Raup's three numbers

David Raup published a model in 1966 that generates coiled shells from three numbers.

W — how much the whorl expands in one turn. D — how far the generating opening sits from the coiling axis. T — how far it travels along the axis per turn.

Gastropods, ammonites, bivalves, brachiopods, nautiloids: essentially every coiled shell in the fossil record and every one alive is a point in that three-dimensional space.

What the compression means #

Three numbers is very few, and the immediate question is whether the model is doing work or whether it is flexible enough to fit anything.

It is not flexible. The model has no freedom to bend a whorl, thicken a wall, or adjust a profile locally. Given W, D and T the shape is fixed, and a shell that does not lie in the space cannot be produced by adjusting the parameters — it is simply outside the model.

So the fact that real shells lie in it is a genuine constraint being satisfied, and it says something specific: shells grow by accretion at an aperture, with the aperture scaled, rotated and translated at fixed rates. That is a claim about the growth process, and the model’s success is evidence for it.

The most useful line in the space #

A whorl at angle θ occupies a radial interval, and the next turn’s inner edge clears this turn’s outer edge exactly when W·D ≥ 1. One product, one line, and it separates two shapes of shell that anyone can tell apart.

Above the line the whorls are free of each other — an evolute shell, like a planispiral ammonite or a ram’s horn, where each turn is visible and separate. Below it the whorls are in contact, each wrapping over the one before — an involute shell, which is what most gastropods are.

The label that was wrong #

The first version of this called the region below the line impossible, and drew it in the colour reserved for failed claims.

That is simply false. Whorls in contact are not a contradiction — they are the ordinary condition of most snails, where each whorl wraps over its predecessor and the shell is a compact spiral rather than an open coil. Nothing about it is geometrically inconsistent.

The error was a specific and instructive kind: taking a boundary that the geometry genuinely computes and attaching to it a meaning the geometry does not supply. The line is real; “impossible” was an interpretation, and it was wrong.

What the model cannot say #

Raup’s actual finding, the one the model is remembered for, is not in the geometry at all.

He plotted where real shells fall in the space and found that large regions of it are empty. Not geometrically excluded — perfectly constructible, and unoccupied. Some of the emptiness is functional, some is developmental, and some may be historical accident.

That is an empirical result obtained by measuring shells, and this site does not have it. Drawing the boundary is geometry and can be computed; drawing the occupancy is a census and cannot. The figures here show where the geometry changes and stop, and saying so is the difference between reporting Raup’s model and claiming his result.

The same distinction applies throughout: the packing measurements compute what a criterion says and cannot say what a plant does, and the dynamical model computes what a rule produces and cannot say what a meristem uses.

The third parameter #

T is what turns a flat coil into a tower.

At T = 0 the whorls stay in one plane and the shell is planispiral — an ammonite, a nautilus, a ram’s horn. Increase T and each whorl steps along the axis, producing the conical spire of an ordinary garden snail or a whelk. Very large T gives a shape like an ice-cream cone with a thread wrapped round it.

The reason the figures here mostly show T = 0 is that a section through a planispiral shell is a plane curve, and a plane curve can be drawn honestly on a page. A high-T shell needs a projection, which introduces choices of its own — which is why the shells here are read as plane curves at a stated growth factor rather than as pictures of solids.

What the three parameters do, one at a time #

Sweeping each while holding the others fixed is the quickest way to see what the model is claiming.

W, the whorl expansion. At W just above 1 the shell is a tight coil of many whorls, like a nautilus or a planispiral ammonite. At W = 2 or 3 the whorls are visibly growing, like a garden snail. Past about 6 the shell opens so fast that a single turn dominates it, which is the limpet-and-bivalve region — a bivalve is a Raup shell with an enormous W and less than one whorl.

D, the distance from the axis. At D near 0 the generating curve passes close to the coiling axis and the whorls stack tightly, which is the involute case. Increase D and the whorls separate, giving the open coil of a ram’s horn.

T, the translation along the axis. At 0 the shell is planispiral, flat. Increase it and the whorls step along the axis into the familiar conical spire.

