Phyllotaxis
the spirals are counted, not admired
Concept

Silent failure — where it appears

A bound that returns a clean, plausible, wrong answer when it is crossed, rather than noise, a refusal or a wide error bar. Both bounds on the counting window are of this kind: one organ too narrow returns the previous rung of the same ladder, and a family past the offset ceiling is reported as a coarser rung at every width there is.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

The fork the cost chooses at a daughter ratio of 1: 37.47 and 37.47 degrees. Three ends held fixed, three weights fixed by the radii, and the branch point put where the total cost is least. At the optimal radius the pumping term is exactly half the upkeep term, so a segment's weight is its own cross-section and the three weights here are 1.5874, 1.0000, 1.0000. Minimising directly over the position — a 41×41 grid re-centred and shrunk 220 times, told nothing about any formula — puts the daughters at 37.4673° and 37.4673° from the parent's own forward direction, against the closed form's 37.4673° and 37.4673°.

The angle the cost chooses

Murray's exponent falls out of minimising a cost over the radius of a tube. The same cost minimised over the position of the branch point instead fixes both fork angles, and 281 networks minimised on a grid told nothing about any formula agree with the closed form to under two ten-thousandths of a degree.

branching · Fork angle 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

Named alongside it

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

Instrument settingClaim testingHonest limitsLadderParastichy pairReading windowSystematic errorThresholdBranch pointClosed formConvergenceCylinder

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