Phyllotaxis
the spirals are counted, not admired
Concept

Involute — where it appears

A coiled shell whose whorls wrap over the one before, in contact rather than free, which is what most gastropods are. It is the region of Raup's morphospace below the contact line, and a real one has its aperture indented where the earlier whorl lies against it, which the model does not carry.

Named by 6 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 A shell expanding by 3.2 a turn, divided by 13 septa to a whorl. A shell expanding by 3.2 a turn at axis distance 0.1, seen down its coiling axis, with 13 septa to a whorl, 27.7° apart; the last whorl's chambers are shaded. Each chamber is the one before it turned and scaled about the apex by 3.2^(1/13), so a length grows by ×1.0936 from one chamber to the next, an area by ×1.1960 and a volume by ×1.307896. Integrated over the tube's own rings, successive chamber volumes grow by 1.307896 to 1.307896, and a chamber and the one a whorl out differ by 32.7680, which is 3.2³ = 32.7680.

What the septa count

A nautilus's chambers are each a scaled copy of the last, and an earlier essay gave their ratio as about 1.3 — what a growth factor of 3.2 gives over a third of a turn. It does not: a third of a turn at 3.2 is 1.474 in length. A ratio of 1.3 is 4.43 septa a whorl as a length, 8.87 as an area and 13.30 as a volume, so the dimension decides the count threefold. And the count is an exponent in any reading of the growth factor taken from one chamber to the next: one septum miscounted at thirteen moves it by 9.14 per cent. A chamber and the one a whorl out give W³ with no count at all.

shells · Nautilus

Named alongside it

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

EvoluteHonest limitsModel scopeMorphospaceWhorlGrowth factorResolutionCensusClaim testingDiscretisationMeasureNegative result

All concepts