Phyllotaxis
the spirals are counted, not admired
Concept

Morphospace — where it appears

A parameterisation of possible forms in which a real organism is a point and the unoccupied regions are the question. Raup's three numbers put nearly every coiled shell that has existed into one, and reading its empty regions as forbidden rather than merely unvisited is where most over-reading of it happens.

Named by 19 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 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 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.

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 One cut along a shell's axis read back for all three of Raup's numbers, the axis found from the cut itself. The seven whorl sections a high spire at W = 2, D = 0.2 and T = 2 shows over three turns on a plane through its axis, alternately right and left of it, each drawn at 48 points moved by noise of 0.003 of the rim. Told only the points and which section each lies on, and started from an axis laid a quarter of the innermost radius off and two degrees out, the fit finds the axis 0.0055 innermost radii and 0.0053 degrees from the true one — the inset magnifies the axis at the apex — and reads W = 2.0000, D = 0.2004 and T = 1.9994: out by 0.002%, 0.205% and -0.029%. The circles are redrawn from those numbers. Held at the laid axis, the same cut reads T = 2.084 and D = 0.1746.

The cut along the axis

A median section cannot see how far a shell travels along its axis, so the third of Raup's numbers seemed to need a second cut. It does not need the first. One cut through the axis shows every whorl twice a turn as a whole circle, and fitted with the axis free it returns W, D and T together — to 0.023, 0.085 and 0.031 per cent on a high spire drawn to a thousandth of its rim, from an axis laid a quarter of a radius off and two degrees out. On the shell the median section was read on, it reads the expansion twice as well over a turn and the distance from the axis as well. Laid by eye and left there, the axis costs T four per cent and a high spire's D thirteen, and the residual says so. What it does not always say is that the saw missed the axis: on a low spire a cut a fifth of the innermost radius off moves W by two and a half times its own noise error with the residual at the noise.

shells · Raup A cut along the axis of a shell whose opening is an ellipse, read with circles and with ellipses. The seven sections a high spire at W = 2, D = 0.2 and T = 2 shows over three turns on a plane through its axis, its opening an ellipse 1.25 times as tall as it is wide and turned 20° against the axis, each drawn at 48 points moved by noise of 0.003 of the rim. Read with Raup's circles, the cut gives W = 1.9995, D = 0.1413 and T = 1.9032, with the points missing their circles by 6.4 times the noise. Read with ellipses, fitted with the rest, W = 1.9994, D = 0.2003, T = 2.0024, the aspect 1.2524 and the turn 19.65°, missing by 0.93 times the noise. The dashed circles and solid ellipses are redrawn from each reading.

An opening that is not round

One cut through a shell's axis returns Raup's three numbers to a few hundredths of a per cent when every section is a circle. A snail's opening is closer to an ellipse. Read with circles, a cut of a shell whose opening is a quarter taller than it is wide gives W exactly and puts the whole misfit into the other two: D comes back 29 per cent low and T 4.9 per cent low, by a first-order form in which the circle takes the ellipse's mean half-axis. A nearly round opening is the dangerous one — an aspect of 0.99 moves D by fourteen of its own noise errors while the points miss their circles by only a quarter more than the noise. Fitted with ellipses, the cut reads all five shape numbers with W, D and T nearly as precise as before, and the opening's height does not trade against the spire's translation.

shells · Raup An axial cut of a shell with round openings, photographed fifteen degrees off the face's normal, squashed at a bearing of 45 degrees. The seven sections of a high spire at W = 2, D = 0.2 and T = 2 with round openings, photographed with the camera tilted 15° so that the picture is squashed to 0.9659 of its size across a bearing 45° from straight across the axis, each section drawn at 48 points with noise of 0.002 of the rim, and the ellipses the opening reading redraws. Read as a shell with elliptical openings photographed square: W 2.0184, D 0.2070, T 1.9943, aspect 1.0193, turned -37.5°, missing the points by 3.34 times the noise. Read as round openings through one squash: W 1.9996, D 0.2002, T 1.9984, the squash 0.9655, missing by 0.92 times the noise.

A photograph squashed along the axis is another shell

A cut shell is photographed, not measured with calipers, and a camera a few degrees off the face's normal squashes the whole picture across one bearing. Read as a shell with elliptical openings, a squash straight across the axis or along it is absorbed without a trace: the picture is exactly that of another shell, its openings ellipses and its translation off by the squash — six per cent at a twenty-degree tilt — and no fit can tell. At any bearing between, the squash turns every opening the same way on both sides of the axis, which a real opening, mirrored across the axis, never does: a ten-degree tilt at fifteen to seventy-five degrees from the axis is caught on every section. The common photograph, the shell upright in the frame and the camera off to one side, is the one that cannot be caught.

shells · Raup An axial cut of a high spire photographed twenty degrees off the face's normal, along the axis, by a camera close enough to see it in perspective. The seven round sections of a high spire at W = 2, D = 0.2 and T = 2, each drawn at 48 points with noise of a thousandth of the rim, photographed by a pinhole 10 rim radii away and tilted 20° along the axis, so that the whorls nearer the lens are larger than those further away. Read as round openings through one squash, this picture gives W 2.0802, D 0.2003 and T 1.8728, missing its points by 3.59 times the noise; through a camera fitted beside the shell, W 2.0007, D 0.2001 and T 1.9975, the camera 10.0 rim radii away at 19.8°, missing by 0.93. Of twenty such pictures, 20 are missed by the squash reading by more than any photograph taken from infinitely far.

