Phyllotaxis
the spirals are counted, not admired
Concept

Slot — where it appears

The gap between the two neighbours a growing organ is placed against, whose walls are the lattice's two contact families. Removing one, the other or both is the design that asks whether the two neighbours act separately, and on most rungs they do not.

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

What is the same at 0.00998 and at 0.00997. Six quantities read on the two rises the transition sits between. Five of them are identical: the two walls the slot has, the divergence the intact stem settles to, the block the cut opens, which of the three removals wreck the stem, and the lags the doubled cut leaves rigid. The sixth is how far the first organ placed after the doubled cut moves, and it goes from 163.59 to -9.14 degrees. The lattice is the same on both sides; where one organ goes is not.

A transition and not a slope

The question was whether a fourth cell's cost declines smoothly to nothing or falls in one step. It falls in one step, and the answer decides whether a word in the collection names something or is a threshold on a continuum.

cylinder · Both walls Where the doubled cut's run finishes, at every rise of the sweep. The divergence the wrecked run ends at, rise by rise. It takes only a handful of values and jumps between them at rises one part in a thousand apart, 4 times inside the 9 rises of the finest sweep alone — where the walls, the block, the intact stem's divergence and the rigid lags are all held. So the end of a wrecked run is not a stable quantity on this rung, and no statement about where the stem finishes is available on either side of the transition. The first organ's displacement is the reproducible half of the same measurement.

The end of a wrecked run

The obvious follow-up to a transition in where one organ goes is whether the stem also finishes somewhere different. On this rung the question has no answer: a wrecked run's final divergence takes four values and changes between rises a thousandth apart, three times inside a nine-rise sweep.

cylinder · Both walls The golden 5/8 rung swept at 27 rises, with each removal's cost. How far the first organ placed after a cut moves, at every rise the sweep visits, coarse on the left. Removing both walls costs far more than removing the larger one alone above a rise of 0.00998, and exactly what the larger one costs below it. The change happens in one step of the grid the ladder is named on: 163.59 degrees at 0.00998 and 9.14 degrees at 0.00997, which is a fall of 154.5 degrees for a change of one part in a thousand in the rise.

Where a slot loses a wall

Two rungs were reported to go free at 84 and 85 per cent of themselves — the second removal stops costing anything over the larger one alone. Three samples a rung cannot say whether that is a transition or a slope, and twenty-nine more say it is a transition one grid step wide.

cylinder · Both walls Where every wrecked run finishes, rung by rung. One row per rung of the ladder, one mark per cut cell of the slot design, placed at the divergence that run ended on. The open mark on each row is the divergence the intact stem of that rung settles to. On the Lucas 1/3 and golden 2/3 rungs every wrecked run finishes at the same value; on the golden 5/8 they finish at 13 values spanning 215 degrees. 27 cells recover, and each of those finishes at its own settled divergence to within 0.03 degrees.

The column nobody read

Every cell of the slot design carries where its run finished as well as how far its first organ moved. One rung's worth had been plotted and called unusable. Reading all eight says the endpoint is exact on two rungs, wanders on six, and is worst on the one it was read on.

cylinder · Both walls Wrecked endpoints against the settling table's destinations. The upper lane is the 15 divergences the settling table reaches, grown from intact stems started at nine arbitrary angles across four falloff exponents and eight rises, with no cut anywhere in them. The lower lane is where the slot design's 63 wrecked runs finish, read without handedness so that a run ending at 209 degrees is placed at 151. 29 of them sit within 1 degree of a destination and 34 do not. The two measurements share no run and no design, so the agreement is not a construction.

A wrecked run goes somewhere

Where a wrecked stem finishes was called unstable. Half of them finish within a degree of a destination measured from intact stems started at arbitrary angles — two tables that share no run, no design and no question.

cylinder · Both walls The twelve wrecked runs that finish at a half turn. Wrecked runs whose final divergence is a half turn, which is two files of organs rather than a spiral. Every one is on a rung at the coarse end of its branch, where the front is short enough that removing one organ reaches past it. The value is within 0.35 degrees of 180 on eight of the twelve.

Two files, and a way back

Twelve wrecked runs finish at exactly a half turn, which is a pattern with no spiral in it, and every one is at the coarse end of the ladder. Three finish at the divergence they would have had anyway, after being thrown a hundred degrees off it.

cylinder · Both walls

Named alongside it

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

Claim testingHonest limitsAblationBoth wallsDisplacementDivergenceMeasurementResolutionAttractorAzimuth gridHandednessInteraction

All concepts