The shape and the law

The previous essay ended with a sentence that reads like a conclusion and is really a prediction about objects nobody has built.

Write the local circumference of a surface of revolution as a power of arc length along the meridian, $C(s) = k s^{a}$, and let the elements arrive under one of two laws. If the organ elongates — a constant meridian step between one element and the next — the rise is $\Delta/C \propto s^{-a}$. If it fills — a constant area per element — the step itself is $A/C$, so the rise is $A/C^2 \propto s^{-2a}$. Either way the rise falls as $s^{-p}$ and the transitions are a factor of $\varphi^{2/p}$ apart, with

$$p = a \ \text{(elongating)}, \qquad p = 2a \ \text{(filling)}$$

A cone that elongates is $a = 1$, $p = 1$, ratio 2.618. A disc that fills is $a = 1$, $p = 2$, ratio 1.618. Those are the two the site already had, and they are the same surface under two different laws.

Building the family #

To test a formula about exponents one needs surfaces at several exponents, and they have to be comparable.

Five were built. All five place their elements at $\theta_i = i\delta$ with $\delta$ the golden divergence; all five start at the same arc length; all five differ only in $a$ and in which law supplies the step.

The worst disagreement is 0.8%, on the disc, and it is a resolution effect rather than a discrepancy: the disc passes its measurable transitions within a factor of three in arc length, so each change-point’s position is a larger fraction of the spacing between them. The cone spreads four change-points over a factor of eighteen and lands within 0.04%.

Every number in the right-hand column comes from the same procedure. The lattice is built. A counter that receives arc length, position around, and local circumference — and no divergence, no exponent and no law — is walked up the axis in a hundred or more narrow bands. Every place its answer changes is recorded. The ratios of consecutive change positions are averaged geometrically.

The two degeneracies, which are the point #

Read the table by rows and it confirms a formula. Read it by pairs and it says something the formula alone does not.

Rows one and two are different objects with the same answer. A cone that gets longer and a paraboloid that fills outward have nothing in common as surfaces — one is developable and the other is not, one has a straight meridian and the other a square-root one, one adds material at a constant rate along its axis and the other at a constant rate per unit area. They have the same $p$, and the blind counter, walked up each, returns the same spacing to three decimal places.

So the transition spacing does not identify the shape.

Rows three and five are the same object with different answers. Both are $a = 1.5$; one elongates and one fills; the measured spacings are 1.914 and 1.380. Filling doubles the exponent, and doubling the exponent takes the square root of the ratio.

So the transition spacing does not identify the growth law either.

What it identifies is the product, and only the product. That is a real limitation on what can be inferred from a specimen, and it is the kind of limitation worth having explicit, because a measured spacing of 2.6 on a real organ invites the sentence “so it is a cone” and the sentence is not available.

What has to be held fixed for a comparison to mean anything #

The five surfaces are compared at the same starting rise, and that is not a cosmetic choice. Getting it wrong produced the most confusing failure of this work.

The obvious way to build the family is to fix the scale constant $k$ and let each surface be whatever it is. Done that way the five surfaces begin on wildly different rungs of the ladder: a trumpet with $C = k s^2$ at the same $k$ as a cone has a circumference a hundred times larger by the time it is measurable, so its rise is a hundred times smaller and its parastichy counts are in the hundreds before the first band contains enough elements to count in.

That is not a disagreement, it is a silence — and it presented as one. The counter compares index offsets up to a stated order and past that order it answers with the shortest offset it is allowed to consider, so every band of the trumpet returned the same truncated answer and the surface reported no transitions at all. A formula about the spacing of transitions, tested on a surface with none, is a formula that has been neither confirmed nor refuted.

The fix is to set $k$ from the rise at the first element rather than to state it: given a wanted starting rise $h_0$, take $k = \Delta / (h_0 s_0^{a})$. Every surface then begins on the same rung, and a comparison between two of them is a comparison of exponents rather than of arbitrary units.

The general form of that lesson has appeared on this site before, in the essay about the bifurcation sweep’s lower bound: a model that stops working over part of its range does not announce it, and the output is well formed the whole way. Here it was a whole surface reporting nothing, and it looked exactly like a surface with nothing to report.

The metric, and why a general surface is harder than a cone #

A cone is developable. Its surface can be unrolled onto the plane without stretching, so distances on it are exactly distances in the unrolled sector, and the previous essays use that exact answer.

Almost nothing else in the family is. A paraboloid has curvature, a trumpet has curvature, and there is no unrolling to appeal to. So the general counter uses the local metric instead: two elements separated by $\Delta\theta$ around and $\Delta s$ along the meridian are

$$\sqrt{\left(\bar{C}\,\Delta\theta / 2\pi\right)^2 + (\Delta s)^2}$$

apart, with $\bar C$ the mean local circumference. That is the first fundamental form, and it is exact in the limit of nearby points.

Whether “nearby” is near enough is a question with an answer, and the cone is where the answer is available: on a four-hundred-node cone the local metric and the exact developable one disagree by at most half a per cent over every parastichy hop of every offset. A cone is the worst case in the family in one specific respect — its circumference grows as fast as any surface here in proportion to the distance the hop covers — so half a per cent is an upper bound rather than a typical value.

And the quantity that has to survive is not a distance but an ordering: which offset is shortest. Half a per cent moves an ordering only where two offsets are within half a per cent of each other, which is precisely at a transition, and precisely where the counted answer is fuzzy for reasons that have nothing to do with the metric.

What the family will not reach #

Two corners of the parameter space are not measurable and both refuse for the same structural reason, which is worth stating because it bounds the claim.

