Phyllotaxis
the spirals are counted, not admired
Concept

Bracket — where it appears

The interval of swept rises a wall is known to lie inside, whose ends are sweep positions rather than error bars. Two of four falloff exponents had one with no upper end at all, and closing every bracket put the four walls inside a factor of 1.111 of one another.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

Ten of the 24 orderings four exponents admit, from 15 readings of the wall. Every arrangement of the four falloff exponents is a cell, and a cell is filled when some level of some sampling puts the four walls in that order. 15 readings of runs that are shared cell for cell give ten of the 24, with the most common occurring three times. The marked cells are the orderings read at a half, the only level any round of this collection has published, and each of them occurs once. Exponent 5 is ranked first in 8 of the 15 readings and each of the others in two or three.

The level was doing the ordering

Four falloff exponents have been put in order by the rise at which half their runs stop settling, and a half is the only level that order has ever been read at. Read at nine levels the same runs give ten different orderings of the same four numbers, and the one comparison in the whole study that resolves runs the other way.

mechanism · Falloff exponent Every bracket on the wall at a half, before the refinement and after. One pair of bars per falloff exponent: above, the rises consistent with that exponent's crossing on forty starting angles and the 5 published rises; below, the same on eighty angles and 9 rises. Two of the four have an open end before — at a half that is the coarse end, where no published rise has a share confidently above the level — and none is open after, the widest closing at 1.57 times. The four walls sit inside a factor of 1.111 of one another, against 1.624 on the reading that could not locate them, so they came closer together rather than further apart.

Four walls closer than they looked

Two of the four falloff exponents had a wall with no upper end at all, and the other two were located to factors of two and a half and nearly four. Nine rises at eighty starting angles close every bracket — and the four walls turn out to sit inside a factor of 1.111 of one another, which is narrower than the narrowest bracket.

mechanism · Falloff exponent The bracket at each of the four corners of the refinement, at a half. The geometric mean bracket at each corner, and under it how much of the joint narrowing each half of the refinement accounts for. It is taken over the two exponents whose bracket is closed at every corner, with the two halves of the refinement crossed against each other. The rule across the bars is 1.54, the narrowest bracket the published rise list can express at all. On forty starting angles and the published rises it is 3.06 times with two of the four brackets closed; doubling the angles alone gives 1.96, halving the rise spacing alone gives 1.46, and both together give 1.44. In the logarithm, where the two effects add, the rises account for 98.8 per cent of the joint narrowing and the angles for 59.3 per cent — summing to 158.1 per cent, which is what overlapping causes look like and not two independent factors of two.

Two refinements that do not multiply

The design that located the wall did two things at once — doubled the starting angles and halved the rise spacing — and the arithmetic behind it assumed each would buy about a factor of two. The finer rises did ninety-nine per cent of the narrowing and the doubled angles added under one, because a bracket's ends are rises and no error bar can move them.

mechanism · Falloff exponent The settling share against the rise at exponent 5, on the refined rises. One line, over 80 starting angles at exponent 5. The rise runs coarse on the left and the level the wall is read at is the dashed rule. Between 0.02 and 0.013 the refined list holds two rises the published one does not, and at exponent 5 the share reaches 0.725 at 0.01732 against 0.550 at the ends of that stretch — 3.13 standard errors above the higher end. It turns in all three readings independently, which is what separates a feature of the curve from a bump in a sample. The ringed point is that maximum.

A maximum in the gap

Four falloff exponents have refused to separate on every quantity this thread has read off them, and the wall that was supposed to tell them apart cannot. Two of the four carry a maximum in the settling share at a rise the published list stepped straight over, and it is there in both halves of the sampling independently.

mechanism · Falloff exponent Six rungs walked at the grid, 10 crossings between them. Each row is one rung, drawn from its coarse end on the left to its fine end on the right and scaled to its own width so positions inside different rungs can be compared. The shaded stretch is the band the ladder grows around the rise it recorded as that rung's handover. five of the six rungs carry exactly one crossing, and on each of those the whole stretch a second one could sit in has been walked at the grid and closed at both ends. The 3/4 rung of the Lucas branch carries five, spaced 22, 12, 15 and 20 grid steps apart, and its band holds three of them. The rule on each row is a located crossing.

Five rungs walked

Six rungs of the ladder carry a handover and only one of them had ever been walked at the resolution its rises are named on. Walking the other five costs 1,224 grown stems and no cuts at all, and it returns a crossing count per rung — five ones and a five.

cylinder · Handover grid Six handovers relocated, in steps of the sweep that recorded them. The ladder finds a handover by stepping at a ratio of one per cent, which never lands on the grid the rises are named on, so a recorded handover is the nearest rise the sweep visited to a crossing nobody had located. This is the difference, in units of the sweep's own step at that rise: 0.082 to 0.835, mean 0.374. All six are positive and all six are inside a single step of the sweep, and both of those are predictions rather than summaries: the ladder reports the first rise it visits at which the ordering has already changed, so the rise it records must sit on the fine side of a crossing and within one of its own steps. The dashed rule is one step of the sweep, which is the bound the sampling predicts.

The handovers corrected

Six recorded handovers, relocated to the grid against where a one-per-cent sweep put them: all six sit on the fine side of a crossing and all six inside a single sweep step. Nothing about the rung explains the size of the discrepancy, which is what a sampling artefact is supposed to look like.

cylinder · Handover grid

Named alongside it

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

Claim testingHonest limitsSamplingResolutionRiseFalloff exponentSettlingDiscretisationInstrument settingMeasurement errorNegative resultError propagation

All concepts