Phyllotaxis
the spirals are counted, not admired
Concept

Ogive — where it appears

5 essays name this object, across one field. What follows is each of them, and the objects they name alongside it.
1231234rise exponent p, where the rise falls as z⁻ᵖratio between consecutive transitions along the axis5 surfaces, 16 measured transitionsworst disagreement 0.8%

The shape and the law

A cone's transitions are a factor of φ² apart and a disc's a factor of φ, and the temptation is to read the ratio as the shape. It is not. Five surfaces built and counted show that the ratio measures one exponent, and that the exponent is the shape multiplied by the way material arrives.

cylinder · shape exponent 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.

LadderTransitionsConeRiseSpecimenThe local exponentParastichySurface of revolutionArea fillingBranch pointCapitulumCountable region

All concepts