A cut seen in perspective is not another shell

A photograph of a cut shell squashed straight along the axis is exactly the picture of another shell, with nothing to tell them apart. A camera close enough to see the cut in perspective is different. The near whorls come out larger than the far ones, and a whorl's size then changes across the picture in a way no shell's own expansion does — so the reading that fitted a distant photograph misses a near one by more than the noise, even straight along the axis. Before it is seen the error is small, W read too large by about the sine of the tilt over the distance. A camera fitted beside the shell, three numbers more, takes it back entirely: W, D and T within a third of a per cent at every bearing and every distance down to three rim radii.

shells · Raup An axial cut of a high spire with turned elliptical openings, photographed in perspective and read with twelve numbers. The seven sections of a high spire at W = 2, D = 0.2 and T = 2 whose openings are a quarter taller than wide and turned 20 degrees, each drawn at 48 points with noise of a thousandth of the rim, photographed by a pinhole 5 rim radii away tilted 20 degrees across the axis. Read with the shell's nine numbers and the camera's three, the fit returns W 2.0001, D 0.1974, T 2.1189, an aspect of 1.313 turned 15.8 degrees, and a camera tilted 4.2 degrees 1.0 rim radii away, missing the points by 0.79 times the noise. At this bearing the camera slides along the line on which the tangent of its tilt over its distance is fixed, and the spire's T, the opening's aspect and its turn move with the tilt it settles on, with the picture fitted to the noise.

A free opening lets the camera trade again

A camera fitted beside a shell recovered every photographed cut whose openings were round. Give the opening its own shape — an aspect and a turn, as a snail's has — and the twelve numbers recover everything only when the camera's tilt runs at a bearing between the axis and the line across it. Straight across or along the axis, a pinhole tilted τ at L rim radii is exactly a pinhole tilted τ′ at L′ followed by a squash, whenever tan τ′/L′ = tan τ/L, and that squash is another shell. Every camera on the line fits a noiseless picture to rounding. The fit keeps W and the perspective's strength, and nothing else about the camera: T and the aspect come back scaled by a ratio of cosines, a turned opening's turn and its D move with them, and on noisy pictures T ran from six per cent low to a quarter high with every picture fitted inside the noise.

shells · Raup Two photographs of one cut shell, read together with fifteen numbers. The seven sections of a high spire at W = 2, D = 0.2 and T = 2, openings a quarter taller than wide and turned 20 degrees, photographed twice with noise of a thousandth of the rim: first by a pinhole tilted 10 degrees straight across the axis at 10 rim radii, then by one tilted 10 degrees with its tilt turned 90 degrees from the first's at 10 rim radii. Read together — the shell's nine numbers shared, three for each camera — the fit returns W 1.9999, D 0.2000, T 2.0013, an aspect of 1.251 turned 19.9 degrees, and a first camera tilted 9.9 degrees, missing the points by 0.86 times the noise. Turned, the second picture pins the shell.

Two photographs pin the shell only from two sides

A photograph of a cut shell whose opening has its own shape, taken straight across or along the axis, fits every camera on a line, and each camera on the line reads a different shell. A second photograph of the same cut shares the shell and brings a camera of its own. Taken at the same bearing — farther away, or steeper — it does not help: the two cameras slide together along a family set by one closed form, cos τ₁′/cos τ₁ = cos τ₂′/cos τ₂, each picture fitted to rounding at every point of it, and the shell slides with them. Turned so that its tilt runs across a different direction, it pins the shell, and on noisy pictures a turn of ten degrees is already enough.

shells · Raup 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.

shells · Morphospace How well a registered pair reads a shell's translation, against how far apart its cameras' bearings are. Nineteen numbers fitted to two photographs. The high spire: 0.0152 at 5 degrees, 0.0054 at 10 degrees, 0.0020 at 20 degrees, 0.0013 at 45 degrees, 0.0023 at 70 degrees, 0.0069 at 80 degrees, 0.0298 at 90 degrees. The low spire: 0.0062 at 5 degrees, 0.0032 at 10 degrees, 0.0017 at 20 degrees, 0.0006 at 45 degrees, 0.0009 at 70 degrees, 0.0071 at 80 degrees, 0.0078 at 90 degrees. With frames given, the same pair at ninety degrees read 0.0013 on the high spire. A camera merely turned in the hand reads 0.0298. With the scale between the pictures known and the rest free, the high spire reads 0.0154, 0.0053, 0.0019, 0.0013, 0.0017, 0.0019, 0.0018 at the same bearings.

Two free frames cost one right angle

Two photographs of one cut shell pin it only when their cameras stand at different bearings, and the pair at right angles read best of all. Each of those fits was handed both pictures in one frame, and no real pair comes that way. Give the second picture a frame of its own, rotated and scaled and shifted, and leave the reading to find it: nineteen numbers, not fifteen. It costs almost nothing. The rotation comes back to a twentieth of a degree, the scale to under a tenth of a per cent, and the shell's numbers as well as before, except at one bearing. At exactly ninety degrees a free scale swallows one camera's squash whole, the fit slides, and the pair that had been best reads the translation twenty-three times worse than a pair at forty-five.

shells · Raup

Named alongside it

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

Honest limitsModel scopeWhorlIdentifiabilityMeasurement errorClaim testingParameter spaceEvoluteGrowth factorInvoluteError propagationResidual

All concepts