Transitions a factor of φ² apart

The rungs of the ladder are equally spaced, in the only sense in which anything about this subject is equally spaced: as a ratio.

The Fibonacci ladder measured the transition rises at the golden divergence and found 0.12472, 0.04767, 0.01822, 0.00696, 0.00265 and 0.00101 — consecutive ratios between 0.3817 and 0.3828 against $1/\varphi^2 = 0.381966$. The ladder is geometric, the common ratio is $1/\varphi^2$, and nothing was fitted to make it so.

Everything in this essay follows from that one fact and from a change of variable.

The change of variable #

Write the rise of a surface as a power of position along its axis,

$$h(z) \propto z^{-p}$$

and the transitions follow immediately. If consecutive transition rises satisfy $h_{k+1}/h_k = \varphi^{-2}$, then the positions at which they occur satisfy $z \propto h^{-1/p}$, so

$$\frac{z_{k+1}}{z_k} = \left(\varphi^{2}\right)^{1/p} = \varphi^{2/p}$$

Two geometries the site already has fall out of this at once.

A cone. The rise is $\Delta/2\pi z\sin\alpha$, so $p = 1$ and the transitions are a factor of $\varphi^{2} = 2.618$ apart along the axis.

A Vogel disc. The rise is $c^2/4\pi r^2$, so $p = 2$ and the transitions are a factor of $\varphi = 1.618$ apart in radius.

A cylinder. The rise does not depend on position at all, so $p = 0$, the ratio is infinite, and there are no transitions. Which is the result the cylinder essays are built on, arriving here as a degenerate case rather than as a separate fact.

That is the whole derivation. What remains is to find out whether it describes anything.

Predicting, then measuring #

There are two quite different things one can do with the relation and both are worth doing, because they fail in different ways.

The first is a conversion. Take the ladder’s transition rises, feed each through the cone’s own $h(z)$, and read off the slant distances. At a flare of 0.35 with unit step that gives 3.64, 9.53, 24.95, 65.34 and 171.12, with consecutive ratios of 2.6190 — against $\varphi^2 = 2.6180$.

Four ratios agreeing to four significant figures looks stronger than it is, and the small discrepancy is the interesting part of it. The ladder’s transition rises are found by sweeping the rise on a geometric grid and recording where the dominant pair changes, so each is located to within one grid interval and each carries the same fractional uncertainty. A ratio of two such numbers inherits a bias that does not shrink as more transitions are taken, which is exactly what the four identical ratios show: the answer is 2.6190 because the grid says 2.6190, not because the geometry does. Refine the grid and it walks toward 2.6180.

So the conversion is a consistency check on arithmetic, not a measurement. It could not have failed except by algebra.

The second thing is a measurement, and it can fail. Build the lattice on the surface. Walk a blind counter up its axis in many narrow bands. Record every position at which the counter’s answer changes. Take the ratios of those positions.

Done that way, on an elongating cone of six thousand nodes, the transitions land at 297, 759, 2084 and 5324, and the mean ratio is 2.617 against $\varphi^2 = 2.618$. Done on a filling disc of the same size, the transitions land at 39, 64 and 104, and the mean ratio is 1.631 against $\varphi = 1.618$.

Nothing in the counting knows either number. The counter is given coordinates and a local circumference; it returns whichever two index offsets have the shortest median hop; and where its answer changes is where it changes.

Why 1/φ² and not something else #

The ratio the whole essay is built on deserves a sentence, because it is the one place the golden ratio enters and it enters for a reason that has nothing to do with packing.

At the golden divergence the lattice vector of offset $j$ has an “around” component $\langle j\delta\rangle$ — the distance from $j\delta$ to the nearest whole turn. For $\delta = 1/\varphi^2$ turns, those are smallest at the Fibonacci offsets, and the sequence of smallest values shrinks by a factor of $\varphi$ at each Fibonacci step, because that is what the continued fraction $[0;2,1,1,1,\dots]$ says. A transition happens when the vertical part of the shorter vector catches up with the horizontal part of the longer one, and since the vertical part is $jh$ with $j$ growing by $\varphi$ and the horizontal part shrinking by $\varphi$, the rise at which they balance falls by $\varphi^2$.

