Phyllotaxis
the spirals are counted, not admired
Concept

Tracking — where it appears

Following a pattern's counts and divergence up a single growing stem rather than reading one number off the whole of it. It is what a lattice with a falling rise requires, since every count is a statement about the band it was taken in.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

The lag between two rates, measured against the rate. Each dot is one rate: the mean gap between where the grown pattern changed its count and where the static ladder puts that transition, in rungs. Over rates from 9 to 135 nodes per rung the worst is 0.087 of a rung. A lag of one rung would put a dot on the top line.

The lag that is not there

A pattern built out of its own history should hold its old parastichy pair past the point where a fresh lattice would have changed, and the gap should grow as the shoot is hurried. Over a fifteenfold range of rate it does not — every transition lands within a tenth of a rung of where the static ladder puts it.

emergence · Tracking The memory of a divergence sequence, at 0.75° of scatter. With no noise at all the lag-one correlation is 0.54: the rule corrects itself, so a lattice arrives with a memory in it. Matched at the same recorded scatter, placement noise leaves -0.10, jostle noise leaves 0.66, field noise leaves 0.47. The band is ±0.13, which is what an uncorrelated sequence of this length gives.

The sequence has a memory

Every measurement this collection has made of a stem's divergence angles throws the order away. A spread is invariant to shuffling. Put the angles back in order and there is a large correlation between one and the next — 0.54 with no noise at all — which is the rule correcting itself, and which nothing had looked at.

cylinder · Sequence The memory belongs to the rise, not to the lattice. The lag-one correlation of a noiseless rising stem, against how fast it climbs the ladder. Below about sixty nodes per rung it is negative; above it, 0.54, 0.74, 0.58, 0.58, 0.55 — flat across a fivefold change in rate. The horizontal line is the same rule with the rise held FIXED, where the correlation is -0.68. So the +0.74 the earlier work called the sequence's own memory is the pattern chasing an equilibrium that is moving under it.

The memory was the rise

The earlier work measured a lag-one correlation of 0.54 in a noiseless divergence sequence and called it the sequence's own memory. Hold the rise fixed and there is no sequence at all — every angle identical — and under a disturbance the correlation is negative. The 0.54 belongs to the pattern chasing an equilibrium that is moving under it.

cylinder · Sequence The memory belongs to the rise, not to the lattice. The lag-one correlation of a noiseless rising stem, against how fast it climbs the ladder. Below about sixty nodes per rung it is negative; above it, 0.54, 0.74, 0.58, 0.58, 0.55 — flat across a fivefold change in rate. The horizontal line is the same rule with the rise held FIXED, where the correlation is -0.68. So the +0.74 the earlier work called the sequence's own memory is the pattern chasing an equilibrium that is moving under it.

A shoot too fast to remember

Sweep the rate at which a stem climbs the ladder and the correlation between one divergence and the next changes sign — negative below about fifty-five nodes per rung, positive above it, with the flip inside one step of the grid. The instrument the earlier work proposed is unavailable on a fast shoot, and nothing said so.

emergence · Rising A window that fits inside a rung. Stems that climb the ladder at four rates, read over a window at the fine end. The condition is a ratio: the window has to be shorter than a rung. 250 internodes at 130 per rung is 1.92 rungs and agrees on 0 of 3; 400 internodes at 130 per rung is 3.08 rungs and agrees on 0 of 3; 250 internodes at 260 per rung is 0.96 rungs and agrees on 3 of 3; 400 internodes at 260 per rung is 1.54 rungs and agrees on 1 of 3; 250 internodes at 520 per rung is 0.48 rungs and agrees on 2 of 3; 400 internodes at 520 per rung is 0.77 rungs and agrees on 3 of 3; 250 internodes at 1040 per rung is 0.24 rungs and agrees on 3 of 3; 400 internodes at 1040 per rung is 0.38 rungs and agrees on 3 of 3. Read over the whole stem instead, every rate returns nothing — 0 of 3, 0 of 3, 0 of 3, 0 of 3 — because the quantity the comb is periodic in changes as the pattern climbs.

A window inside a rung

A stem that climbs the ladder has no comb in it at any rate, because the quantity the comb is periodic in changes as it goes. Read a window instead and it comes back, on one condition: the window has to be shorter than a rung — which makes the shoot's rate the thing that decides whether a plant can be asked.

emergence · Rising Two windows on a shoot at 400 nodes per rung. A stem grown at 400 nodes to the rung with a disturbance of 0.25, its rise falling from 0.4 to 0.004 over 1914 nodes. The upper window is the last 250 internodes — where a count would be made on a real plant — and the lower is the same length shifted down 125. The upper reads 8/13; the lower reads 8/13. The verdict is agree, and the window holds 0.63 of a rung.

Two windows on one stem

A pair read off a climbing shoot can only be read through a window, and a window can straddle a transition. Read a second window half a length lower and the outcomes fall into four kinds — and agreement between them never happens on a shoot whose rung is shorter than the window, which turns the most awkward of the four refusal causes into something a reading can certify.

emergence · Two windows

Named alongside it

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

AutocorrelationDivergence angleMeasurementEnsembleEquilibriumRiseCylinderLadderNoiseParastichy pairThe placement ruleRung

All concepts