Phyllotaxis
the spirals are counted, not admired
Concept

Measure — where it appears

The size of a set of parameter values, used here to say how much of a range gives a stated outcome. It is what turns how often is it Fibonacci into a question with a number rather than an impression.

Named by 9 essays across 5 fields — each of them below, with the objects they name alongside it.

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.

How often is it Fibonacci

The claim that plant spirals come in consecutive Fibonacci numbers is stated as a near-universal. Asked of the geometry, the answer collapses with scale — at a coarse rise 67% of divergences give Fibonacci pairs, and at a fine one 15%, with whorled and unnamed pairs taking the rest.

wrong · Frequency Two measures of one boundary at W = 2.5: overlap slope 1, buried area slope 1.4999. The linear overlap and the buried area, against how far below the boundary the shell sits, both axes logarithmic. The buried fraction rises as 1.499945 — three halves, the lens between two nearly tangent circles — so at a ten-thousandth below the line 0.0011410 per cent of the whorl is under its successor and at 0.30 below it 86.83 per cent is. The linear overlap over the same range is exactly 2.5 times the distance below and falls straight to zero, so the two measures disagree about whether the boundary is sharp and the area is the one that answers the question. A shell just inside the boundary is not a different kind of shell; one well inside is.

A boundary with no edge

Two continuous measures cross the line where a coiled shell's whorls begin to touch, and they disagree about whether it is sharp. One falls to zero as a straight line and makes the line a kink; the other leaves it as a three-halves power and makes it a tangency.

shells · Morphospace One geometry, six boxes: 4.642 per cent to 52.81 per cent forbidden. The share of each box that the coiling geometry excludes, across six boxes that differ only in where their edges were put and in how one axis is sampled. The spread is a factor of 11.4, from 4.642 per cent in the widest box to 52.81 per cent in the tightest, and sampling the same two decades of expansion geometrically rather than uniformly multiplies the answer by 4.59 on its own. The forbidden area is the integral of 1/W and grows as a logarithm while a box grows as a line, so the fraction has no value of its own to quote.

A fraction of nothing

Six boxes differing only in where their edges were drawn give between 4.64 and 52.81 per cent for the same geometry, and sampling one axis geometrically rather than uniformly multiplies the answer by 4.60. The share tends to zero as the box widens, because the region under a hyperbola is a logarithm and a box is a line.

shells · Morphospace The disorder of a head against its divergence angle, 900 points. μ₂ is the mean squared departure of a Voronoi cell's side count from six, measured on 900 points inside 86% of the radius. Swept across 1.32° it is a staircase: flat over stretches of a few hundredths of a degree, with sharp steps between them and narrow deep dips wherever a rational falls. At 137.143° — which is 360 × 8/21 — it is 0.025; At 137.882° — which is 360 × 18/47 — it is 0.125; At 138.002° — which is 360 × 23/60 — it is 0.089. The golden angle is marked and sits at 0.253, in the middle of a flat stretch and nowhere near the largest value on the range.

The most irrational is not the most disordered

If rational angles make ordered tissue, the most badly approximable angle should make the most disordered — which would at last give the golden angle a criterion it wins. Swept across the interval it is the most irrational point of, μ₂ peaks at 138.42° and the golden angle sits unremarkably in the middle.

wrong · Second statistic Two shapes, two ranges, one contrast. The exponential's lattice ends at 3.63 spacings and the gaussian's at 2.25 — ranges 47% apart — and at those two ranges the contrast is 5.74 and 6.09, 6% apart. The band is what the exponent route leaves: 3.98 at p = 1, where the uncut rule makes nothing, and 7.63 at p = 1.25, where it makes a lattice.