So $\varphi^2$ is not the spacing of the transitions because the golden angle is optimal at anything. It is the spacing because $\varphi$ is the growth rate of the Fibonacci numbers and it appears twice — once in the index and once in the offset. At the Lucas divergence the same argument gives the same factor, for the same reason, with Lucas numbers in place of Fibonacci ones. The claim that survives is about approximability, and this is not it.

The disc, restated #

The disc’s factor of $\varphi$ was already in A disc is a cylinder, stated as “every time a head’s radius grows by 62%, its counts advance one rung”. It is worth restating with the cone beside it, because the pair of numbers says something neither says alone.

A sunflower’s countable region — outside the undifferentiated middle, inside the packed rim — spans perhaps a factor of four in radius. That is $\log_\varphi 4 \approx 2.9$ rungs, so three transitions, which is what the counts-change-with-radius figure finds by counting.

A conifer cone’s countable region spans perhaps a factor of four in length, from the first fertile scale to the base. That is $\log_{\varphi^2} 4 \approx 1.4$ rungs, so one transition, sometimes two.

The same organ size, the same divergence, half as many changes. A cone is countable once and a head is not, and the ratio between “once” and “three times” is the ratio between one power and two in the rise law.

Where the two curves cross #

There is a case where the cone and the disc coincide exactly, and it is worth having because it shows what the exponent is really measuring.

The rise at the $i$-th node of an elongating cone is $1/(2\pi i \sin\alpha)$ — the step cancels, as the previous essay shows. The rise at the $i$-th element of a Vogel disc is $c^2/4\pi r^2$ with $r = c\sqrt i$, which is $1/(4\pi i)$ — and $c$ cancels the same way.

Both fall as one over the element number, and they differ only by the constant $2\sin\alpha$. So in element number the two geometries have identical ladders, and a cone with a flare of exactly 0.5 has the same rise at its hundredth node as a disc has at its hundredth element.

What differs is where those elements are. A cone’s are spread along a length, so element number is proportional to distance; a disc’s are spread over an area, so element number is proportional to radius squared. The exponent $p$ is not a statement about the ladder at all. It is a statement about how an organ maps element number onto space, and everything about transition spacing follows from that map.

What a disagreement would have looked like #

A measurement that cannot fail is not a measurement, so it is worth saying what failure would have been, since the arrangement here is unusual enough that the reader is entitled to suspect it of being circular.

The counter is walked up the axis in narrow geometric bands and reports two index offsets per band. Three separate things could have gone wrong, and each has a distinctive signature.

The answer could have failed to be geometric at all. Nothing in the counting enforces a constant ratio between the positions of successive changes. Had the rise law been misderived — had the circumference been taken as $2\pi z$ rather than $2\pi z \sin\alpha$, say, or the meridian step confused with the axial one — the transitions would still have been found, and their spacing would have drifted rather than held. The signature of that is ratios that trend, and the measured ratios do not: on the cone they are within a per cent of each other across a factor of eighteen in position.

The answer could have been geometric with the wrong ratio. This is the failure that would have been most informative, because the ratio is the thing the derivation predicts and the derivation has an exponent in it that is easy to get wrong by one. A cone whose transitions came out a factor of 1.6 apart would have said that the rise falls as one over position squared, which would have meant either that the model was filling rather than elongating or that the surface was not the surface intended.

The counter could have run out. It compares index offsets up to a stated order, and past that order it silently answers with the shortest offset it is allowed to consider. That failure did occur during this work and it produced a picture that argued the opposite of the truth: with the order limit at its default of 120, the disc’s ladder appeared to stop at element 537 while the cone’s continued, so a figure comparing how many transitions each geometry passes drew the disc as the poorer of the two. Raising the limit to 400 removed it. The symptom was not an error message; it was a plausible-looking curve, and it was caught because the two geometries were known in advance to have the same ladder in element number and the figure said they did not.

That third one is the shape of failure this site keeps finding, and it is worth naming again: the machinery gives an answer, the answer is well formed, and it is the answer to a smaller question than the one asked.

Measuring this on something real #

The relation is cheap enough to use on a specimen, and it is worth writing down what that would take, because the ingredients are unusually modest.

