Phyllotaxis
the spirals are counted, not admired
Depth

Series — page 2

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.
Ten of the 24 orderings four exponents admit, from 15 readings of the wall. Every arrangement of the four falloff exponents is a cell, and a cell is filled when some level of some sampling puts the four walls in that order. 15 readings of runs that are shared cell for cell give ten of the 24, with the most common occurring three times. The marked cells are the orderings read at a half, the only level any round of this collection has published, and each of them occurs once. Exponent 5 is ranked first in 8 of the 15 readings and each of the others in two or three.

Falloff exponent

  1. 3 The level was doing the ordering
  2. 4 The exponent that barely matters
  3. 4 Four walls closer than they looked
  4. 5 Two shapes, one threshold
  5. 5 Two refinements that do not multiply
  6. +1 more
6 essays · mechanism One seed, two rates, two ladders. Both stems begin as forty nodes of Lucas lattice at a rise of 0.12. At 65 nodes per rung the divergence stays at 99.5° and the counts walk 1/3 → 3/4 → 4/7 → 7/11. At 131 it leaves for 137.7° and walks 1/3 → 2/3 → 3/5 → 5/8 → 8/13 instead.

Branch choice

  1. 4 The rate decides the branch
  2. 5 A stem on the other branch
  3. 6 Seven rises and two seeds
  4. 7 The front that reads one short
  5. 8 The hole on the other branch
5 essays · emergence Which rises are a lattice, from 0.04 to 0.13. How much the divergence wanders over the last sixty organs, at each rise up the coarse ladder. 13 of the 19 settle, scattering between 0.0000 and 0.3356 degrees. six do not: from 0.09 to 0.115 the divergence sticks on exactly 135.0000 degrees, which is three eighths of a turn, and wobbles about it by 0.79 to 1.60 degrees. A counter shown either kind returns the same pair, so the counts cannot tell them apart. The line is the threshold a rise has to pass before a cut is made on it, and it sits in the gap rather than among the measurements.

Coarse rung

  1. 1 A stem coarse enough to cut
  2. 2 The shallower front turns over
  3. 3 Half a turn, four at a time
  4. 4 The band was not the sampling
  5. 5 The slide a counter holds constant
5 essays · cylinder More organs removed, more stems that never come back. The share of arrangements at the 5/8 rung that never return to the divergence they were cut from, against how many organs the cut removed. One organ wrecks 2 of 8 arrangements and five wreck 63 of 64. The number of arrangements differs from bar to bar because a cut of five organs has more ways of being placed than a cut of one, and it is printed on each bar for that reason. What the dose decides is whether a stem falls off its lattice; where it lands when it does is decided by something else.

Dose

  1. 1 Three organs and no mirror
  2. 2 The share was not the thing
  3. 3 A wreck has a short list
  4. 4 A count with a factor in it
  5. 5 A file has to close
5 essays · emergence Every rung has a window it can be read in. For each rung of the ladder, the range of lag windows in which the pair can be read: at least three times the smaller parastichy number, so the main comb has three teeth inside the window, and less than three times the larger, so the larger number cannot itself be a candidate spacing. The band is [15, 24) at 5/8, [24, 39) at 8/13, [39, 63) at 13/21. It is non-empty at every rung, because it is empty only when the larger number is no larger than the smaller. The line at 30 is the window the earlier work used everywhere: it sits inside the 8/13 band and outside the 13/21 one, which is what was mistaken for a ceiling.

Instrument ceiling

  1. 1 The rung was not the instrument
  2. 2 What a sample grid decides
  3. 3 The grid was in the number
  4. 4 A period the grid invented
  5. 5 Matching instead of correcting
5 essays · cylinder The 2-jugate forks converge on 68.7539°. Every fork sits at a rational divergence, with denominator 4(m² + mn + n²) — 10/28, 30/76, 74/196 and so on. The limit is 68.7539°, which is 137.5078 divided by 2, and it is at none of them.

