Phyllotaxis
the spirals are counted, not admired
Concept

Offset ceiling — where it appears

The largest hop a counter is allowed to measure, set by the width of its window or by an explicit limit. A family beyond it cannot be returned at all. A count below the ceiling can still look entirely normal, which is how the limit decides answers without anyone seeing it.

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

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 members of the two sequences the essays are 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 A stem at a rise of 0.003 unrolled, with each organ moved by 0.5 of a spacing. Two hundred and forty organs of an ideal lattice, azimuth across and height up, each moved in both directions by a gaussian with a standard deviation of 0.5 of the lattice's shortest vector. Open circles mark where each organ belonged, and lines join it to where it went. The pair 8/13 is the lattice's own.

A wider window buys back a shaken stem

On exact stems the counting window decides nothing once it clears a small floor, which leaves it looking like a setting nobody needs. Move every organ and it becomes the instrument. A pair that a twenty-organ window reads on an ideal lattice needs one or two hundred organs with each organ moved by half a spacing, four to eight hundred at four fifths, and up to twelve hundred at a whole spacing, where the organs have left their places altogether and the pair is still there to be counted. The price is steep, close to the two-and-a-half power of the error, and at one and a half spacings no window up to sixteen hundred organs holds the pair. When it goes, it goes upward: the wrong answer is the next pair, not the one below.

lattices · Read window

Named alongside it

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

Honest limitsInstrument settingMeasurement errorParastichy pairReading windowThresholdLadderNegative resultRefusalRungSamplingSilent failure

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