What is needed is the positions of two transitions along one organ. Not the divergence angle, not the scale count, not the shape: two places where the parastichy pair changes, and their distance from the tip.

Take their ratio. If it is near 1.6, the organ’s rise falls as one over position squared, which for a surface whose circumference grows linearly means it is filling by area. If it is near 2.6, the rise falls as one over position, which for the same surface means it is elongating at a steady rate. If it is near 6.9, the exponent is a half, which means either a much blunter surface or a much slower one.

That is a measurement of a growth law made by counting spirals in two places, and it does not require watching the plant grow. It is the same trick as recovering a growth factor from a drawn shell: a developmental history leaves a geometric signature in a finished object, and the signature can be read off without the history.

The honest caveats are three. A single ratio from two transitions has no error bar, and the transitions themselves are fuzzy — near one, three offsets are nearly equally short and the counted answer depends on where the counting band’s edges fall. Real organs change shape along their length, so a single exponent is a fit rather than a fact. And an organ with only one transition in its countable region gives no ratio at all, which is the usual case for a cone and the reason a seed head is the better specimen for this particular measurement despite being the worse one for everything else.

Two ways to be wrong about this #

The relation is simple enough to be misused in two directions, and both misuses are the kind that produce a confident wrong sentence.

The spacing does not identify the shape. A ratio of $\varphi^2$ means $p = 1$, and $p = 1$ is a cone that elongates or a paraboloid that fills, or anything else whose rise happens to fall as one over position. Measuring the spacing on a real organ constrains the product of its shape and its growth law, not either separately. The next essay builds five surfaces that make this concrete, two of which are different objects sharing a ratio.

The spacing is not a property of the plant’s divergence. It is $\varphi^{2/p}$ at every divergence whose ladder is geometric with ratio $1/\varphi^2$, which includes the Lucas branch and every other noble one. A cone at 99.5° has its transitions a factor of $\varphi^2$ apart too, at different rises and between different pairs. So a measured spacing of 2.6 says something about the organ’s geometry and nothing about which branch it is on.

That second point matters because the golden ratio appearing in a measured spacing is exactly the kind of result that gets reported as evidence for the golden angle. It is not. It is evidence that the organ’s rise falls as one over its axis, which is a statement about how it grows.

What the measurement does not resolve #

Two limits, both of which are visible in the numbers rather than argued around.

The disc’s measured ratio is 1.631 against 1.618, and the cone’s is 2.617 against 2.618. The disc is out by 0.8% and the cone by 0.04%, and the difference between them is not a difference in the geometry. It is a difference in how well a transition’s position can be located: the disc passes its transitions within a factor of three in radius, so its three change-points are crowded and each one’s position is a larger fraction of the spacing. The cone spreads four change-points over a factor of eighteen. More room, better ratio.

Both are measured on lattices, not on plants. Every number here comes from a point set built by a rule. A real cone’s scales are not points, its shape is not a cone, and its internodes are not constant — three departures taken up separately in the cone essay’s closing section and in the rising-phyllotaxis essays.

What has been established is narrower than “conifer cones have transitions a factor of 2.6 apart” and more useful than a restatement of the algebra: two independent calculations — a periodic lattice’s ladder, and a blind count walked up a built surface — agree on a number that neither was fitted to, at two different exponents, and the disagreement between them is a resolution effect that shrinks when the lattice is made longer.

The generalisation, stated #

Everything above is the special case $p \in \{0, 1, 2\}$ of one sentence, and the sentence is worth putting down before the next essay tests it.

Let the local circumference of a surface of revolution be $C(s) = k s^a$ in terms of arc length along the meridian. Then:

• if the organ elongates — a constant meridian step between elements — the rise is $\Delta/C \propto s^{-a}$, so $p = a$;
• if it fills — a constant area per element — the step is $A/C$ and the rise is $A/C^2 \propto s^{-2a}$, so $p = 2a$.

A cone that elongates: $a = 1$, $p = 1$, ratio 2.618. A disc that fills: $a = 1$, $p = 2$, ratio 1.618. Filling doubles the exponent, and doubling the exponent takes the square root of the ratio.

Which is a prediction about surfaces nobody has built yet, and therefore something to build.