Jugate limit

  1. 4 Half the golden angle
  2. 5 A grown stem halves its ladder
  3. 6 A Lucas seed counts whorls
  4. 7 A seed measured in whorls
  5. 8 The window was not carrying it
5 essays · lattices An L-system after 4 rewrites of two rules. X → F[+X]F[-X]+X and F → FF, walked by a turtle turning 22.5°. 130 segments, and not one of them knows anything about light, water or auxin.

Lsystem

  1. 2 L-systems describe, they do not explain
  2. 3 A count set by a delay
  3. 4 A count that loses its growing points
  4. 5 What a scar is worth
  5. 6 Two ways to die, three things to count
5 essays · branching The settled divergence down the golden branch. Every rise from 0.07 down to 0.00482, plotted against the divergence the rule settles on, with each rung drawn in its own stroke and the branch's limit angle marked. The curve does not slide: it turns three times in four rungs, climbing across one and falling across the next, so a value it takes on one rung it takes again on another. That is what makes a matched pair possible — two rises, different counted pairs, one angle — and it is the whole reason the design exists on this branch. The widest excursions from the limit angle, coarse rung first, are 3.195°, 3.352°, 0.961°, 0.422°.

Same angle

  1. 1 The angle the ladder returns to
  2. 2 Two rungs, one angle
  3. 3 The angle is not the actor
  4. 4 Where the survivors meet
  5. 5 The last of three quantities
5 essays · cylinder Two windows on a shoot at 400 nodes per rung. A stem grown at 400 nodes to the rung with a disturbance of 0.25, its rise falling from 0.4 to 0.004 over 1914 nodes. The upper window is the last 250 internodes — where a count would be made on a real plant — and the lower is the same length shifted down 125. The upper reads 8/13; the lower reads 8/13. The verdict is agree, and the window holds 0.63 of a rung.

Two windows

  1. 1 Two windows on one stem
  2. 2 A refusal with a reason
  3. 3 A spread that grows with its window
  4. 4 The window nobody moved
  5. 5 Three rows a window moves
5 essays · emergence three stems: 1, 2, 3 primordia at a time. 1-jugate at 137.51° counts 2 and 3; 2-jugate at 68.75° counts 2 and 4; 3-jugate at 45.84° counts 3 and 6. Every count shares the factor k, and dividing it out leaves an ordinary lattice: what a k-jugate stem is, exactly, is k copies of an ordinary one wrapped k times round.

Jugacy

  1. 2 Two at a time
  2. 3 What a cut costs a whorl
  3. 4 The symmetry that is not there
  4. 5 A whorl that misses its share
4 essays · lattices A golden spiral and a nautilus spiral over 2.5 turns, from the same start. After 2.5 turns the golden curve is 7× larger. The growth factors are 6.85 and 3.2, a factor of 2.14 apart.

Nautilus

  1. 3 The nautilus question
  2. 4 What the septa count
  3. 5 The error budget for a nautilus
  4. 6 Three entries and one span
4 essays · shells Only noise that arrives before the choice can change what is chosen. Intact runs only, from the whole amplitude sweep. Placement noise displaces the node after the rule has picked an azimuth: 44 runs, none of which changed branch at any amplitude that left a lattice. Field noise perturbs the energy profile the rule picks over, so it can move the minimum into a neighbouring gap: 1 of 66 did.

Noise entry

  1. 5 Where the noise gets in
  2. 6 The noise that arrives through the neighbours
  3. 7 Which minimum was chosen
  4. 8 The neighbourhood was already settled
4 essays · emergence All 19 destinations, counted at 17 windows from 20 to 800 organs. One row per destination, one cell per counting window, coarsest on the left. A dark cell is a window returning the same pair as the standing 200; a warm cell is a window returning a different one; a pale cell is a window that reaches below the run's own settling organ and was dropped. 5 of the 19 rows differ anywhere, and every difference sits left of the floor its own pair sets — 20 organs for a pair under 14 parastichies and the larger count plus 6 above it. The census read at 25 organs and at 800 is the same census.

