Phyllotaxis
the spirals are counted, not admired
Concept

Bisection — where it appears

Finding where a quantity crosses a value by repeatedly halving a bracket that contains the crossing. It is only an answer when the ends of the bracket straddle the value; a bracket that does not returns one of its own ends, which looks exactly like a result.

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

The boundary located at 481 expansions, against D = 1/W. 481 expansions log-spaced across the range, each boundary found by bisecting the two drawn circles to two hundred steps rather than read off a grid. The hyperbola D = 1/W lies on the located curve to 2.220e-16 over the whole of it — the last bit of a double — so the textbook line is the boundary and not an approximation of it. The closed form and the bisection agree to 2.81e-15 at worst over all 3,200 located boundaries, which is what makes a residual of zero a reading.

The line was already exact

The boundary between shells whose whorls run into one another and shells whose whorls run free is quoted everywhere as D = 1/W, and a survey designed to measure how far off it sits found that it is not off at all. Located by bisecting the drawn circles at 481 expansions, the residual is 2.2 × 10⁻¹⁶ — the last bit a double holds, over the whole range.

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.

Closed formHonest limitsModel scopeMorphospaceWhorlAxis translationClaim testingDiscretisationEvoluteGrowth factorInvoluteMachine epsilon

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