Phyllotaxis
the spirals are counted, not admired
Generator

same-counts-different-symmetry

Both are counted 2 and 4 by machinery shown only their positions. Rotating the left one by half a turn maps it onto itself and rotating the right one does not, so the left is bijugate and the right is not — and no count could have said so.
Same counts, different patternsBoth are counted 2 and 4 by machinery shown only their positions. Rotating the left one by half a turn maps it onto itself and rotating the right one does not, so the left is bijugate and the right is not — and no count could have said so.two at a time68.75°, bijugatecounted 2 and 4half a turn maps it onto itselfone at a time180.5°, ordinarycounted 2 and 4half a turn does notboth counted 2/4symmetry order 2 against 1

Drawn at its defaults, in whorls, and patterns with k at a time. It takes no options at all, so every essay calling it gets this exact drawing.

Called by 6 essays

the blast radius of changing it

The claims, measured

Fibonacci is a branch, not a law

Fibonacci counts come from one branch of the model. The Lucas branch — which the same rule reaches at a different growth rate — gives 47 and 76, and neither is a Fibonacci number. The sequence is a consequence of an angle rather than a property of plants.

 The pattern itself

Counting the spirals

Almost every claim about phyllotaxis is a claim about how many spirals run through a pattern, and the count is almost never done. It can be done from the points alone, by machinery that is never told what angle built them — and then a count of 34 is evidence rather than a restatement.

 The claims, measured

What "whorled" was hiding

The expansion phase's census put 35% of divergences in a bucket labelled whorled and moved on. Opened, every pair in it is k and 2k — the coarsest rung of the ladder, repeated k times — and reading the census up to jugacy takes the Fibonacci share from 14.7% to 50.1% without describing a single extra plant.

 The pattern itself

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.

 The pattern itself

Counting without an index

A person counting spirals on a cone puts a finger on one scale, follows a family round, and counts how many distinct chains there are. That needs no order of arrival — and building it turns out to be strictly more general than the counter the site had, and to find a bug in the counting of a bijugate stem that no assertion would have caught.

 The claims, measured

The survey this site cannot do

Four phases of asking for a dataset, and it is still not here. What this phase can do instead is specify it — the fields, the sampling, the sizes, and which of this collection's claims each one would settle. Two of the four fields asked for turn out to be worth less than the asking implied, and one was never asked for at all.

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