The remarkable thing is that a bivalve, a snail and an ammonite differ in three numbers and in nothing else — the same generating process with different constants. That is a strong claim about growth and it survives being checked against a great many shells.

Why this counts as a model rather than a fit #

The distinction matters and it is easy to get wrong, since three parameters sounds generous.

A fit has enough freedom to accommodate whatever it is shown. Raup’s model does not: given W, D and T the entire shape is determined, and there is no term left over to bend a whorl or adjust a profile. A shell whose section is not a scaled, rotated, translated copy of itself at every turn simply cannot be produced.

That means the model makes a falsifiable claim about every shell it is applied to — that the aperture is transformed by fixed amounts per turn — and shells overwhelmingly satisfy it. That is evidence about the growth process rather than evidence about the flexibility of the parameterisation.

Real departures exist and they are informative. Shells whose expansion rate changes with age, or whose aperture profile changes at maturity, show up as systematic residuals rather than as a slightly different point in the space — which is the same diagnostic the spiral fit uses, where a residual that is not arithmetic noise means the model does not apply.

The census this site does not do #

Raup’s famous result is not the model. It is what he found when he plotted where real shells fall in it: large regions are empty.

Not geometrically excluded — perfectly constructible, and unoccupied. Some of the emptiness has functional explanations (a shell whose whorls do not touch is mechanically weak; one that expands too fast cannot be carried), some is developmental, and some may be historical accident — a region nothing happened to evolve into.

That is an empirical claim obtained by measuring shells, and this site does not have it. The figures compute the boundary where the geometry changes character and stop, and the distinction between what the geometry says and what a census says is the same one the packing essays keep.

Saying so is not modesty. A site that drew an occupancy map it had not measured would be doing the thing the whole subject suffers from.

Morphospace as a way of thinking #

The idea outlived the shell model and is worth naming for itself.

A morphospace is a parameterisation of possible forms, in which real organisms are points. Once one exists, a set of questions become askable that were not before: which regions are occupied, whether occupation has changed through time, whether lineages move through it or stay put, whether the empty regions are empty for a reason.

That is a considerably more productive framing than describing shapes in words, and it is the reason Raup’s paper is still cited. The specific parameterisation matters less than the move from adjectives to coordinates — which is, in a smaller way, what counting spirals instead of admiring them does for the other half of this site.

The fourth parameter, and why it is usually omitted #

Raup’s original formulation has a fourth: the shape of the generating curve itself, which the version here takes to be a circle.

Real apertures are not circles. They are ovals, D-shapes, slots, and in ammonites they can be elaborately lobed. Allowing an arbitrary generating shape adds as many parameters as the shape needs and makes the space unbounded, which defeats the purpose of having a morphospace.

So the standard practice is to fix the generating curve and accept that the model describes the coiling rather than the shell. That is a real limitation and it is worth stating: two shells at the same point in W–D–T space can look quite different if their apertures differ, and the model says nothing about which aperture a species has.

What it does say is that the coiling and the aperture are largely independent — the same coiling parameters appear with many aperture shapes across unrelated groups, which suggests the two are controlled separately in development. That is a claim the model makes by being incomplete in a specific way, and it is testable.

Where the model came from #

Raup was working in the early 1960s with an analogue computer and later a digital one, and the model’s shape reflects the constraint: three parameters is what could be swept and plotted with the equipment available.

That is worth knowing because the model’s virtue is partly an accident of its limitations. A more flexible parameterisation would have described more shells and would not have produced a picture anyone could look at, and the picture is what made the argument. A three-dimensional space can be drawn, sampled and shown to be mostly empty; a twenty-dimensional one cannot.

The same trade appears throughout this collection: Vogel’s rule has one parameter and describes a seed head, and its inflexibility is exactly why an agreement with a real head means something.

What a point in the space is not #

One last clarification, because a compression this good invites over-reading.

A shell’s position in W–D–T space is a description of its coiling geometry and nothing else. It says nothing about the animal’s biology, ecology, relatedness or history, and two shells at nearly the same point can be from unrelated groups separated by hundreds of millions of years.

