Phyllotaxis
the spirals are counted, not admired
Concept

Divergence — where it appears

The angle between one organ and the next round the axis, which is the single number a lattice is usually named by. It is an output of the placement rule here rather than an input, and an angle names a lattice only once the rise is named beside it.

Named by 11 essays across 3 fields — each of them below, with the objects they name alongside it.

g005 at offset 6, read against two levels. One wrecked cut's four residue classes, drawn twice. On the left the level is the median of the four class means, which falls on the class at -5.5 degrees sitting on its own; three classes are then exceptions and the row is set aside as having no balanced pair, summing to -173.6 degrees. On the right the level is the pair of classes that agree, at -138.6 degrees; two classes are then exceptions, they are adjacent and equal and opposite at 133.1 and 134.2 degrees, and the row is an exchange. The exchange table goes from 20 rows to 21.

The twenty-first row

Recomputing the level moves one row out of the set the exchange sets aside and into the exchange itself. Its hop is four times larger than any the correction was fitted over, and the correction fails on it in the one way it had never failed.

mechanism · Damage shape 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 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 Where a stem started at each of twenty angles ends up, at a falloff exponent of 3. One column per starting angle and one row per rise, coarse at the top. A filled cell is a run that reached a lattice, its tone the destination it reached; a pale cell is a run that never settles. Runs of one tone across neighbouring columns are basins, and the widest of them spans 7 consecutive angles. The angles are 6.25 to 10 degrees apart, so a basin narrower than that cannot be seen here and a single filled cell says nothing about how wide its basin is.

A basin has a width

A destination reached from one starting angle is a presence. A destination reached from seven consecutive starting angles spanning forty-five degrees is a basin with an extent, and nine angles could not have measured one — they were too far apart to have two of them land in the same place.

emergence · Settling How often each group of starting angles reaches a lattice. The share of runs that settle, for the nine angles the table was grown from, for eight angles placed halfway between them, and for three angles below the nine's lowest. The nine settle far more often than either. So it is the refinement rather than the extension that drags the share down, and the settling share this collection reports is biased upwards by the choice of angles rather than by their range.

Round numbers are not a sample

The nine starting angles the settling table was grown from reach a lattice four times in ten. Eight angles placed exactly halfway between them reach one a quarter of the time. The difference is not noise and it is not the range — several of the nine sit next door to somewhere a stem could settle.

emergence · Settling Where a stem started at each of twenty angles ends up, at a falloff exponent of 3. One column per starting angle and one row per rise, coarse at the top. A filled cell is a run that reached a lattice, its tone the destination it reached; a pale cell is a run that never settles. Runs of one tone across neighbouring columns are basins, and the widest of them spans 7 consecutive angles. The angles are 6.25 to 10 degrees apart, so a basin narrower than that cannot be seen here and a single filled cell says nothing about how wide its basin is.

Twenty angles instead of nine

Every claim in this collection about where a stem ends up rests on nine starting angles a cell, and the file that uses them says so — it computes a binomial error of 0.17 and declines to read a spread against it. Eleven more angles halve that error and change what several of the numbers were.

emergence · Settling 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 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 testingBasinSettlingStarting angleAttractorResolutionSamplingAblationAzimuth gridFalloff exponentMeasurement error

All concepts