Phyllotaxis
the spirals are counted, not admired
Depth

Series — page 4

A field says what an essay is about. A series follows one idea essay by essay — from the question that introduces it to the one that assumes all the others.
The rise falls as one over the distance from the apex. A straight line of slope −1 on log axes. The dashed horizontals are the transition rises of the cylinder's ladder, computed with no cone anywhere in them; where they cross, the count changes. Consecutive crossings are 2.62, 2.62, 2.62, 2.62 apart — φ² is 2.618.

Cone transitions

  1. 3 Transitions a factor of φ² apart
1 essay · cylinder A cell's neighbours are its spiral families. Left: part of a 900-point golden, 137.508° head, with every contact between two cells drawn and coloured by the difference between the two nodes' placement indices. Right: the share each difference takes, across all 1903 contacts between interior cells. They are the parastichy numbers — 34, 55, 21, 89, 13, 8 — and the pair a person would count is 34 and 55. The six sides Euler forces are shared out among four of them, 5.72 edges per cell.

Contact network

  1. 1 The six are the spirals
1 essay · tissue Fibonacci at a rise of 4.8e-3, asked two ways. Choose a divergence at random and a static lattice at this rise gives a consecutive Fibonacci pair 10.8 per cent of the time. Start coarse at a divergence nobody chose, grow the stem down to the same rise, and it is 100 per cent of 16 runs. The geometry is not generous; continuity is.

Continuity

  1. 5 Continuity from a coarse start
1 essay · emergence A periodicity reports a different partner every time. eight kinematic lattices, differing only in the seed of their disturbance, each read by the same instrument. The disturbance repeats every 8 organs at a weight of 0.9: it puts a strong comb at spacing 8 — 0.75 against a band of 0.07 — and the partner it names is 8/10, 8/12, 8/11 across the 8 stems and never 8/13, which is what the position counter finds in every one of them. There is no placement rule in any of these arrangements.

Control spec

  1. 1 The control a survey would need
1 essay · wrong Six stems built, forgotten and recovered. Each row is a lattice built from a divergence and a rise, counted by machinery shown only the coordinates, and reconstructed from the counts and the two hop lengths. The worst error in the recovered angle is 3.0e-13°.

Cylinder recovery

  1. 3 Two numbers out of the points
1 essay · cylinder The spiral counts four different divergence angles produce. Fibonacci counts come from one angle. The Lucas angle — which the same dynamical model reaches on a different branch — gives 47 and 76, and neither number is a Fibonacci number.

Fibbranch

  1. 3 Fibonacci is a branch, not a law
1 essay · wrong One exponent fitted to an organ that has 4 of them. Each dot is one step between consecutive rings, reporting 2 ln φ / ln(s′/s) — the exponent that step would have if the organ had one. They run from 1.980 to 1.697. The line is what a single fit returns, 1.891, which is their harmonic mean of 1.880 and sits below their plain average of 1.887.

Fitted exponent

  1. 5 What one exponent reports
1 essay · cylinder The plane of stems: divergence across, rise up. Each shade is one parastichy pair. The marked points are the lattices where three families are equally short — the forks — and the Fibonacci ones run up the middle towards 137.51°.

Forks

  1. 4 The forks are exact
1 essay · cylinder What a divergence picked at random gives, at a rise of 0.100. Fibonacci pairs take 59.6% of the circle at this rise, and the share falls as the rise does. The claim that Fibonacci counts are what nature "prefers" needs the preference to come from somewhere, and it is not from the geometry being generous.

Frequency

  1. 4 How often is it Fibonacci
1 essay · wrong What the "whorled" bucket contains, at a rise of 0.008. 7 pairs, sharing 7 different factors, and every one of them is k and 2k. The bucket the previous census called whorled is the coarsest pattern the ladder has, repeated k times around the stem — not a residue of odd arrangements.

Jugate census

  1. 4 What "whorled" was hiding
1 essay · wrong Every transition as the rise falls. The pair climbs 1/2 → 2/3 → 3/5 → 5/8 → 8/13 → 13/21. Consecutive transitions are 0.382, 0.382, 0.383, 0.382 of the previous rise — 1/φ² is 0.3820.

Ladder

  1. 3 The Fibonacci ladder
1 essay · cylinder A head of 200 primordia at a divergence of 137.51°. Nothing is placed by hand: the nth point sits at n·137.51° and radius √n. The closest any two points come is 1.60 of the mean spacing.

Lattice

  1. 1 A head is a set of points
1 essay · lattices Where this implementation stops converging. Below about G = 0.18 the settled angle wanders over 113° however long the run. That is the model's limit, not a fact about plants.

Modellimit

  1. 3 Where the model stops
1 essay · emergence What is visible in the outer part of a 4000-element organ. Both surfaces have the same ladder in element number — the rise is 1/(2πi·flare) on a cone and 1/(4πi) on a disc, and c and the internode step both cancel. What differs is where the elements are. Counting outside 50 per cent of the extent, a cone shows 1 change and a disc 2, because half a cone's length holds half its elements and half a disc's radius holds three quarters of them.

Organ size

  1. 4 Why a cone can be counted once
1 essay · cylinder Which patterns grow on a ring of circumference 1.40. Modes 4 to 15 have positive growth rates and mode 8 is fastest. Integrating the full equations from a disordered start gives 8 peaks.

Reaction–diffusion

  1. 1 Turing's last problem
1 essay · mechanism At 1.0 per cent on each ring, 5 rings show the drift. The gaps between consecutive rings are 1.626, 1.631, 1.657, 1.763. Two rings give one gap and no way to disagree with itself; three give two gaps and a fit with nothing left over. The question is how many gaps it takes for their spread to exceed what the measuring error can explain, and the answer depends on the error as much as on the organ.

Ring identifiability

  1. 6 How much of a cone to measure
1 essay · cylinder The angle between the first two peaks, for six starting disorders. The same equations, the same ring, six different starting perturbations: 177°, 47°, 109°, 151°, 47°, 133°. A spiral needs the angle between successive elements to be the same one each time, and a stationary ring does not supply that.

Ring limit

  1. 2 A ring cannot make a spiral
1 essay · mechanism Where the disc's counts change, predicted from a cylinder. The dashed lines are the transition radii the cylinder's ladder gives through h = c²/4πr², with nothing fitted. The dots are what the blind counter returns from the disc: 15 of 16 bands agree, and the ones that do not sit on a transition.

Rise and radius

  1. 4 A disc is a cylinder
1 essay · cylinder The exponent sets the spacing, and only the exponent. The curve is φ^(2/p), drawn from the ladder's ratio of 1/φ² and nothing else. The dots are measured: each surface built, a blind counter walked up its axis, the places its answer changed recorded. A cone that elongates and a paraboloid that fills sit on the same point at p = 1 — so the spacing identifies neither the shape nor the way material arrives, only their product.

Shape exponent

  1. 4 The shape and the law
1 essay · cylinder The same counter, on a stem and on a disc. The stem returns 2 and 3 in all three bands. The disc returns 21/34, 34/55, 55/89 — three answers to one question, which is why a published count needs to say where it was taken.

Stem count

  1. 2 Counting up the stem
1 essay · cylinder

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