Phyllotaxis
the spirals are counted, not admired
Depth

Series — page 3

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.
Which patterns grow on a ring of circumference 0.80. Modes 2 to 8 have positive growth rates and mode 4 is fastest. Integrating the full equations from a disordered start gives 4 peaks.

Mechanism claims

  1. 3 What a mechanism would have to show
  2. 4 A growing organ is part of the rule
  3. 5 A comb is evidence of a rule
3 essays · mechanism What becomes of a seeded branch at 65 nodes per rung. Each bar is 10 runs at one amplitude, divided by what the blind counter found at the top of the stem. With no noise this rate ends on the Lucas branch. Placement noise displaces the node after the rule has chosen; field noise perturbs the energy the rule chooses over. Across 160 runs, 1 reached the Fibonacci branch with the lattice intact.

Noise amplitude

  1. 5 Noise is not a slow rate
  2. 6 The fragility belonged to the window
  3. 7 A disturbance the organs share
3 essays · emergence A round trip on four heads of 400 primordia: the divergence angle recovered from each. The counter is shown the points and nothing else. The worst recovery across the four is 0.035°.

Recovery

  1. 3 Recovering the angle from the counts
  2. 4 A head displaced before it is counted
  3. 5 A twist is a divergence
3 essays · lattices A stem grown at 42 nodes per rung. 167 nodes, each placed where the repulsion from the ones below it was least, with the rise falling from 0.2 to 0.0045. Counted blind in a sliding window the pattern walks 1/2 → 2/3 → 3/5 → 5/8, and the marks are where its answer changed.

Rising

  1. 3 A pattern with a rate
  2. 4 A shoot too fast to remember
  3. 5 A window inside a rung
3 essays · emergence 19 changes and 2 at the coarse step, 21 and 2 at half of it. The changes of surviving family each band makes, at the step the sweep has always used and at half of it. The golden 8/13 band goes 19 to 21 and the golden 5/8 band 2 to 2. Halving the step cannot lose a change, since the fine grid holds every coarse rise with the same answer; what it can do is find one, and it does so on one band and not on the other. So whether a tally is a floor or a total is a property of the band rather than of the sweep — a floor where there are islands narrower than the step, and exact where there are none.

Sweep resolution

  1. 1 A count or a floor
  2. 2 New islands or old edges
  3. 3 The last unmoved setting
3 essays · cylinder How the largest gap behaves as the head fills, on the interior's scale, from 150 to 2000 organs. On the interior's scale, the rational angle's gap grows by a factor of 3.8 over this range, from 2.48 to 9.38 spacings; the golden angle's runs from 0.841 to 0.844, decided at radius 0.871 at every size, and 137.3° reaches 0.863. A rational angle's gap is unbounded and an irrational one's is not, which is a claim about growth rather than about a value at any one head.

Gapgrowth

  1. 3 The gap that grows
  2. 4 One over root two
2 essays · tissue Which neighbours decide where an element goes. Each line is one exponent: how much each shell of neighbours makes the energy profile vary around the circumference, divided by what the nearest shell contributes. At p = 0.5 the nearest shell leads the next by a factor of 1.1 and a node is placed against the whole neighbourhood at once. At p = 3 it leads by 9.7e+3, and a node is placed against its immediate neighbours — which is what a lattice is.

The range of the interaction

  1. 3 How far a primordium reaches
  2. 4 A neighbourhood is a hypothesis
2 essays · emergence The 10 sequences the settling table's destinations belong to. Every pair the counter returns is two consecutive terms of a sequence in which each term is the sum of the two before it, which is the ladder's own rule. two of these are the sequences plants are observed to follow. The rest are not, and they are not rare: the 1, 4, 5, 9, 14 sequence and the 2, 5, 7, 12, 19 sequence each supply several destinations, at every falloff exponent the table is grown at. The right-hand column is the divergences that land on each.

Off-ladder

  1. 1 Ten sequences, two of them the ladder's
  2. 2 A list that was a rounding
2 essays · emergence Closest pair across 120–155° at 400 organs, read both ways. On the interior's scale the golden angle reads 0.9027 and ranks 1st of 72, against 0.9026 for the best grid angle at 137.5°, with the window running from 0.0668 to 0.9026; counting the rim's cells the golden angle reads 0.7076 and ranks 2nd of 72, against 0.7129 for the best grid angle at 137.5°, with the window running from 0.0331 to 0.7129. The dashed upright is the golden angle, which a grid of decimal degrees never lands on and which is therefore read separately.

Packing

  1. 3 Packing, measured four ways
  2. 4 Packing, measured against the interior
2 essays · tissue A shell section at W = 2.40, D = 0.42. W·D = 1.01, so the whorls are free of each other — an evolute shell, like a ram's horn or a planispiral ammonite. Both are things animals grow.

Raup

  1. 3 Raup's three numbers
  2. 4 What the axis distance costs
2 essays · shells A lattice survives about 1.8° of scatter, whichever way the noise arrives. The largest divergence scatter at which a run is still a lattice, from each kind of noise at the largest amplitude that leaves one. The two routes share no code below the placement rule: one displaces the node after the choice, the other perturbs the energy the choice is made over. They agree to 0.33°.

Scatter tolerance

  1. 5 Two degrees of scatter
  2. 6 The boundary belongs to the pattern
2 essays · emergence The same rule at p = 1, cut off at two distances. The top 220 nodes of two stems grown by an identical rule whose energy does not converge. Allowed to see 3/√h neighbours it produces 8/13 at 137.62° with 0.58° of scatter — a lattice no test here would question. Allowed 12/√h it produces 44° of scatter and no pattern. The truncation was doing the work.

Truncation

  1. 5 A window that makes a pattern
  2. 6 A hard edge is not a falloff
2 essays · emergence Two runs of the same rule from unrelated starting angles. Both settle at 137.0°, within 0.5° of the golden angle, from seeds 166° apart.

Analogue

  1. 3 Droplets with no biology in them
1 essay · emergence 48 cells with a carrier that pumps auxin up the gradient. Each short line is one cell's polarisation — the neighbour it pumps towards, which is always the richer one. 10 peaks come out, at a contrast of 93%, from a start that was uniform to within 6%.

Auxin

  1. 2 A pump that works uphill
1 essay · mechanism What the model settles on, against how fast the meristem grows. A broad golden branch, a transition, and then the two-whorl regime at exactly half a turn. 9 of 27 converged settings land within 4° of the golden angle; 15 land more than 20° away.

Bifurcation

  1. 3 The bifurcation diagram
1 essay · emergence Two paths down the same tree. Both start at the same first fork. Keeping the larger family every time reaches 137.473°; one different choice reaches 99.549°. Neither angle is in the arithmetic — both are limits of a path.

Branch tree

  1. 4 The tree and the attractor
1 essay · emergence Two laws, two tilings, and they disagree about which tiling is tissue. Lewis's law wants disorder: its slope is 0.231 on the random set and 0.009 on the golden head. Aboav's relation wants order: a = 1.18 on the head, 0.59 on the random set.

Cell laws

  1. 4 Two laws that want opposite tissue
1 essay · tissue A cone unrolled: 105 nodes at 137.51°, sector 126°. The counter is shown these coordinates and the sector angle and nothing else. Near the apex it finds 3 and 5; near the base, 8 and 13. The rise falls as one over the distance from the apex, so the pair has to climb.

Cone

  1. 2 A cone has a rise that falls
1 essay · cylinder 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

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