The counts a surface reaches grow as $h^{-1/2}$, which is $s^{p/2}$, while the number of elements available to count them in grows with the arc length. For an elongating surface the element number is proportional to $s$, so the counts grow as $i^{a/2}$ and the elements as $i$ — which is fine as long as $a < 2$. At $a = 2$ the two grow at the same rate and the pattern outruns the counter: by the time a band holds enough elements to count in, the parastichy numbers are in the hundreds and every band returns the counter’s order limit.

For a filling surface the arithmetic is friendlier — the counts grow as $i^{a/(a+1)}$, which is always slower than $i$ — so the filling half of the family is measurable at every exponent tried.

The other corner is small $p$. At $p = 0.5$ the predicted ratio is $\varphi^4 = 6.854$, so consecutive transitions are nearly seven times apart in position, and a lattice long enough to hold three of them past the point where bands become countable needs tens of thousands of elements. The prediction is not in doubt there; it is simply not tested, and the table says $p \geq 1$ for that reason rather than by preference.

Where the cylinder went #

The cylinder is missing from the table and its absence is not an oversight; it is the degenerate member and it is worth taking properly, because it explains what the exponent means better than any of the five entries do.

A cylinder has $a = 0$: its circumference does not depend on position along its axis. Under elongation that gives $p = 0$, a rise that does not fall, and a transition spacing of $\varphi^{2/0}$, which is to say no transitions anywhere. Under filling it gives the same thing, because a constant area per element on a surface of constant circumference is a constant step. A cylinder cannot be made to climb the ladder by changing how it adds material, and that is the sharpest statement of what is special about a stem.

Every other surface here climbs, and the only question is how fast. So the family is not five cases plus a cylinder; it is a one-parameter continuum in which the cylinder sits at one end and the ratio runs from infinite down through 6.854, 2.618, 1.900, 1.618, 1.378 and onward toward 1 as the exponent grows. A ratio approaching 1 means transitions arbitrarily close together, which means an organ whose counted pair is never stable anywhere — a surface flaring fast enough that the pattern is always in transit.

Nothing biological is at that end, and there is a reason to think nothing could be: a pattern in permanent transition is a pattern in which three parastichy families are always nearly equally short, which is the condition under which small disturbances change which two win. The stable end of the family is the countable end, and the countable end is where plants are.

The counter, and what it is not told #

Three counters now exist on this site and it is worth putting them side by side, because the differences between them are exactly the differences between the geometries.

The disc counter takes points with $(x, y)$ and a radius window, measures Euclidean hop lengths for each index offset, normalises by the local spacing and takes local minima. It needs the local-minimum rule because on a disc offset 1 is always shortish and a bare “take the shortest” would answer 1 and 2 for every pattern ever made.

The cylinder counter takes points with a position around and a height, measures hops with the wrap, and takes the two shortest outright. It must not impose a local minimum, and the essay that records why is Counting up the stem: at a rise of 0.09 the two shortest offsets are 2 and 3, but 3 is not a local minimum of the hop curve, so the rule skipped it and reported a lattice that does not exist.

The surface counter here takes arc length, position around, and the local circumference. It uses the local metric, takes the two shortest, and is otherwise the cylinder counter. What it receives about the surface is one number per element — how big the circle it sits on is — and nothing about the rule that put it there.

That last point is what makes the measured column of the table a measurement. The counter cannot know whether it is on a cone or a paraboloid; it cannot know whether the elements arrived by elongation or by filling; it cannot know the divergence angle, and it has no representation of Fibonacci numbers anywhere. It reports which two index offsets have the shortest median hop in a band, and the essay is about where that answer changes.

Which real organs sit where #

The family is a parameterisation, not a catalogue, and it is worth being careful about how much of a plant is in it.

The two anchored cases are real. A shoot apex that elongates at a steady rate on a conical receptacle is $a = 1$ elongating; a capitulum whose florets are added at a steady rate per unit area on a flat receptacle is $a = 1$ filling, which is Vogel’s model and is where the site’s seed-head figures come from.

Beyond those the mapping is looser than the arithmetic. A real shoot apex is a dome, closer to $a = 1/2$ near the tip and flattening below it; a real capitulum is not flat but slightly convex and becomes more so as it fills; a real cone is an ogive rather than a cone. Each of those is a surface whose exponent changes along its own axis, and a single $p$ is a fit rather than a description.

That is not a defect the model can be repaired out of. It is the reason the measurement proposed at the end of the previous essay — take the ratio of two transition positions — gives a local exponent rather than a global one, and why it is worth taking at more than one place on the same organ if the organ is large enough to allow it.

What the five surfaces buy #

The formula $\varphi^{2/p}$ was derivable in two lines from a fact the site already had, and a reader is entitled to ask what building five lattices adds to two lines of algebra.

Three things.

It turns an identity into a measurement. The conversion from transition rises to transition positions cannot fail, because it is a change of variable. Walking a counter up a built lattice and finding the same spacing can fail, and would have failed had the rise law been misderived, had the metric been too coarse, or had the counter’s order limit been left where it was. All three of those were live and one of them actually bit.

It exhibits the degeneracy rather than deducing it. That two different surfaces share a ratio follows from the formula, but a formula’s consequences are easy to skim past. Two dots sitting on the same point of a curve, labelled with two different objects, are harder to skim past — and the inferential mistake they prevent is one a reader will otherwise make in a sentence.

And it found the exponent the family cannot reach, which no amount of algebra would have volunteered. $a = 2$ elongating is not a case the formula excludes; it is a case the instrument cannot see, and knowing which cases those are is part of knowing what the instrument is worth.