Phyllotaxis
the spirals are counted, not admired
Concept

Azimuth grid — where it appears

The finite set of directions an organ may be placed at, here 1,536 samples of the circle at a quarter of a degree apart. Every angle this collection reports is a multiple of that step, so a quantity smaller than one step is the grid rather than a measurement, and a block that is the grid rounding a constant is a period nothing has.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

Three chains, each moved a third of a turn, on g008 cut at offset 6. The eight chains of a stem that kept a lag of 8, set round the circle, with the three that are displaced marked and their displacements written on. Each is about a third of a turn the same way round, and they sum to -359.6 degrees — one whole turn to within half a degree, on an azimuth grid whose step is a quarter of a degree. A rearrangement that closes by going once round sums to a turn, not to nothing, which is why the test it was set had the wrong number in it.

A cycle sums to a whole turn

The test proposed for whether three displaced chains are a three-cycle was that their displacements sum to zero. Three chains rotating into one another's places each move about a third of a turn the same way round, and a third of a turn three times is a whole turn — which the unfolded test calls the worst row in the census.

mechanism · Damage shape 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 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 1.75° of starting angle between two basins that reaches neither, and a 3° void beside it. 20.25 degrees of starting angle swept unbroken at 0.25°, from inside the basin at 101.5° to inside the widest basin at 139.3°, one cell per sampled angle. The last angle reaching 101.5° is 114.5° and the first reaching 139.3° is 116.25°; the 1.75° between them holds 6 sampled angles and 0 of them settle anywhere at all. The widest run of angles reaching nothing is 3° wide, at 118–120.75°, and a detached island of 139.3° sits between the two. Starting angle is left over. The stretch that reaches neither basin is bracketed above the strip, from the sweep at 1200 organs a run.

The angles left over

A stem started anywhere on the circle was assumed to end up in one basin or another, so that the settled destinations divided the starting angles between them. Twenty and a quarter degrees swept without a hole at a quarter of a degree find 1.75 degrees between two basins that reaches neither of them and nothing else, and a three-degree void beside it.

mechanism · Attractor 41.75° in the middle of the widest basin, sampled only at the table's own 3.12–4.38°. The widest basin from its located lower boundary at 124.375° to the reflection point at 180°, with the two ten-degree windows swept at 0.25° shaded and the stretch between them left open. Nothing has looked inside that stretch more finely than the forty-angle table's own 3.12–4.38° spacing. The narrowest feature this sweep found anywhere is the 0.75° wedge inside same-rise, and a feature that size falls between the table's angles 79% of the time — 75% under an even 3° sampling. So an unmeasured sliver or puncture could sit anywhere in the middle of this basin and nothing here would have seen it. The two swept windows are shaded and the stretch between them is left open, from the sweep at 1200 organs a run.

What a quarter degree cannot see

Six boundaries were located to an eighth of a degree, three basins were named and one width was quoted, and every one of those readings has the same floor under it. The sweep's grid is one step of the grid the stems are placed on, so nothing here bounds a basin narrower than half a degree — and the widest basin's own middle was never swept at all.

mechanism · Attractor 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 A reading that steps against an ordering that slides, on six rungs. Every hop length here is arithmetic on the settled divergence, and the settled divergence is a mean of angles read off a lattice placed on a fixed set of azimuths — so it is read in steps rather than continuously. Between two steps the ordering slides with the rise; at a step it jumps. The bar is how much a jump is worth in slides on each rung, measured inside the stretch that was walked at the grid so that the six are read over comparable regions. The account is a threshold at one and it is right on all six: the 3/4 rung sits at 12.71 and carries five crossings, and on three rungs the reading does not step inside the window at all.

One crossing or two

One rung of the ladder changes hands five times where the other five change hands once, and the difference is not in the geometry. It is a divergence read in steps against an ordering that slides, and the account is a threshold at one that is right on all six rungs.

cylinder · Handover grid

Named alongside it

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

Honest limitsClaim testingResolutionAttractorDivergenceBasinMeasurement errorSamplingSettlingStarting angleAblationArtefact

All concepts