Phyllotaxis
the spirals are counted, not admired
Concept

Rewriting — where it appears

Replacing every symbol of a string by a string, all at once, which is the step an L-system repeats to grow its description of a plant. Its rules say what each part becomes in one season and nothing about why, so a count that follows from them belongs to the grammar until a rule is tied to something a plant has.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

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.

L-systems describe, they do not explain

Two rewriting rules and a turtle produce something indistinguishable from a plant, and there is no plant in it — no light, no water, no auxin, no mechanics. That is worth demonstrating precisely because the output is so convincing.

branching · Lsystem The number of growing points after each season, for buds that wait no season, one, two, three or four. From one mature apex, each season every mature apex makes a new bud, and a bud branches only after it has waited its delay. With no delay the counts run 1, 2, 4, 8, 16, 32, 64, 128, 256, 512 and settle into growing by 2.0000 a season; with one season the counts run 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 and settle into growing by 1.6180 a season; with two seasons the counts run 1, 2, 3, 4, 6, 9, 13, 19, 28, 41 and settle into growing by 1.4656 a season; with three seasons the counts run 1, 2, 3, 4, 5, 7, 10, 14, 19, 26 and settle into growing by 1.3803 a season; with four seasons the counts run 1, 2, 3, 4, 5, 6, 8, 11, 15, 20 and settle into growing by 1.3247 a season. On a logarithmic axis each settles into a straight line whose slope is its growth rate, the positive root of x^(d+1) = x^d + 1.

A count set by a delay

An L-system describes a plant and forbids nothing, because none of its parameters is anything a plant has. One branching grammar is the exception: a mature apex makes a new bud every season, and a bud waits d seasons before it branches. Its counts grow at the root of x^(d+1) = x^d + 1, a delay of one season gives Fibonacci's numbers and nothing else does, and the fourth count already separates a one-season wait from every longer one. So a Fibonacci count in a branching plant is a measurement of how long its buds wait. It is also a fragile one: if one bud in ten waits two seasons instead, eleven counts in a row come out Fibonacci's three times in a thousand.

branching · Lsystem The rate a branching count grows at, against the chance a growing point dies. Every point dies with probability q each season and the survivors rewrite as before, so the expected counts obey x^(d+1) = (1 − q)·x^d + (1 − q)^(d+1). Substituting x = (1 − q)y returns the deathless equation exactly, which makes every line here straight: the rate is the deathless root multiplied by the survival. No delay runs from 2.0000 to one at q = 0.5000; one season runs from 1.6180 to one at q = 0.3820; two seasons runs from 1.4656 to one at q = 0.3177; three seasons runs from 1.3803 to one at q = 0.2755; four seasons runs from 1.3247 to one at q = 0.2451. Below the marked line a lineage shrinks.

A count that loses its growing points

The branching grammar behind the Fibonacci claim has no deaths in it, and a stem that loses shoots is the common case. Giving every growing point a chance q of dying each season leaves the counts a linear recurrence and does exactly one thing to it: the growth rate becomes the deathless root multiplied by 1 − q, at every delay and every death chance, to the last bit a double holds. So each waiting time has a death chance above which its lineage shrinks — a half with no wait, 0.3820 at one season, 0.2451 at four — and a longer wait tolerates less. What does not survive is the count itself: a plant losing one growing point in ten a season shows eight Fibonacci counts in a row one time in ten thousand, against one time in eight for a bud that occasionally waits an extra season.

branching · Lsystem

Named alongside it

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

BranchL-systemsClaim testingDescription versus mechanismFalsifiabilityFibonacciHonest limitsPredictionSample sizeUnderdeterminationEvidenceExtinction

All concepts