Phyllotaxis
the spirals are counted, not admired
Concept

Reading window — where it appears

The stretch at the top of a run a measurement is taken over, here a hundred and twenty organs. It is a tail rather than a slice, so a wide window on a short run reaches back into the disturbance and reads the healing as the pattern.

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

All 19 destinations, counted at 17 windows from 20 to 800 organs. One row per destination, one cell per counting window, coarsest on the left. A dark cell is a window returning the same pair as the standing 200; a warm cell is a window returning a different one; a pale cell is a window that reaches below the run's own settling organ and was dropped. 5 of the 19 rows differ anywhere, and every difference sits left of the floor its own pair sets — 20 organs for a pair under 14 parastichies and the larger count plus 6 above it. The census read at 25 organs and at 800 is the same census.

The window nobody varied

Every parastichy pair this collection reports about the settling table was counted over the top two hundred organs of a run, and that number had never been moved. Moved seventeen ways across a factor of forty, on all three hundred and eighty-four settled runs, it changes nothing at all.

lattices · Read window What a 200-organ window allows against what the offset ceiling allows. Two settings decide what the counter can return, and at this window the offset ceiling is the tighter of them. A window of 200 organs holds a family of at most 194, since the counter scores an offset only where it has 6 hops of it inside the band; the largest offset it looks at is 60, a factor of 3.2 between them. The ticks are the rungs of the two ladders this collection is built on, and the largest counted number anywhere in the settling table is 19 — inside both bounds, which is what makes the table inert.

A plateau the instrument should have had

The counting window decides nothing on the settling table, and it has two bounds that decide everything outside it. A window one organ too narrow returns the previous rung of the same ladder, a family past the offset ceiling is reported as a coarser rung at every width there is, and neither failure produces a refusal, noise or a wide error bar.

lattices · Read window The window 16 known pairs need, against the 20 organs the counter refuses under. Cylinders built at a stated divergence and rise, counted at every window from 20 organs to 200, so the pair is known before the counter sees it. The smallest window that reads it is 20 organs for every pair up to 14 parastichies and the larger count plus 6 above that — 13 of 16 measured exactly, on both branches, with nothing fitted. A pair of 76 and 123 needs 129 organs where 2 and 3 needs 20; the open marks are pairs the counter's own offset ceiling refuses at every window, read only once that ceiling is raised.

How many organs a pair needs

A count taken over too few organs does not fail. It returns the rung below, which is a perfectly good pair, and nothing anywhere says so. The window that avoids it is not a constant but the counter's own arithmetic, and 384 settled runs sit exactly where that arithmetic puts them.

lattices · Read window The five readings of 384 whose two counted families wind the same way. Each row is an additive sequence with the term the counter stepped over drawn open between the two it returned. 4/9 is 1, 4, 5, 9, 14 with 5 skipped, and the pair a crossing requirement would have returned is 4/5 — a sequence the same destination's other runs already sit on. Dropping these readings takes the census at 40 starting angles from 31 arrangements and 15 sequences to 28 and 12, on 5 of 384 runs.

A sequence that was a reading

One of the settling table's additive sequences rested on a single count that skipped a term, and the account of it was that the count had been taken over the wrong patch. It was not. The counter works out whether its two families wind opposite ways and the reading throws the answer away, on five runs in three hundred and eighty-four.

lattices · Read window Each cut's class spread at three reading windows. The worst spread within a residue class, measured over the last 60, 120 and 180 organs of the same runs. A quantity that is steady with noise on it would give a flat line; a quantity drifting steadily along the run gives a line through the origin. Nearly every row gives the second, so the classes this thread calls steady are sliding slowly and the number that says how much depends on how much of the run is read. The dashed guide is exact proportionality.

A spread that grows with its window

A spread over a sample of a steady quantity does not depend on how big the sample is. These spreads triple when the window triples, on nearly every row of the census, which means the classes this thread calls steady are sliding — slowly, and invisibly at any single window.

mechanism · Two windows Which cuts count as periodic, at each reading window. One row per wrecked cut and one column per window. A filled cell is a cut whose worst class spread is under the ten-degree line and is therefore called periodic. At 60 organs 28 of the 30 cuts are, at 120 organs 25, and at 180 organs 25. three rows change side, all of them losing their periodicity as the window widens, and they are marked.

The window nobody moved

Three instrument settings sit between the ablation census and every statement it makes. Two have been varied and both decided answers. The third is a hundred and twenty organs at the top of a run, it has never been moved, and moving it changes the verdict on three rows.

mechanism · Two windows Where each window begins, against where the pattern begins. The run is drawn left to right, one bar per cut. The pale stretch is the disturbance still healing, up to the organ from which the profile holds its levels for the rest of the run; the dark stretch is the pattern repeating. The marks are the organ each of the three windows starts at, since a window is the last N organs of a run. On these rows the widest window starts inside the pale stretch, so it is reading the healing and calling it the pattern — which is why widening the window takes them off the periodic list.

Three rows a window moves

Three of the census's thirty cuts are periodic when a hundred and twenty organs are read and not when a hundred and eighty are. Their spreads do not grow in proportion to the window, they grow by twenty and forty times, and the reason is that a window is the tail of a run.

mechanism · Two windows

Named alongside it

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

Honest limitsClaim testingInstrument settingMeasurement errorParastichy pairArtefactDamage profilePeriodicityResidue classResolutionCensusCensus design

All concepts