Phyllotaxis
the spirals are counted, not admired
Concept

Tangency — where it appears

The condition of two curves or surfaces meeting at a point without crossing. For a coiled shell it is the boundary between tubes whose successive whorls overlap and tubes that stand free, the setting at which a whorl just meets the one before it.

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

Two measures of one boundary at W = 2.5: overlap slope 1, buried area slope 1.4999. The linear overlap and the buried area, against how far below the boundary the shell sits, both axes logarithmic. The buried fraction rises as 1.499945 — three halves, the lens between two nearly tangent circles — so at a ten-thousandth below the line 0.0011410 per cent of the whorl is under its successor and at 0.30 below it 86.83 per cent is. The linear overlap over the same range is exactly 2.5 times the distance below and falls straight to zero, so the two measures disagree about whether the boundary is sharp and the area is the one that answers the question. A shell just inside the boundary is not a different kind of shell; one well inside is.

A boundary with no edge

Two continuous measures cross the line where a coiled shell's whorls begin to touch, and they disagree about whether it is sharp. One falls to zero as a straight line and makes the line a kink; the other leaves it as a three-halves power and makes it a tangency.

shells · Morphospace An axial section of a spire at W = 2.4, D = 0.3, T = 1, with the angle that decides contact. The discs where the plane holding the axis cuts three whorls, on both sides of the axis. Every disc subtends the same half-angle from the apex, γ = 17.065°, about the line of the disc centres at β = 33.024° from the axis, so the envelope's apical angle is 100.178° whatever the expansion. Whorls with that angle touch below W = 1.8307, and at 2.4 they run free. Successive discs on one side are 2.4 times farther from the apex, and one disc gives back D = 0.3000 and T = 1.0000.

One angle decides contact

Seen from the apex of its coiling axis, every whorl of Raup's shell subtends the same half-angle, and two whorls touch exactly when the sine of that angle exceeds (W − 1)/(W + 1) — with no disagreement against the drawn discs at 400,000 random shells. Two of the three numbers enter only through the angle and the third only through the threshold, which decides which picture of a shell can answer the question: a spire's outline carries no W, a plan carries no T, and an axial section carries all three.

shells · Morphospace An ellipse twice as tall as it is wide at W = 2.4, T = 0.5, beside a circle at T = 0.25. Each panel is an opening, in colour, and the same opening one whorl on, 2.4 times larger about the apex, drawn at the axis distance where the two just meet. An ellipse twice as tall as it is wide at a translation of 0.5 meets at D = 0.391257; a circle at a translation of 0.25 meets at D = 0.391257, the same number, because stretching the axis by 1/2 turns the ellipse into the circle and 0.5 into 0.25.

The fourth number divides the third

Every boundary on Raup's cube was located for a circular opening, and three essays ended on the same hedge: the numbers would move with a differently shaped aperture by an amount nothing had measured. Measured on the drawn outlines of eleven openings, the boundary with no translation does not move at all for any convex opening symmetric about the plane of coiling; an ellipse's height divides the translation and does nothing else; the square law in the translation belongs to a round tip; and a turned opening frees ground only in the D a plan reads.

shells · Morphospace

Named alongside it

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

Claim testingHonest limitsModel scopeMorphospaceWhorlAxis translationClosed formParameter spaceSpireBisectionBuried fractionEvolute

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