Two shapes, one threshold

Read in the same unit, an exponential falloff and a gaussian one disagree about where the lattice ends by half. The quantity they agree on turns out to be one the earlier work measured for an unrelated reason — and it agrees with a bracket left by a sweep of a completely different parameter.

emergence · Falloff exponent Four ways to count the rule's neighbourhood, and one ordering. How many organs the placement rule is looking at, as its falloff exponent is swept from 1.5 to 6, by four different definitions. Counting the organs that carry half the profile gives 33, 12, 3, 2, 1. Counting the organs that carry nine tenths gives 182, 120, 30, 8, 3. Counting the organs that carry ninety-nine hundredths gives 284, 263, 149, 54, 9. Counting the organs that carry the weighted mean lag gives 68, 45, 21, 14, 11. They disagree about the size of the neighbourhood by a factor of 10 — the widest moves by 61 times across the sweep and the narrowest by 6.1 — and every one of them falls as the exponent rises. So no choice among them changes which rule is the deeper, and no redefinition can turn a result about the depth around.

Four ways to count a neighbourhood

How deep does the placement rule look? Counting the organs that carry nine tenths of its profile gives 182 down to 3 as the falloff steepens. Counting the ones that carry half gives 33 down to 1. The weighted mean lag gives 68 down to 11. The four disagree by an order of magnitude about the size and agree exactly about the order.

mechanism · Noise colour Where each kind's lattice gives way. The largest amplitude at which every run still has a lattice, and the scatter it produces there. The amplitudes are incomparable — field 0.015 (fraction of the barrier), jostle 1 (degrees of azimuth), placement 0.8 (degrees of azimuth) — and the scatters agree to 19%. The boundary belongs to the pattern rather than to the disturbance: a lattice fails at about a degree and a half of scatter, and which of three mechanisms produced it does not move where.

The boundary belongs to the pattern

Three kinds of noise, in three incommensurable units, destroy a lattice at the same place — about a degree and a half of divergence scatter. The earlier work measured that of two kinds and called it a scale rather than a constant. With a third it looks less like a coincidence and more like a property of what a lattice is.

emergence · Scatter tolerance Four ways to count the rule's neighbourhood, and one ordering. How many organs the placement rule is looking at, as its falloff exponent is swept from 1.5 to 6, by three different definitions. Counting the organs that carry half the profile gives 33, 12, 3, 2, 1. Counting the organs that carry nine tenths gives 182, 120, 30, 8, 3. Counting the organs that carry the weighted mean lag gives 68, 45, 21, 14, 11. They disagree about the size of the neighbourhood by a factor of 10 — the widest moves by 61 times across the sweep and the narrowest by 6.1 — and every one of them falls as the exponent rises. So no choice among them changes which rule is the deeper, and no redefinition can turn a result about the depth around.

The nearest organ is not the nearest neighbour

Rank the terms of the sum the rule minimises and read off which organs the biggest ones belong to. At every falloff exponent from 1.5 to 6 the answer is the same five: lags 13, 8, 5, 21 and 26. The organ placed immediately before is not among them, and counting the neighbourhood in organs was the wrong unit.

mechanism · Noise colour Everywhere a cut of one to five organs can send a 5/8 stem. Every settled divergence reached by any arrangement of up to five organs removed from a stem at the 5/8 rung, on one axis. There are six of them and no more. three are slips of the lattice the stem was cut from: each keeps the lag-5 family intact and sits a whole number of turns of it from the next, which is the ladder marked below the axis with rungs 72.0 degrees apart. The other three keep no lag at all and sit near a fraction of a turn, marked above: 175.0 degrees near 1 of 2 turns, 190.0 degrees near 1 of 2 turns, 235.0 degrees near 2 of 3 turns. A stem that is cut either slides along the ladder it was on or leaves it for a lattice with files in it.

A file has to close

The three destinations counted with a shared factor sit near a half turn, a half turn and two thirds. Measuring how near is the trap: by distance from the fraction, the golden angle is closer to two fifths than two of them are to anything, and would be reported as having five files it does not have.

cylinder · Dose

Named alongside it

The objects these essays reach for when they reach for this one.

Divergence angleHonest limitsMeasurementThe placement ruleThe range of the interactionModel scopeNearest neighbourNeighbourhoodParastichy pairSummary statisticToleranceCensus

All concepts