within one per cent of the golden angle the first ratios run from 1.5610 to 1.6389.

Forty-six angles within a per cent #

Forty-six angles between 20° and 180° come within one per cent of the golden angle’s resistance over that window. The five nearest are the golden angle itself; 99.50° at 99.989 per cent; 106.45° at 99.931; 151.14° at 99.919; and 132.18° at 99.907. Within a tenth of a per cent there are five.

They are not clustered near the golden angle. The forty-six run from 33.90° to 162.42°, twenty-four of them below 90°, and only two lie within two degrees of 137.5°: 136.73° and the golden angle itself, with 139.53° just outside. A head counted from 34 to 144 whose counts resist approximation this well could sit almost anywhere in the range a divergence angle can usefully take, and the resistance alone would not say where.

The golden angle is the highest of all 550, so the claim still picks it out. But it picks it out by a ten-thousandth of the score, over a window spanning four counts, and a measurement of any head that could resolve that difference would have to count spirals to four significant figures of their arithmetic.

What the family shows #

The golden angle shows 34, 55, 89 and 144. The Lucas angle shows 47, 76 and 123, the counts the model’s second branch gives. The angle at 106.45° shows 44, 71 and 115; 151.14° shows 50, 81 and 131; 132.18° shows 49, 79 and 128; 158.14° shows 41, 66 and 107; 77.96° shows 37, 60 and 97; 162.42° shows 51, 82 and 133.

On a logarithmic axis every one of those sequences steps by about the golden ratio, which is why they are equally hard to approximate. They differ in which numbers they are, and that is the only thing about them a head’s counts over this window can tell apart.

The family grows outward #

Over counts from 13 to 55, six angles come within one per cent; over 21 to 89, twenty; over 34 to 144, forty-six; over 55 to 233, a hundred and twenty-one; over 89 to 377, two hundred and eighty-seven. Each step outward multiplies the number by 3.33, 2.30, 2.63 and 2.37, about $\varphi^2$.

The reason is counting. The ratios that qualify are the same narrow band at every window, and the number of coprime pairs of counts inside a band of ratios grows as the square of the counts, which is $\varphi^2$ per step. So a larger head does not narrow the claim’s choice. It widens it.

The lead shrinks too #

Over counts from 13 to 55 the golden angle leads the Lucas angle by $3.3 \times 10^{-4}$. Over 34 to 144 it leads the Lucas angle by $4.8 \times 10^{-5}$, and over 89 to 377 by $7.0 \times 10^{-6}$, with the runner-up alternating between the Lucas angle and 151.14°. The lead falls by a factor of about forty-seven from the innermost window to the outermost.

Two windows apart the runner-up is the same angle, and the lead has fallen by exactly $\varphi^4 = 6.8541$ each time, to four decimal places. That number has a reason. The score at a convergent is $1/(\varphi + q_{k-1}/q_k)$, and in any sequence of counts that add up, the ratio of consecutive counts closes on $1/\varphi$ by a factor of $\varphi^2$ with every term, alternating above and below it. The runner-up’s first ratio in a window is one term further along two windows later, twice over, so its shortfall is $\varphi^4$ smaller.

That is the finite form of the statement that every noble number reaches Hurwitz’s bound in the limit. The golden angle reaches it soonest, and the others are catching up at a rate that is itself a power of the golden ratio.

The Lucas angle, specifically #

The Lucas angle is the runner-up over three of the five windows, and it is the one member of the family a growing stem has been followed onto here. Its continued fraction is the golden angle’s with the first quotient three instead of two, so its convergent counts are 1, 3, 4, 7, 11, 18, 29 where the golden angle’s are 1, 2, 3, 5, 8, 13, 21. From 4 onward they add up exactly as Fibonacci numbers do.

Everything that separates the two is in that first quotient, and it shows at a count of 4: the Lucas angle’s best approximation there scores 0.4223, while the golden angle’s worst below 34 is 0.4377, at a count of 3. A head at the Lucas angle and a head at the golden angle, counted anywhere from 29 or 34 outward, are within a ten-thousandth of each other in how resistant their counts are; counted at the centre, where the Lucas head shows 3 and 4 and the golden head 2 and 3, they differ by 0.015.