Read window

  1. 1 The window nobody varied
  2. 2 A plateau the instrument should have had
  3. 3 How many organs a pair needs
  4. 4 A sequence that was a reading
4 essays · lattices Where the pattern starts, measured at two run lengths. One row per wrecked cut. The small mark is the onset a 300-organ run reports and the ring is what a 600-organ run reports; a row with only one mark reports an onset at only one length. 19 of the 24 rows that report both move by more than twenty organs, and the largest move is from 135 to 435. The range the thread has been quoting, 7 to 303 organs, becomes 28 to 473.

Run length

  1. 1 Twice the run
  2. 2 An onset at the end of the run
  3. 3 Three rows change sides
  4. 4 A window nobody aligned
4 essays · mechanism How many sides the cells actually have. The mean is 5.908, which Euler's formula forces. The spread around it is not noise — a tiling of only hexagons cannot close up on a finite patch.

Sixsides

  1. 2 Why the average cell has six sides
  2. 3 The second moment is the measurement
  3. 4 A second moment that goes to zero
  4. 5 Nothing in the staircase moves
4 essays · tissue 135 fives and 129 sevens among 1631 interior cells. The side counts of every cell strictly inside a golden, 137.508° head of 2400 organs, cut at 86% of the radius. The fives and the sevens are counted apart rather than summed into a spread, because they are opposite charges and the sum hides them. Summed over the interior, 264 cells that are not hexagons carry a charge of +6. Over the whole patch the charge is 294, which is exactly 6 + 2·144 — a number fixed by the 144 cells on the patch's own boundary and carrying nothing whatever about the interior. The defects are not scarce; they are balanced.

Topological charge

  1. 1 An interior that is nearly neutral
  2. 2 The defects lie on rings
  3. 3 Every five is bound to a seven
  4. 4 The rings are not the transitions
4 essays · tissue Tracing one family: 4 chains. Every node is joined to the node one repeated displacement away, and the chains that result are drawn separately. There are 4 of them, which is the parastichy number of this family. No index of arrival was used anywhere, which is what lets the same count be made on a pattern where 2 primordia appear at once.

Chain counting

  1. 3 Counting without an index
  2. 4 A counter that cannot be slid
  3. 5 A band that follows the rise
3 essays · lattices A count of m and n pins the divergence to 221°/mn. Each dot is one reported pair, and its height is the total width of the divergence angles that could have produced it at some rise. 2/3 leaves 38.8° open; 34/55 leaves 0.118°. The line is 221°/mn, taken from the three highest pairs and drawn back through the rest.

Count value

  1. 5 What a count is worth
  2. 6 A count that can be wrong by one
  3. 7 A count that drifts by two
3 essays · wrong Six rungs walked at the grid, 10 crossings between them. Each row is one rung, drawn from its coarse end on the left to its fine end on the right and scaled to its own width so positions inside different rungs can be compared. The shaded stretch is the band the ladder grows around the rise it recorded as that rung's handover. five of the six rungs carry exactly one crossing, and on each of those the whole stretch a second one could sit in has been walked at the grid and closed at both ends. The 3/4 rung of the Lucas branch carries five, spaced 22, 12, 15 and 20 grid steps apart, and its band holds three of them. The rule on each row is a located crossing.

Handover grid

  1. 1 Five rungs walked
  2. 2 One crossing or two
  3. 3 The handovers corrected
3 essays · cylinder The version of the claim that does survive measurement. The golden angle scores 0.4377, against 0.3462 for the best of 938 other angles sampled. The dashed line is Hurwitz's 1/√5, which no number can exceed.

Hurwitz

  1. 4 The claim that survives
  2. 5 What a head can mean by most irrational
  3. 6 The first three hundred organs
3 essays · wrong

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