Phyllotaxis
the spirals are counted, not admired
Series

Morphospace — the series

9 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · shells
  2. 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.

    part 2 · shells
  3. 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.

    part 3 · shells
  4. One geometry, six boxes: 4.642 per cent to 52.81 per cent forbidden. The share of each box that the coiling geometry excludes, across six boxes that differ only in where their edges were put and in how one axis is sampled. The spread is a factor of 11.4, from 4.642 per cent in the widest box to 52.81 per cent in the tightest, and sampling the same two decades of expansion geometrically rather than uniformly multiplies the answer by 4.59 on its own. The forbidden area is the integral of 1/W and grows as a logarithm while a box grows as a line, so the fraction has no value of its own to quote.

    A fraction of nothing

    Six boxes differing only in where their edges were drawn give between 4.64 and 52.81 per cent for the same geometry, and sampling one axis geometrically rather than uniformly multiplies the answer by 4.60. The share tends to zero as the box widens, because the region under a hyperbola is a logarithm and a box is a line.

    part 4 · shells
  5. 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.

    part 5 · shells
  6. 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.

    part 6 · shells
  7. The band of shells a measured growth factor cannot place. The boundary D = 1/W is exact — located by bisection to the last bit a double holds. A specimen is not: its growth factor arrives with an error, and carrying that error onto the line turns it into a band, running from 1/(W(1+b)) to 1/(W(1−b)). At an expansion of 3.2 and an error of 54.4% the band runs from 0.2024 to 0.6853 around an exact 0.3125. A shell inside it has whorls that a measurement cannot say are in contact or free.

    The band nobody can be placed in

    The boundary between shells whose whorls run into one another and shells whose whorls run free was located here to the last bit a double holds. A specimen is not a point on that line, it is a measurement with an error, and carrying the whole measured error budget onto the boundary turns the line into a band running from 1/(W(1+b)) to 1/(W(1−b)). At the budget with the dividers set aside that band covers 48.2 per cent of the box the morphospace figure here is drawn on, and its share runs from 6.7 to 59.5 per cent across the six boxes in use — the same box-dependence the contact region itself showed. The angle criterion carries the same error better above an expansion of 1 + √2 and worse below it, exactly.

    part 7 · shells
  8. The band the boundary's two numbers make together. The undecidable band at an expansion error of 54.4%, drawn twice: carrying that error alone, and carrying it with the 16.1% the distance from the axis costs at the same assumed centre. At an expansion of 3.2 the one-error band runs 0.2024 to 0.6853 and the two-error band 0.1743 to 0.8172, around an exact 0.3125 — wider by a factor of 1.33. The wider band is not a worse measurement; it is the same measurement with the half of it that was missing put back.

    A floor no better fit can lift

    The band of shells nobody can place was built from one of the two numbers the contact boundary relates, and the other has now been priced. Carried together they widen the band by a third and take 48.2 per cent of the morphospace box to 59.7. The number that matters is further down: with the whorl expansion measured perfectly, 12.6 per cent of the box is still undecidable, and at an expansion controlled to a hundredth 94 per cent of what remains belongs to the second number. And the two errors are not independent — they come out of one guessed centre, which traces a curve across the boundary rather than a rectangle around it.

    part 8 · shells
  9. Nine hundred shells whose whorls run free, read from a centre guessed to within 0.1 of the innermost radius. A population uniform in the logarithm of the expansion and in the distance from the axis, every shell's whorls running free, each read once by fitting its section from a centre displaced in a direction of its own. 20 of these 900 are read on the wrong side of the contact line, where a shell that was seen to run free cannot be, and 2.9% of all twenty thousand.

    A displaced census leaves a hole, not a pile

    A mislocated centre moves a shell's place in Raup's space rather than blurring it, and the prediction was that a whole census would be compressed against the contact line, in a pile that could pass for shells preferring the line. Twenty thousand shells read from guessed centres say otherwise. Nothing piles up. The density beside the line falls, by about a sixth in the nearest bin, because a shell near the line is as likely to be carried over it as along it. And that gives a census a test that needs no outside knowledge: a section seen to run free cannot be read below the line, and 2.9 per cent are, at a tenth of a radius of error. Seventy-nine specimens are enough for one to show up.

    part 9 · shells

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