Phyllotaxis
the spirals are counted, not admired
Concept

The neighbour graph — where it appears

Which organs count as adjacent, obtained from the positions rather than assumed. Every claim about inherited errors depends on it, since who a disturbance is shared with is decided by which organs touch.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

Only noise that arrives before the choice can change what is chosen. Intact runs only, from the whole amplitude sweep. Placement noise displaces the node after the rule has picked an azimuth: 44 runs, none of which changed branch at any amplitude that left a lattice. Field noise perturbs the energy profile the rule picks over, so it can move the minimum into a neighbouring gap: 1 of 66 did.

Where the noise gets in

Ninety runs of noise applied after the rule has chosen, and not one changes branch. Fifty-six of noise applied to the choice itself, and one does. Only a disturbance upstream of the decision can restructure which nodes are neighbours of which — which is what a branch is.

emergence · Noise entry Which neighbours decide where an element goes. Each line is one exponent: how much each shell of neighbours makes the energy profile vary around the circumference, divided by what the nearest shell contributes. At p = 0.5 the nearest shell leads the next by a factor of 1.1 and a node is placed against the whole neighbourhood at once. At p = 3 it leads by 9.7e+3, and a node is placed against its immediate neighbours — which is what a lattice is.

How far a primordium reaches

The placement rule's repulsion falls as an inverse cube because that is what two magnetised droplets do, and nothing about a plant supplies the exponent. Asking what it controls produced one tidy wrong answer and one measured right one — and the difference between them is the difference between a total and a variation.

emergence · The range of the interaction The same rule at p = 1, cut off at two distances. The top 220 nodes of two stems grown by an identical rule whose energy does not converge. Allowed to see 3/√h neighbours it produces 8/13 at 137.62° with 0.58° of scatter — a lattice no test here would question. Allowed 12/√h it produces 44° of scatter and no pattern. The truncation was doing the work.

A window that makes a pattern

A rule whose energy has no well-defined minimum produces a clean 8/13 lattice at 137.62°, with half a degree of scatter, when its neighbourhood is cut at three node spacings. Let it see twelve and the pattern is gone. Every simulation of this kind truncates something, and truncation manufactures exactly the result it is used to look for.

emergence · Truncation 135 fives and 129 sevens among 1631 interior cells. The side counts of every cell strictly inside a golden, 137.508° head of 2400 organs, cut at 86% of the radius. The fives and the sevens are counted apart rather than summed into a spread, because they are opposite charges and the sum hides them. Summed over the interior, 264 cells that are not hexagons carry a charge of +6. Over the whole patch the charge is 294, which is exactly 6 + 2·144 — a number fixed by the 144 cells on the patch's own boundary and carrying nothing whatever about the interior. The defects are not scarce; they are balanced.

An interior that is nearly neutral

Give every cell a charge of six minus its number of sides and the total over a tessellated head is fixed by its own boundary, exactly, with nothing left over for the interior. On a golden head that freedom is spent on 264 exceptions among 1,631 cells which cancel to six.

tissue · Topological charge A cut-off would have to exceed 2.236 and not exceed 1.441, and nothing does both. Each of the 608 interior cells contributes two marks: its furthest wall, and its nearest partner that is not a wall, both in units of that cell's own shortest lag. A single cut-off would have to sit to the right of every mark of the first kind and to the left of every mark of the second, and the two clouds overlap — the extreme cases are 2.236 at 0.0 per cent of the radius and 1.441 at 60.0 per cent. So the interval is empty by a factor of 1.55, while 606 of the 608 cells have a cut-off that works for themselves.

No cut-off makes them one

Two different relations on a head have both been called neighbour: the shortest index lags a count keeps, and a shared Voronoi wall. A cut-off that turns the first into the second exists for almost every cell taken alone, and for no whole head at any size.

tissue · Neighbour definition

Named alongside it

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

The placement ruleRiseDelaunayDivergence angleMeristemNoiseRepulsionRim effectTruncationVoronoi cellsArtefactAuxin

All concepts