Phyllotaxis
the spirals are counted, not admired
Concept

Monotonicity — where it appears

Whether a quantity only ever moves one way as a setting is turned. It is assumed wherever a wall is located by bracketing a crossing, and the settling share is now known not to have it at two of four falloff exponents, at a rise inside the stretch those brackets cross.

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

The site's hero shell comes free at a translation of 1.1066, and a slower one at 10.49. Setting the located boundary to zero and solving leaves T_free = √W/(W−1), drawn here across the whole expansion range on logarithmic axes. Above the curve the shell is free at every distance from its axis and the contact region has left the plane rather than merely shrunk in it; a slowly expanding shell at an expansion of 1.1 needs 10.4881 turns of translation to buy that and one expanding twentyfold needs 0.2354. The tower is what buys a shell the right to coil close to its own axis, and the faster it expands the less tower it takes.

What a spire buys

Translation along the coiling axis enters the contact boundary as its square, so a shell that has only just begun to walk along its axis has not moved the boundary at all. It is also strictly one-way, and it has a threshold above which no distance from the axis whatever puts the whorls in touch.

shells · Morphospace 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

Named alongside it

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

Honest limitsArtefactAxis translationBracketClaim testingDiscretisationEvoluteFalloff exponentGrowth factorInvoluteLocal maximumMeasurement error

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