That is a feature rather than a defect: it means the space is a genuine morphospace, a description of form independent of ancestry, and it is what makes questions about convergence askable. Two lineages arriving at the same coiling parameters from different starting points is a fact that needs the space to be stateable at all.

But it also means a position in the space is not a classification, and the model should not be read as a taxonomy. It describes what a shell is shaped like, and what a shell is shaped like turns out to be a poor guide to what it is.

Why this belongs on a site about phyllotaxis #

The connection is not the shells. It is the method.

Both halves of this site take a family of biological forms, find a small parameterisation that generates them, and then use the parameterisation to turn claims into measurements. A seed head is one angle; a coiled shell is three numbers; a branching network is one exponent.

Once the parameterisation exists, the same set of moves becomes available in each case. The parameter can be recovered from a form, so a claim about the form becomes a claim about a number. Regions of the space can be compared, so “unusual” becomes a position rather than an adjective. And the model’s inflexibility means an agreement is evidence rather than a fit.

That is the whole method, and Raup’s paper is one of its cleanest early examples in biology — which is a reason to have it here beyond the shells themselves.

What the model made possible #

A last note on consequences, since the paper is sixty years old and still cited.

Before it, describing shell form meant words and drawings, and comparing two shells meant comparing descriptions. After it, a shell is a point, a species is a cloud, a lineage is a trajectory, and the question of whether a group’s morphology has expanded or contracted through time becomes a measurement.

That reframing is worth more than the specific parameterisation, and it has since been done for many other groups with quite different models. The move from adjectives to coordinates is the transferable part.

What to measure first #

For anyone with a shell and an interest, the parameters come in a useful order.

W first. The growth factor per turn is the parameter with the widest range and the easiest measurement: log radius against unwrapped angle, least squares. It also settles the only famous claim about shells.

T second. Whether the coil stays in a plane or walks along an axis is visible at a glance and separates the two biggest groups of shells without any arithmetic.

D last. The distance of the generating curve from the axis is the fiddliest to measure and the one that decides whether the whorls touch, so it matters most for the shells where W and T have already put them near the boundary.

Two of the three take a photograph and a spreadsheet, which is the practical form of the claim that this is a model rather than a description.

The ordering carries one more instruction that is easy to lose. Each parameter should be reported with the range of the data it was fitted over, because a fitted exponent means different things over half a whorl and over five. That is not a general caution about care: it is the specific way this kind of fit fails, and this collection has measured it happening in a neighbouring subject, where a growth exponent fitted on the part of a branching tree a person can reach comes back precise and wrong. A shell has the same hazard in a milder form — the inner whorls are small, hard to measure and where W is least well determined, so a measurement taken where the shell is convenient is a measurement of the outer whorls.

The remedy is the same one: state the arc the fit used, and fit the same shell twice over different arcs. Two values of W that disagree can be a finding about the animal’s growth through life, which is the thing the space’s first caution below says it cannot represent — but only once the centre is known. A centre wrong by a quarter of the innermost radius makes the two halves of a spiral that does not change disagree by 4.1 per cent, the split a genuine 8.35 per cent change produces, so the disagreement has to be larger than the centre’s before it says anything about growth.

What the space does not have #

A closing caution, since the parameterisation is presented approvingly.

Raup’s space has no time in it. A shell is a point, not a trajectory, and the animal’s growth through life — which may change W as it matures — is compressed away. That is a real loss and it is the same loss Vogel’s model takes on a seed head: a description of the finished form, with the process discarded.

It has no function in it. Two shells at the same point may live entirely differently, and the model has nothing to say about why one region of the space is occupied and another is not.

And it has no mechanism. The parameters describe the coiling; they do not say what the animal does to produce it, which is the same gap that separates a grammar from an explanation.

Those three absences are exactly what makes it a morphospace rather than a theory, and being clear about that is what stops the compression from being over-read — which, as with the nautilus, is the usual fate of a good description of a shape.