Half the golden angle

The sharpest result the expansion phase produced was about where the golden angle is not.

The forks are exact found that the branch points of the van Iterson diagram — the lattices where three parastichy families are equally short — have a closed form, and that every one of them sits at a rational divergence: $5/14$, $15/38$, $37/98$, $99/258$, $257/674$, $675/1766$ of a turn. Those converge on 137.50776° and reach it at no step, because each is rational and the limit is not.

This essay does the same thing to that result that the previous two essays did to the counters. Divide the tree by $k$ and the whole structure comes with it.

Halving the tree #

The scaling identity is the whole derivation. A $k$-jugate lattice at $(\delta, h)$ is, under the map that wraps the cylinder $k$ times, the ordinary lattice at $(k\delta, kh)$ with every count multiplied by $k$ and every distance divided by $k$.

So every feature of the ordinary tree has a $k$-jugate image:

• an ordinary fork between families $m$ and $n$ becomes a $k$-jugate fork between $km$ and $kn$;
• its rise $\sqrt3/2q$ becomes $\sqrt3/2qk$, with $q = m^2 + mn + n^2$ as before;
• its spacing $1/\sqrt q$ becomes $1/k\sqrt q$;
• and its divergence, an exact multiple of $1/2q$ turns, becomes an exact multiple of $1/2qk$.

The last of those is the one worth stating twice. Every multijugate fork sits at a rational divergence too, with a denominator $k$ times the ordinary one. The bijugate forks run

$$\frac{10}{28},\ \frac{30}{76},\ \frac{74}{196},\ \frac{198}{516},\ \frac{514}{1348},\ \frac{1350}{3532},\ \frac{3530}{9244}$$

of a turn, between the families 2/4, 4/6, 6/10, 10/16, 16/26, 26/42 and 42/68. In degrees: 64.2857, 71.0526, 67.9592, 69.0698, 68.6350, 68.7995 and 68.7365, alternating above and below and closing on

$$\frac{137.50776}{2} = 68.75388°$$

which is at none of them.

The family of limit angles #

The golden angle is the $k = 1$ member of a family, and the family is short enough to write out.

$$\delta_k = \frac{137.50776}{k}$$

so $68.7539°$ for $k = 2$, $45.8359°$ for $k=3$, $34.3769°$ for $k=4$ and $27.5016°$ for $k=5$.

These are not approximations and they are not coincidences of the construction. Each is the limit of a genuine fork sequence in its own tree, computed by solving for the lattices where three families are equally short and dividing by $k$ — and the closed form and the solver are required to agree to the last digit, which is the check the ordinary forks essay already carries.

Where the plants are #

The bijugate number is not a curiosity of the arithmetic. It is a measurement that exists in the literature.

Dipsacus — teasel — and Cephalaria are the standard bijugate genera, and the divergence reported for them is close to 68.75°. Bijugate sunflowers are documented, and they are counted 42 and 68, which is $2 \times (21, 34)$: exactly the family the sixth bijugate fork sits between.

That is a satisfying agreement and it is worth being careful about what kind of agreement it is. The theory does not predict that teasel exists; it predicts that if a plant is bijugate and if it is deep in its own Fibonacci-analogue branch, then its divergence is 68.754° and its counts are twice consecutive Fibonacci numbers. Both halves of that were known descriptively before anything here was computed. What the derivation adds is that the number is not an independent fact requiring its own explanation — it is the golden angle divided by two, and it follows from the jugacy alone.

Modulo what #

The reported number needs a qualification that the ordinary case does not, and it is not a technicality.

Adding $360/k$ to a $k$-jugate divergence leaves the point set identical — the same coordinates, not a congruent pattern. So a bijugate plant’s divergence is defined modulo 180°, and “teasel is at 68.75°” means “teasel is at 68.75° or 248.75°, and nothing in the specimen distinguishes them”.

The convention here is to report the representative in $[0, 360/k)$, which is why the limit angles come out as 137.5078 over $k$ rather than as some other member of each class. It is a convention and it is stated as one.

There is a version of this that is not merely bookkeeping. Because the period shrinks with $k$, a trijugate pattern’s divergence lives in a 120° window and a quinquejugate one’s in 72°. The higher the jugacy, the less room there is between distinct divergences — and correspondingly the more the geometry of a multijugate pattern is determined by $k$ rather than by $\delta$.

The ladder, halved #

The forks are the skeleton; the ladder is what a specimen actually walks, and it halves in the same way.

