Phyllotaxis
the spirals are counted, not admired
Depth

Series

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.
Take away the organ eight places back, and the next one goes into the hole. The last 34 organs of a stem at a rise of 0.005, unrolled. The open circle is the organ removed — eight places before the tip. The ring at the top is where the rule puts the next organ with every organ present; the filled mark beside it is where the rule puts it with that one missing. The two are 16.4° apart, against a local spacing of 25°, and the vacancy itself is 22.7° from the undisturbed answer. Nothing else differs between the two runs: same rise, same history, same rule.

Ablation

  1. 1 The organ that was taken away
  2. 2 An experiment a needle could run
  3. 3 A rule that cannot heal a hole
  4. 4 A front with no middle
  5. 5 The response with a hole in it
  6. +22 more
27 essays · mechanism Which offsets give short hops, at a rise of 0.013. The two lowest points are at 5 and 8, and those are the parastichy numbers. Offset 1 is high because a hop of one node is at least the rise, which is what makes a stem easier to count than a disc.

Rung interior

  1. 1 Where a handover sits
  2. 2 Two lines that cross once
  3. 3 A band that holds the angle still
  4. 4 Four crossings nobody visited
  5. 5 A band that moves nothing
  6. +20 more
25 essays · cylinder A period of 5, and the two classes that are not with the rest. The same wrecked stem, folded on the lag it kept: one row per residue class, each drawn at the mean displacement of its own organs against the level the rest of them share. The bar through each row is the spread inside that class, and the widest of them is 10.28° — so within a class the displacement is a constant. three classes sit at the common level. The two that do not sit at 143.0° and -147.6°, equal and opposite to within 3.2 per cent, and they are neighbouring residues. The stem's own divergence is 137.97°, so an exception is one organ's step.

Damage shape

  1. 1 The damage has a period
  2. 2 One level and two exceptions
  3. 3 A step of one organ
  4. 4 The plateau was a prediction
  5. 5 Two regimes above a hole
  6. +11 more
16 essays · mechanism How many organs a stem needs before it is on a lattice. One row per rise, one mark per starting angle, placed at the organ from which every later divergence stays within a degree and a half of the run's own final value. Where a stem settles at all it does so between 0 and 290 organs in, against the 400 every ablation run here grows before it cuts anything. Not one row needs the length it is given. What changes down the table is the count on the right: how many of the nine starting angles reach a lattice at all, which falls from 7 at the coarse rises to 1 at the finest.

Settling

  1. 1 How long a stem takes to settle
  2. 2 A wall and not a budget
  3. 3 A steeper rule walls nowhere else
  4. 4 The clock a share cannot see
  5. 5 Destinations only a steep rule reaches
  6. +11 more
16 essays · cylinder A lattice with independent errors. The autocorrelation of 759 divergence angles from a kinematic lattice at a divergence of 137.8261° and a rise of 0.005, with 0.5° of scatter on each azimuth. There is no placement rule anywhere in it: node i is put at exactly i times the divergence and then displaced. The only thing that differs between this figure and the control is the structure of the displacement — here, independent errors — that earlier work's control at 0.5° of independent scatter. The largest comb mean is 0.029 against a sampling band of 0.073, and the readout refuses.

Noise colour

  1. 1 A disturbance with a memory
  2. 2 A periodicity is not a lattice
  3. 3 The disturbance that travels
  4. 4 A disturbance that is not passed on
  5. 5 What the rule does to a drift
  6. +10 more
15 essays · mechanism Removing one wall of the slot, then the other, then both. The growing tip sits in a slot between its two chain-neighbours — the organ 5 places back and the organ 8 places back. Each bar is how far the next organ placed moves when those are removed, against a control sharing the same history. Either alone is a cheap removal: 26.3° and 4.9°, both inside the 45° that separates taking a neighbour from taking anything else. Together they move it 164.1°, against 31.2° for the two effects added, so the interaction is +132.9°. The slot is not two independent walls.

Both walls

  1. 1 Both walls of the slot
  2. 2 A removal that changes nothing
  3. 3 The rung that two organs wreck
  4. 4 Six lattices were not enough
  5. 5 When the second wall is free
  6. +8 more
13 essays · mechanism The statistic everybody reports is the one that cannot vary. Six arrangements of 900 points, from a whorled lattice to a set with no rule in it. The mean number of sides per cell is 5.97–6.04 on all six, because Euler's formula forces it. The mean squared departure from six runs from 0.023 to 1.83 — a factor of 79 — and the most hexagonal tissue in the set is the whorled one, at a rational angle.

Second statistic

  1. 1 What a summary throws away
  2. 2 The disorder is a staircase
  3. 3 A dip belongs to the head
  4. 4 The most irrational is not the most disordered
  5. 5 The background is not one sample
  6. +8 more
13 essays · wrong 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
  6. +7 more
12 essays · branching A branching tree in which every junction obeys the cube law. r₀³ = r₁³ + r₂³ at all 63 junctions, to 2e-16. The widths in the drawing are the radii the law gives, not widths chosen to look right.

Murray

  1. 2 The cube law
  2. 3 A cube law with a lever arm
  3. 4 A crown that carries its own wood
  4. 5 A crown sized for how far it bends
  5. 6 Three rules, one exponent
  6. +6 more
11 essays · branching The rule, 26 steps in, at a growth of 0.40. The next primordium goes where the repulsion is least — the marked minimum at 216°. Nothing in the rule refers to any particular angle.

