Phyllotaxis
the spirals are counted, not admired
Concept

Specimen — where it appears

6 essays name this object, across 3 fields. What follows is each of them, and the objects they name alongside it.
1 at a time2 and 3symmetry order 12 at a time2 and 4symmetry order 23 at a time3 and 6symmetry order 3divergences 137.51° · 68.75° · 45.84°counted 2/3 · 2/4 · 3/6

Two at a time

Every counter on this site asks how far it is from element i to element i plus m, and that index is a claim that the elements arrived one at a time. For teasel, for Cephalaria and for a real minority of plants the claim is false — and what those patterns turn out to be is an ordinary lattice, wrapped twice.

lattices · jugacy 00.2500.5000.75015101520specimenschance of detecting it14 specimens90%exact binomial · one-sided at α = 0.0514 specimens, cut at 5

How many plants would it take

Fourteen specimens separate the geometry's Fibonacci share from a coin weighted to a half. Four separate it from what a grown history gives. One fir cone measured at three rings settles whether its ladder is spaced as a cone's or an ogive's. The sample sizes are small, and that is the uncomfortable part.

wrong · sample size 024680.2000.4000.6000.8001counting only outside this fraction of the organ's length or radiustransitions inside the counted partdisc: 2 beyond 50%cone: 1 beyond 50%flare 0.12 · 4000 elements1 against 2 in the outer 50%

Why a cone can be counted once

A pineapple is described as 8 and 13 and the description holds. A sunflower is described as 34 and 55 and the description is a statement about one annulus. Both organs have the same ladder in element number — what differs is where an organ puts its elements.

cylinder · organ size the meridiana conean ogivea convex head00.50011.50200.2000.4000.6000.8001along the meridian, as a fractionlocal exponent a(s)a = 1, a conedifferenced from each profile, not looked upogive 2.00 → 0.23

An organ has no single exponent

The previous phase measured that a surface whose circumference grows as a power of arc length puts its transitions a fixed factor apart, and checked it on five surfaces. Every one of them had a single exponent, and no organ does — a fir cone is an ogive, whose exponent runs from 2 at the tip to nearly 0 at the shoulder.

cylinder · varying exponent 1.701.801.9020123which step of the ladderexponent reportedresidual 0.045of one stepan ogive · 4 stepsfitted 1.891 against a harmonic mean of 1.880

What one exponent reports

Fit a single shape exponent to an organ that has four of them and it returns a real quantity — the harmonic mean of what its individual steps report. Harmonic means sit below arithmetic ones, so the fit understates, systematically, in a known direction, and invisibly.

cylinder · fitted exponent error per ringrings neededspread vs bound1%4 of 5 rings1.9% > 1.4%1%5 of 5 rings8.2% > 2.8%2%5 of 5 rings8.2% > 5.7%3%not on this oneunder 8.5%5%not on this oneunder 14.1%8%not on this oneunder 22.6%an ogive · 5 rings available5 rings at 1%

How much of a cone to measure

Two rings cannot show a varying exponent — not with difficulty, but in principle, because one gap determines one exponent with nothing left to disagree. Four or five can, if each is found to within a per cent. At three per cent this specimen cannot be told from a power law however many of its rings are recorded.

cylinder · ring identifiability

Named alongside it

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

LadderRiseConeOgiveTransitionsThe local exponentCensusIdentifiabilityJugacySurveyWhorlBijugate

All concepts