An ordinary stem at the golden divergence passes its transitions at rises of 0.12472, 0.04767, 0.01822, 0.00696, 0.00265 and 0.00101, changing its pair 1/2 → 2/3 → 3/5 → 5/8 → 8/13 → 13/21 → 21/34. A bijugate stem at 68.754° passes the same transitions at half those rises — 0.06236, 0.02384, 0.00911 and so on — changing its pair 2/4 → 4/6 → 6/10 → 10/16 → 16/26.

Two consequences, and the second is the one worth having.

The first is that a bijugate stem reaches a given pair at half the rise, so it needs to be twice as compressed to look the same. That is a statement about internode length and stem thickness and it is in principle checkable on the two forms of a species that has both.

The second is that the spacing between rungs is unchanged. The rises are all halved, so their ratios are identical, and the ratio is $1/\varphi^2$ in both trees. Everything the cone essays derive about transition spacing — $\varphi^{2/p}$, the factor of two between a cone and a head — therefore applies to multijugate organs unchanged. Jugacy shifts the ladder; it does not stretch it.

That is the sort of invariance worth noticing, because it means the two halves of this phase compose. A bijugate conifer cone, if there is such a thing, would have its transitions a factor of $\varphi^2$ apart along its axis, between families twice the ordinary ones, at half the ordinary rises.

What this does to the golden angle’s status #

The site has been chipping away at the golden angle’s specialness for three phases and this is the fourth strike, so it is worth putting them in one place and being clear about which ones land.

No packing criterion singles it out at fixed head size. Measured: closest-pair, largest-gap and area-evenness pick three different angles, all nearby, and area-evenness is won by rational angles.

It is one branch of a dynamical model, not its output. Measured: sweeping the one parameter of the Douady–Couder rule gives a golden branch, a transition and a two-whorl regime.

It is the limit of a sequence of rational forks and is at none of them. Measured: the tree is exact and the angle is not on it.

And it is one member of a family indexed by jugacy. Which is this essay. A bijugate plant is at half of it, a trijugate one at a third, and each of those is as much an attractor of its own tree as the famous one is of the ordinary tree.

Against all four stands one claim that survives, and it is worth restating because this essay does not touch it. The claim that survives is arithmetic: the golden angle resists rational approximation better than any other angle sampled — 0.4377 against 0.3306 for the best of 1,500 — approaching Hurwitz’s bound of $1/\sqrt5$.

And that claim transfers too, which is the interesting part. $\delta_k / 360 = 1/(k\varphi^2)$, and dividing a number by an integer does not change the tail of its continued fraction, so the $k$-jugate limits are noble numbers as well. They resist approximation for the same reason and to within the same asymptotic constant. The property that makes the golden angle worth having is not a property of 137.5° specifically; it is a property of a class, and every member of the class is a limit divergence of some jugacy.

Nobility, and what it does not buy #

The claim about approximation deserves more than a sentence, because it is the one property of the golden angle that survives every test this site has run, and the multijugate version of it is not quite the trivial extension it looks like.

A number is noble when the tail of its continued fraction is all ones. The golden ratio’s expansion is $[1;1,1,1,\dots]$, and $1/\varphi^2 = 0.381966$ is $[0;2,1,1,1,\dots]$ — all ones after a finite head. Noble numbers are the hardest to approximate by rationals, and $1/\varphi^2$ approaches Hurwitz’s bound $1/\sqrt5$, the best any number can do.

Divide by an integer $k$ and the head of the continued fraction changes completely — $1/(2\varphi^2) = 0.190983$ is $[0;5,4,1,1,1,\dots]$ — but the tail does not. The tail is what decides the approximation constant asymptotically, so every $\delta_k$ is noble, and the property that makes 137.5° worth having is a property all of them share.

What that does not buy is any claim that a bijugate plant is therefore “as optimal” as a single-jugate one. Approximation resistance is a statement about how far the multiples of an angle stay from whole turns, which on a lattice is a statement about how the largest gap behaves as the pattern fills. The gap that grows measured exactly that: at a rational divergence the largest gap grows without bound as the head fills, by a factor of 2.9 from 200 to 1600 points, and at an irrational one it does not, growing by 1.08 over the same range.

A bijugate pattern at $\delta_2$ inherits that. What it does not inherit is any advantage over a single-jugate one, because the comparison is not between them — a $k$-jugate pattern at $\delta_k$ is a single-jugate pattern at $\delta_1$, wrapped. They are the same lattice. Asking which is better packed is asking which of two identical objects is better packed.

