Phyllotaxis
the spirals are counted, not admired
Concept

Lattice vectors — where it appears

The short displacements a lattice repeats, recovered by clustering each node's nearest-neighbour offsets rather than derived from a rule. Counting parastichies from them needs no order of arrival, and it is the ordering of the three shortest, not the two, that decides where a handover falls.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Tracing one family: 4 chains. Every node is joined to the node one repeated displacement away, and the chains that result are drawn separately. There are 4 of them, which is the parastichy number of this family. No index of arrival was used anywhere, which is what lets the same count be made on a pattern where 2 primordia appear at once.

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.

lattices · Chain counting The defect rings are not where the counts change — they are √φ further out. A logarithmic radius axis for a golden, 137.508° head. The lower marks are the radii at which the counted parastichy pair changes, where the two shortest lattice vectors change places; the upper marks are the radii at which a cell's neighbours change, where the third-shortest does. They interleave, and the ratio of each ring to the transition inside it is 1.2715, 1.2723, 1.2723, 1.2719, 1.2719, 1.2723 against √φ = 1.27202. Consecutive rungs are a factor of φ apart in radius and √φ is their geometric midpoint, so a defect ring sits exactly halfway between two parastichy transitions. Anyone looking for the defect line at the radius where the counts change will not find it there.

The rings are not the transitions

A seed head has two ladders on it — the radii where the counted parastichy pair changes, and the radii where the exceptional cells sit — and the obvious guess is that they are the same ladder. They are not: the second sits a factor of the square root of phi outside the first at every rung of two different divergence ladders, which is exactly halfway between two consecutive transitions.

tissue · Topological charge

Named alongside it

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

ArtefactCounting blindChain countingCylinderDefect ringDislocationDisorderGeometric ladderφ, the golden ratioHandoverHonest limitsInstrument setting

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