Attractor

  1. 2 The angle is an output
  2. 3 The pattern the cut leaves behind
  3. 4 The block is the count it was cut from
  4. 5 Two-ranked, by two different routes
  5. 6 The stem that changed hands
  6. +5 more
10 essays · emergence The fork the cost chooses at a daughter ratio of 1: 37.47 and 37.47 degrees. Three ends held fixed, three weights fixed by the radii, and the branch point put where the total cost is least. At the optimal radius the pumping term is exactly half the upkeep term, so a segment's weight is its own cross-section and the three weights here are 1.5874, 1.0000, 1.0000. Minimising directly over the position — a 41×41 grid re-centred and shrunk 220 times, told nothing about any formula — puts the daughters at 37.4673° and 37.4673° from the parent's own forward direction, against the closed form's 37.4673° and 37.4673°.

Fork angle

  1. 1 The angle the cost chooses
  2. 2 A rule that predicts everything
  3. 3 An optimum too flat to reach
  4. 4 The trees drawn at no angle
  5. 5 One constant for every fork
  6. +5 more
10 essays · branching 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
  3. 5 The outline finds its own centre
  4. 6 The cut along the axis
  5. 7 An opening that is not round
  6. +5 more
10 essays · shells 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
  4. 6 The count sees the twist first
  5. 7 The flag reads the angle, not the twist
  6. +5 more
10 essays · lattices 14 specimens separate 14.7% from 50%. The exact binomial power against sample size, for a one-sided test at 5 per cent. It is a staircase rather than a curve because the decision rule is a whole number of specimens: at 14 the cut sits at 5 and the power is 91.0 per cent. A normal approximation smooths that staircase away and reports a different answer.

Sample size

  1. 5 How many plants would it take
  2. 6 The census wants a low count
  3. 7 Counting it again
  4. 8 The recount aims where the counter expects
  5. 9 The high heads the geometry rarely makes
  6. +5 more
10 essays · wrong A logarithmic spiral growing by 3.20× per turn. Fitting log r against angle on the drawn points returns 3.2000, and the model's worst residual is 9e-16 in log r.

Spiral

  1. 2 Growth as a rule
  2. 3 A spiral with no clock
  3. 4 What the growth lines carry
  4. 5 A shell that changed its law
  5. 6 A law that never stopped changing
  6. +5 more
10 essays · shells What a quarter-radius centre error costs a 3.2× spiral, against what the collection publishes. Root-mean-square error in the recovered growth factor when the assumed centre is displaced by a quarter of the innermost whorl's radius, against how much arc is measured. It is 4.56 per cent at two turns, 2.52 per cent at two and a half, 1.91 per cent at three and 1.34 per cent at three and a half. It first falls under one per cent at 4.25 turns — and at 4.25 turns at all six of the growth factors surveyed, so the span rather than the factor is what decides it.

Spiral fit

  1. 1 What the centre costs
  2. 2 How far a centre must move
  3. 3 The residual is not the test
  4. 4 A measurement in steps
  5. 5 One number for a shell that changes
  6. +5 more
10 essays · shells Every open question here needs under 34 specimens. The sample size at which each comparison reaches 90 per cent power at a 5 per cent false-positive rate, from the exact binomial rather than a normal approximation. The census question — do plants show consecutive Fibonacci pairs far more often than the geometry does — needs 4: 14.7% is the share of divergence angles giving a consecutive Fibonacci pair at a fine rise; 90% is what a grown history gives.

Survey spec

  1. 6 The survey this site cannot do
  2. 7 What a quiet plant is worth
  3. 8 What the pair costs
  4. 9 What a refusal does not say
  5. 10 The survey loses its second outcome
  6. +5 more
10 essays · wrong A tree built at 3, measured to 2%, reads 2.957 on the informative band. The exponent recovered from 100 junctions of a tree built at exactly 3, against the relative error placed independently on the parent and on both daughters — half a per cent is a machined section under a microscope, one to two per cent is callipers on a clean branch, five is a branch that is not round, ten is a radius read off a photograph. Each line is a band of daughter ratio and the whiskers are the central 90% of 300 replicate samples. Every band is displaced downward at every error and never upward: at 2% the informative band read 2.957 and the informative band read 2.957. Below, the same rows with the displacement and the spread drawn as separate bars, because only one of them falls when more junctions are measured.

Exponent error

  1. 1 The exponent an error moves
  2. 2 The fragile junctions are the informative ones
  3. 3 The window that closes
  4. 4 Where three and two become one
  5. 5 A swelling at the fork
  6. +4 more
9 essays · branching 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
  6. +4 more
9 essays · lattices The boundary located at 481 expansions, against D = 1/W. 481 expansions log-spaced across the range, each boundary found by bisecting the two drawn circles to two hundred steps rather than read off a grid. The hyperbola D = 1/W lies on the located curve to 2.220e-16 over the whole of it — the last bit of a double — so the textbook line is the boundary and not an approximation of it. The closed form and the bisection agree to 2.81e-15 at worst over all 3,200 located boundaries, which is what makes a residual of zero a reading.

Morphospace

  1. 1 The line was already exact
  2. 2 What a spire buys
  3. 3 A boundary with no edge
  4. 4 A fraction of nothing
  5. 5 One angle decides contact
  6. +4 more
9 essays · shells

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