Why the forks are rational, again #

The rationality of the fork divergences is the part most worth understanding rather than accepting, and the multijugate version makes the reason clearer than the ordinary one did.

At a fork the lattice is equilateral: three families are equally short, so every node has six equidistant neighbours, so the lattice is the triangular one. A triangular lattice on a cylinder of circumference 1 is a very constrained object — it is determined up to rotation by which pair of its lattice vectors wraps the cylinder, and that is a pair of integers.

The Loeschian number $q = m^2 + mn + n^2$ is the norm of the corresponding Eisenstein integer, and the denominator $2q$ falls out of requiring the wrap to close. Multiply the circumference by $k$ — which is what a $k$-jugate pattern does to the underlying lattice — and the denominator picks up the same factor.

So the forks are rational because a triangular lattice on a cylinder has to close up, and closing up is an integer condition. Nothing about the golden ratio is involved; the golden ratio appears only in which forks the Fibonacci path visits, and in the limit of their angles.

The census question, asked and dissolved #

There is an obvious question to ask once a whole family of trees exists: how often does a bijugate pattern come out Fibonacci? The expansion phase asked the ordinary version and got a number — 14.7% of divergences at a rise of 0.008 give a pair of consecutive Fibonacci numbers — and the multijugate version looks like a separate measurement waiting to be made.

It is not, and the reason is the scaling identity again.

Sweeping a bijugate pattern’s divergence over its whole period, 180°, sweeps the underlying ordinary divergence over the full circle. So the census over bijugate lattices at rise $h$ is the ordinary census at rise $2h$, with every count doubled. Same shares, same shape, same answer — relabelled.

There is no separate question. How often a bijugate pattern is “Fibonacci” (in the sense of being twice a consecutive Fibonacci pair) at a given rise is exactly how often an ordinary one is Fibonacci at twice that rise, and the site already has that curve.

That is a small result and it is the kind worth recording, because the alternative was to run the sweep and report a second table of numbers that would have looked like new evidence. A relabelling that produces a table is indistinguishable, from the outside, from a measurement that produces one.

What the jugacy does change is a different census, and it is not the one about bijugate patterns. It is the ordinary one: the bucket the expansion phase labelled “whorled” turns out to contain something specific, and reading it up to jugacy moves the headline share by a factor of three.

Two numbers a specimen could settle #

The derivation makes two statements about bijugate material that are sharp enough to be wrong, and it is worth separating them from the parts that are arithmetic.

A bijugate plant deep in its branch is at 68.754°, not at 68.75° or 69°. The limit is exact and the fork sequence approaches it alternately from above and below, so a specimen near the sixth fork should be within a fiftieth of a degree. Measuring a divergence to that precision on a plant is hard and has been done, and the reported values for teasel are consistent with it without being precise enough to confirm the last digit.

A bijugate plant’s counts are twice consecutive Fibonacci numbers and are never anything else on the Fibonacci branch. 4 and 6, 6 and 10, 10 and 16, 16 and 26, 26 and 42, 42 and 68. Note what is absent: 2 and 6, or 4 and 10, or any pair whose halves are not adjacent in the sequence. A specimen counted 4 and 10 would be bijugate and not on the Fibonacci branch, which is a distinguishable and reportable thing.

Neither of those is checkable here, and the reason is the same one recorded across this site’s wrong field: there is no dataset. Every frequency and every angle on these pages is a statement about the geometry, and the one thing this site cannot do and most wants to is compare them with a published survey that states its counting radii.

What is not settled #

Two honest limits, both about the relation between the arithmetic and the plants.

Nothing here says why a plant is multijugate. The construction takes $k$ as given. Whether an apex produces one primordium at a time or two is a question about the mechanism, and the mechanism essays on this site are careful to say that a model that reproduces a pattern has not explained it. A ring of cells with a selected spacing will produce some number of peaks; whether it produces them in pairs is not something the linear analysis says.

And nothing here says why a bijugate plant should be deep in its own branch rather than shallow. The limit 68.754° is where the fork sequence goes; a real plant is at a particular fork or between two of them, at whatever rise its geometry gives it. Teasel being near the limit is the same kind of fact as a sunflower being near 137.5°, and the same explanation is owed for both — which is a question about histories rather than about geometry, and it is the subject of the last group of essays in this phase.

What has been established is narrower and, on a site about over-claimed numbers, more useful: the famous constant is one member of an indexed family, every member is a noble number, every member is the limit of a rational fork sequence that never reaches it, and which member a plant is at is decided by how many primordia it makes at a time.