Phyllotaxis
the spirals are counted, not admired
Concept

Ogive — where it appears

A profile that is convex near its tip and concave further down, which is the shape of a fir cone rather than a straight-sided cone. A single fitted growth exponent on one reports the harmonic mean of what its individual steps are doing.

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

The exponent sets the spacing, and only the exponent. The curve is φ^(2/p), drawn from the ladder's ratio of 1/φ² and nothing else. The dots are measured: each surface built, a blind counter walked up its axis, the places its answer changed recorded. A cone that elongates and a paraboloid that fills sit on the same point at p = 1 — so the spacing identifies neither the shape nor the way material arrives, only their product.

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 What is visible in the outer part of a 4000-element organ. Both surfaces have the same ladder in element number — the rise is 1/(2πi·flare) on a cone and 1/(4πi) on a disc, and c and the internode step both cancel. What differs is where the elements are. Counting outside 50 per cent of the extent, a cone shows 1 change and a disc 2, because half a cone's length holds half its elements and half a disc's radius holds three quarters of them.

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 Three organs, and the exponent each has at each place. Left: the meridian of a cone, an ogive of arc radius 12, and a spherical cap of radius 40, each scaled to its own length. Right: d log r / d log s along it. The cone sits at 1 the whole way; the ogive starts near 2 at its blunt tip and falls to 0.23 by the end of the 100 per cent shown; the cap starts at 1 and falls slowly.

An organ has no single exponent

The earlier work 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 One exponent fitted to an organ that has 4 of them. Each dot is one step between consecutive rings, reporting 2 ln φ / ln(s′/s) — the exponent that step would have if the organ had one. They run from 1.980 to 1.697. The line is what a single fit returns, 1.891, which is their harmonic mean of 1.880 and sits below their plain average of 1.887.

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 At 1.0 per cent on each ring, 5 rings show the drift. The gaps between consecutive rings are 1.626, 1.631, 1.657, 1.763. Two rings give one gap and no way to disagree with itself; three give two gaps and a fit with nothing left over. The question is how many gaps it takes for their spread to exceed what the measuring error can explain, and the answer depends on the error as much as on the organ.

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