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 The experiment this site can specify
  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 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 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 The memory of a divergence sequence, at 0.75° of scatter. With no noise at all the lag-one correlation is 0.54: the rule corrects itself, so a lattice arrives with a memory in it. Matched at the same recorded scatter, placement noise leaves -0.10, jostle noise leaves 0.66, field noise leaves 0.47. The band is ±0.13, which is what an uncorrelated sequence of this length gives.

Sequence

  1. 1 The sequence has a memory
  2. 2 What one angle says about the next
  3. 3 The test a plant could settle
  4. 4 The order carries the count
  5. 5 The memory was the rise
  6. +4 more
9 essays · cylinder 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. +3 more
8 essays · branching 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. +3 more
8 essays · shells A lattice with an error inherited from the two contact neighbours. 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, an error inherited from the two contact neighbours at coupling 0.7. The largest comb mean is 0.514 against a sampling band of 0.073, and the readout returns 8/13.

Noise transport

  1. 1 Errors that pass between organs
  2. 2 What a forgery has to know
  3. 3 The comb was never the rule
  4. 4 The ratio was the floor of a curve
  5. 5 The ratio was never about the rule
  6. +3 more
8 essays · mechanism 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. +3 more
8 essays · shells The two spiral families a counter finds between 0.68 and 0.92 of the radius. 34 spirals one way and 55 the other, found from the point positions alone — the counter is never told the divergence angle.

Counting

  1. 2 Counting the spirals
  2. 3 A counter that sees no positions
  3. 4 Every family but two is a sum
  4. 5 The angles name the branch
  5. 6 A period that is not a count
  6. +2 more
7 essays · lattices 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. +2 more
7 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. +2 more
7 essays · branching The exponent fitted from the junctions, rather than assumed. Sweeping k and asking where r₀ᵏ = Σ rᵢᵏ holds best gives 3.000 — Murray's 3, recovered rather than imposed.

Exponent

  1. 3 Fitting the exponent
  2. 4 Which junctions say anything
  3. 5 A sample that is confidently wrong
  4. 6 The band decides the answer
  5. 7 A wall that stopped moving
  6. +1 more
6 essays · branching 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 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. +1 more
6 essays · shells

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