Where the golden angle still differs #

Below the window, over the counts from 2 to 33, the golden angle scores 0.4377, set at a count of 3. The best of the other 203 angles within five per cent is the Lucas angle at 0.4223, set at a count of 4. Every one of the family falls behind the golden angle there, because each has a larger partial quotient early in its continued fraction: the Lucas angle’s first quotient is three where the golden angle’s is two, and 151.14°'s second quotient is two where the golden angle’s is one.

So what makes the golden angle the extreme case is real, and it lives at the centre of the head. It is a property of its smallest counts — 2, 3, 5 and 8, the rows nearest the centre — and not of anything a rim count can see.

Why this does not withdraw the claim #

The theorem stands, and nothing here disputes it. The golden angle is the most resistant angle over every window measured and over the whole range of denominators. What changes is what the claim can be evidence for. A head counted from 34 outward that matches the golden angle’s resistance is matching the property that its counts add up from a near-golden pair, and forty-five other angles have that property as closely.

This is not the first time the most irrational angle has failed to be singled out by a measurement taken where the measurement could be made. The most irrational is not the most disordered found the disorder of a head’s cells peaking at 138.42° with the golden angle unremarkable beside it, and packing measured four ways found criteria that each crown a different angle. What is new here is that the claim which does survive, measured on the counts a head shows, shares its crown too.

The property that is special to the golden angle sits where the gap that grows and the square cell at every flip found the geometry of noble angles deciding things: near the centre, in the first few organs, where a head is smallest and hardest to count.

What a person would have to measure #

To use the claim as evidence that a head sits at the golden angle rather than at another member of its family, a count of the rim is not enough, by the argument above. Two things are. The counts themselves, since only one member of the family shows the Fibonacci numbers — which returns the question to how often a head is Fibonacci. Or the arrangement of the first dozen organs, where the golden angle’s resistance is visibly different from its family’s.

The second is the harder measurement, since the centre of a head is where organs are youngest and least regular, and it is the one the claim that survives actually requires.

What this does not say #

It does not say that a head’s window has sharp edges, or which window a head of a given size and rise shows; the essays on counting are about that. It does not say that any plant sits at 106.45° or 132.18°: the rate at which the rise falls decides which branch a stem takes, and the essays here have followed growing stems onto the golden and Lucas angles; which other members of this family a stem can reach has not been measured.

The arithmetic of continued fractions and noble numbers belongs to number theory, and nothing here is new to it. What is measured is its consequence for the counts a head can show.

The claim, reduced #

Over the counts a head shows, resisting approximation like the golden angle is the same as having counts that add up from a near-golden pair. Over counts from 34 to 144, forty-six angles come within one per cent of the golden angle and the best of them within a ten-thousandth; the number within a per cent grows by about $\varphi^2$ per step outward, and the golden angle’s lead over the runner-up falls by exactly $\varphi^4$ every two steps. The golden angle is distinguished from its family only below the counts they share, at the centre of the head.

What would withdraw it #

An angle scoring above the golden angle over a window of counts. A fraction in lowest terms off an angle’s convergents scoring below a half. A family member matching the golden angle below its window. A stem at a family member’s angle that does not show the counts that add up. Each is checked whenever the windows are measured.

Still open: the centre, measured on heads #

Everything here is about denominators, and the conclusion points at the place a head’s geometry has to take over: the first few dozen organs, where the golden angle and its family part company. Whether that difference is visible in the positions of real organs, rather than only in a continued fraction, has not been measured.

The measurement is a head at each of the eight closest angles, grown to a few hundred organs, read by the instruments already used on golden heads — nearest-neighbour distances and the largest empty circle — at the centre and at the rim separately: whether the family members are indistinguishable at the rim and distinguishable at the centre, as the arithmetic says they should be, and how many organs from the centre the difference extends.