Phyllotaxis
the spirals are counted, not admired
Concept

Basin boundary — where it appears

The border between two sets of starting angles that reach different settled patterns. It need not be a line: at three of six boundaries measured here the destination changes across a stretch that settles at nothing, and separating that fade from a clean wall took a stated instrument rather than a picture.

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

Six boundaries on three basins, five kinds between them, and three that are edges. Each basin's stretch of starting angle, with both its boundaries located to ± 0.125° by sweeping ten degrees at a quarter of a degree. The pale bar behind each is the interval the forty-angle table could bracket it in, three to five degrees at a time. The widest basin, at rise 0.03 and exponent 2, is a fringe at 124.375° and a puncture at 179.625°. The same-rise basin, at rise 0.03 and exponent 3, is a sliver at 124.875° and a fade at 167.625°. The narrow basin, at rise 0.02 and exponent 3, is a wall at 144.875° and a fade at 161.625°. Only 3 of the 6 are edges in the sense of a side: the widest basin's upper boundary is a hole 0.75° wide centred on 180°, with the same destination beyond it, so that basin runs out of basin at the reflection point rather than reaching an edge. The interval the forty-angle table bracketed each basin in is drawn behind it, from the sweep at 1200 organs a run.

A basin with no upper edge

The widest basin in the settling table had a width bracketed between 47.5 degrees and about 57, and closing a bracket means sampling near an edge rather than everywhere. Three basins cut at a quarter of a degree located all six of their boundaries, and the widest turned out to run out of basin at 180 degrees rather than reach an edge on that side at all.

mechanism · Attractor 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

Named alongside it

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

AttractorBasinDivergenceFalloff exponentHonest limitsSettlingStarting angleAzimuth gridDiscriminationHandednessInstrument settingMeasurement error

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