Phyllotaxis
the spirals are counted, not admired
Concept

Settling time — where it appears

How many organs a stem takes before its divergence stops moving, measured from a starting angle that is not the answer. It is between nothing and 290 organs where it happens at all, and a run that has not settled quickly does not settle at any length.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

How far above the hole the damage becomes a pattern. One row per wrecked cut, drawn at the first organ from which every residue class stays at its own level for the rest of the run. On the 25 rows that reach it at all, it runs from 7 to 303 organs above the removed one; five rows never reach it inside the 300 organs each run is continued for. Below that point the stem is still moving, and the displacement of the first organ after the cut — the quantity that tells a cheap removal from an expensive one — is measured there. Above it, nothing changes again.

Two regimes above a hole

Below the repeating pattern there is a transient, and the boundary between them is measurable: the first organ from which every class stays at its own level runs from 7 to 303 organs above the hole on twenty-five of thirty cuts, and five never reach it inside the run.

mechanism · Damage shape How many organs a stem needs before it is on a lattice. One row per rise, one mark per starting angle, placed at the organ from which every later divergence stays within a degree and a half of the run's own final value. Where a stem settles at all it does so between 0 and 290 organs in, against the 400 every ablation run here grows before it cuts anything. Not one row needs the length it is given. What changes down the table is the count on the right: how many of the nine starting angles reach a lattice at all, which falls from 7 at the coarse rises to 1 at the finest.

How long a stem takes to settle

Every result here is grown on a stem that has settled onto a lattice, and settling has always been tested for and never timed. Timed, it takes between nothing and two hundred and ninety organs — against the four hundred every ablation run grows before it cuts anything, and the nine hundred the noise runs carry.

cylinder · Settling The same table, grown 2.7 times as long. Every rise and every starting angle, grown to 1200 organs and then to 3200. The two middle columns are how many starting angles reached a lattice at each length, and they are the same column: of the 72 pairs of runs, 72 are identical organ for organ and 0 settle at the longer length after failing at the shorter one. Tripling the budget buys nothing anywhere. What the fine rises are short of is not run length: the share of starting angles that reach a lattice at all falls from 7 of 9 to 1, so the arrangements a stem could fall into have mostly stopped existing.

A wall and not a budget

Below a rise of about 0.005 this collection's stems stop settling onto a lattice, and the limit has been written up four times without anybody asking which kind of limit it is. Grown three times as long, the table is identical row for row: not one stem that failed to settle succeeds. The floor is a wall.

cylinder · Settling Two instruments at six boundaries: the clock cannot tell a wall from a fade and the tail spread separates them by a factor of 327. The clock is the ratio of the slowest settling in the last degree inside a basin to the middle of the rest of its window. It rises towards every boundary and diverges at none: the one clean wall rises by 1.313× and the two clean fades by 1.00× and 1.57×, so the wall sits between them and no threshold on a clock separates the two kinds. The tail spread is how far a run's own last two hundred organs wander, and every run has one whether it settles or not. Past the wall it is 0.101°, as steady as the basin just left; past the two fades it is 33.042° and 39.712°. Pooled over all 574 runs everything that settles spreads by 0–0.442° and everything that does not by 30.828–39.712°. The two instruments are drawn side by side on one row per boundary, from the sweep at 1200 organs a run.

A wall or a fade

A basin's border is either a change of destination or a stretch where the angles stop settling at all, and nothing here could tell the two apart. Two instruments were pointed at the question: the settling clock, which looked obviously right and fails, and the tail spread, which was already being computed on every run and had never been read.

mechanism · Attractor How many starting angles reach a lattice, by rise and by falloff. Every cell is 9 stems grown from 9 starting angles spanning 40° to 180°, at one rise and one exponent in the placement rule's falloff, for 1200 organs each. The number is how many of them settle onto a lattice. Reading down a column, the basin closes as the rise falls, which is the wall. Reading across a row, nothing much happens: the exponent-three column is the published settling table digit for digit, and the other three are the same column inside their own sampling error. The account that a steeper falloff should wall somewhere else does not survive the table it predicted.

A steeper rule walls nowhere else

The account of the wall at the fine end was that the basin narrows because the neighbourhood deepens, which predicts a steeper falloff walling somewhere else. Grown at four exponents, the four columns settle 30, 31, 29 and 27 of 72 — a spread of 0.056 against an error of 0.058.

emergence · Settling How many organs settling takes, by falloff exponent. One mark per stem that settles, placed at the organ from which every later divergence stays within a degree and a half of the run's own final value. At the exponent every other measurement here uses, the slowest is 290 organs — the number four rounds of this collection have carried as though it belonged to the rule. It belongs to the exponent: at 4 the slowest is 808 and at 5 it is 674. A table of shares cannot see this, because a stem that settles at organ 800 and one that settles at organ 8 are the same entry in it. And it is still not a shortage of time: of 72 pairs of runs grown to 1200 organs and then to 3200, 0 settle at the longer length after failing at the shorter.

The clock a share cannot see

Settling takes nothing to 290 organs, and four rounds of this collection have carried that as a property of the rule. At a steeper falloff the slowest is 808 — and not one of the 72 pairs of runs settles at 3,200 organs after failing at 1,200, so the fine end is still a wall.

emergence · Settling

Named alongside it

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

AttractorBasinControlHonest limitsMeasurementNegative resultClaim testingThe placement ruleSample sizeTransientCounting blindExponent

All concepts