Phyllotaxis
the spirals are counted, not admired
Concept

Evolute — where it appears

A coiled shell whose whorls run free of one another, each turn visible and separate, as a planispiral ammonite or a ram's horn is. It is the region of Raup's morphospace above the line where the product of expansion and displacement is one, and that line is exact rather than approximate.

Named by 8 essays across one field — 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 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 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 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 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 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.

InvoluteMorphospaceHonest limitsModel scopeWhorlGrowth factorParameter spaceClaim testingError propagationIdentifiabilityMeasurement errorNegative result

All concepts