The fields
A field says what an essay is about. The order below is not alphabetical and is not a ranking: it is the order the argument builds, and each field leans on the ones above it. A seed head is a set of points and the spiral counts come out of the points alone — but on a disc every count is a statement about an annulus, so the next field is the geometry where that qualification disappears and which turns out to predict the disc's behaviour rather than merely contrast with it.
Then where the angle comes from, which is a result rather than a constant, and what a plant might actually be doing, which is a different kind of question and is careful to say so. Every geometric field here promises that it is not explaining anything, and that promise has to be paid somewhere.
Threads cut the other way: they say what an essay rhymes with, across fields. Concepts cut a third way, from the objects the essays name rather than from any grouping anybody chose.
The pattern itself
A seed head is a set of points, and nearly every claim about one is really a claim about how many spirals run through it. The counting can be done from the points alone, and the answer is not what the captions say.
Stems and cones
A stem is a cylinder, and on a cylinder the spiral counts are the same the whole way up — which on a disc they never are. Two numbers fix the lattice, both come back out of the points exactly, and lowering one of them climbs the Fibonacci ladder.
Where the angle comes from
137.5° is not a constant of nature. It is where a rule settles — a rule that places each new element as far as it can from the others, and which settles somewhere else when one parameter changes.
What a plant might be doing
Turing's last work was on this, and it was unpublished when he died: a ring of cells, two diffusing substances, and a spacing that selects itself. The modern account uses auxin and a pump that works uphill. Both are made to predict a number and then to produce it.
Shells and growth
A logarithmic spiral is what a thing grows into when it adds material without changing shape. Its one parameter is recoverable from any drawn curve, which is how a century-old argument gets a number attached to it.
Branching and transport
A network built to move fluid for the least work obeys a cube law at every junction. That is checkable on a real tree — and an L-system, which produces something that looks like a plant, explains nothing at all.
Packing and tiling
How evenly a pattern fills its disc is a statement about cell areas, and it can be measured four ways that disagree. Cells average six sides because Euler's formula leaves them no choice.
The claims, measured
The nautilus, the sunflower and the golden angle arrive with more confident wrong statements attached than any other subject on this fleet. Each one gets a test and a number — including the one that turns out to be right.