Phyllotaxis
the spirals are counted, not admired
Concept

Parameter space — where it appears

The set of values a model's parameters can take, treated as a region to be surveyed rather than as a point to be fitted. Asking what fraction of it gives a stated pattern is a different question from asking whether some point in it does.

Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.

A shell section at W = 2.40, D = 0.42. W·D = 1.01, so the whorls are free of each other — an evolute shell, like a ram's horn or a planispiral ammonite. Both are things animals grow.

Raup's three numbers

Nearly every coiled shell that has ever existed is a point in a three-dimensional space — how fast the whorl expands, how far the opening sits from the axis, how far it travels along it. That is a remarkable compression, and the most useful line in the space is the one where the whorls come apart.

shells · Raup 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.

shells · Morphospace Every assumed centre from 0.01 to 500 innermost radii, at 19 spans, against a spiral drawn at 3.2. One row per span of arc, one cell per assumed displacement on a logarithmic grid from 0.01 to 500 innermost radii, shaded by the highest growth factor any of 180 directions returns there. A displaced centre reaches 6.854 at every span up to 1.15 turns and at no span from 1.2 upward, so the boundary is a span rather than a displacement. The dashed rule is the two-turn span floor, and the cheapest golden fit anywhere leaves a residual of 0.163 against a threshold of 0.15 — so a golden reading is refused twice over.

How far a centre must move

Four hundred and eighty-two thousand assumed centres, at nineteen spans and a hundred and eighty directions each, asked whether a spiral drawn at 3.2 can be made to read as the golden 6.854. It can, at every span up to 1.15 turns and at none from 1.2 upward, and every centre that manages it is refused twice over.

shells · Spiral fit 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 Two shapes, two ranges, one contrast. The exponential's lattice ends at 3.63 spacings and the gaussian's at 2.25 — ranges 47% apart — and at those two ranges the contrast is 5.74 and 6.09, 6% apart. The band is what the exponent route leaves: 3.98 at p = 1, where the uncut rule makes nothing, and 7.63 at p = 1.25, where it makes a lattice.

Two shapes, one threshold

Read in the same unit, an exponential falloff and a gaussian one disagree about where the lattice ends by half. The quantity they agree on turns out to be one the earlier work measured for an unrelated reason — and it agrees with a bracket left by a sweep of a completely different parameter.

emergence · Falloff exponent The distance from the axis is one ratio on one ray, read from a centre displaced by 0.25 of a whorl. A section at W = 2.40 and D = 0.42, with four rays cast from a centre displaced 0.25 of the read whorl's outer radius to the right. Along each ray the reading is the inner wall's distance over the outer wall's — from the true centre that ratio is 0.42 at every azimuth, to the last bit a double holds. From the displaced centre the same four rays give 0.227, 0.351, 0.582, 0.347: the ray pointing at the displacement reads low and the ray opposite reads high, because subtracting the same length from both distances moves their ratio toward one. The spread is 0.227 to 0.582 on a shell whose distance from the axis is 0.42.

What the axis distance costs

Raup's contact boundary is a relation between two numbers and only one of them has ever been priced here. The second was expected to be the cheaper — a length against another length. It is not: an assumed centre costs it 79.3 per cent where the same centre costs the expansion 6.88, because a ratio of two distances is first order in the centre and a fitted rate is second. But a tilted camera costs it nothing at all, exactly, and averaging the reading round one whorl is free and worth a factor of 4.91. The two numbers fail at opposite ends, and they cross at 1.12 turns of arc.

shells · Raup 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.

shells · Morphospace A drawn section read back for its expansion and its distance from the axis, the centre found from the drawing itself. Both walls of a planispiral shell at W = 2.4 and D = 0.42 over 1.5 turns, every drawn point moved by noise of 0.003 of the rim radius. Told only the points and which wall each lies on, and started from a centre guessed a quarter of the innermost radius off, the fit finds a centre 0.0076 innermost radii from the true one — the inset magnifies the three — and reads W = 2.4016 and D = 0.4199: out by 0.067% and -0.020%. The lines are the walls redrawn from those numbers. Read at the guess itself, without letting the centre move, the same drawing gives W = 2.550 and D = 0.4117.

The outline finds its own centre

A worker with a sawn shell has an outline and nothing else: the centre, the expansion W and the distance from the axis D all have to come off it at once, and three numbers fitted to one outline can trade against each other. Fitted together, they do not trade where it matters. Started from a centre guessed a quarter of the innermost radius off, the drawing locates its own centre to about a thousandth of that radius, and over a turn of section drawn to a thousandth of the rim it returns W to 0.12 per cent and D to 0.045 — where reading the same drawing at the guessed centre gives 16.8 and 0.98. W's error rides the centre's; D's does not, and W and D do not trade against each other at all. The trade the round trip was set to watch for appears only where the model is wrong: a view five degrees off the section's plane moves W by three times its own noise error while the fit's residual stays within the noise, because the free centre moves to absorb the squash.

shells · Raup

Named alongside it

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

Model scopeMorphospaceClaim testingGrowth factorHonest limitsWhorlEvoluteInvoluteIdentifiabilityError propagationMeasurement errorAxis translation

All concepts