Growth as a rule

A shell has a problem that a bone does not. It is mineral, it is dead once laid down, and it cannot be remodelled. Whatever the animal adds has to go on the open end, and everything already there stays exactly as it was.

That constraint has one solution, and the solution is a shape.

Why the spiral follows #

If the animal is to keep the same proportions while getting larger, the shape it adds must be a scaled copy of the shape it has. Scaling a shape about a fixed point and rotating it is the only transformation that keeps a curve looking like itself, and the curve invariant under that combination is the logarithmic spiral, $r = a\,k^{\theta/2\pi}$.

The single parameter k is how much the spiral grows in one full turn. Everything else — the overall size, the starting angle, which way it winds — is a choice of units and orientation.

This is sometimes called gnomonic growth, after the gnomon: the shape one add to a figure to make a larger figure of the same proportions. It is the reason the same curve appears in a ram’s horn, a snail, an ammonite, a hurricane and a spiral galaxy, and its appearance in all of them says only that all of them add material proportionally, which is a weak and true statement.

Recovering the parameter #

Take the logarithm of the radius and it becomes a straight line in the angle, with slope log(k)/2π. So fitting a growth factor to a drawn spiral is ordinary least squares on log r against θ, once the angle is unwrapped so that it increases monotonically along the curve.

Done on a spiral generated at 3.2 per turn, it returns 3.2 to fifteen digits, with a worst residual of 2 × 10⁻¹⁵ in log r. That is arithmetic noise, and it is the right answer: a logarithmic spiral fits a logarithmic spiral exactly.

The value of that is not the precision. It is that the same machinery, pointed at a curve that is not logarithmic, returns a factor with a residual that is not noise — so the fit reports whether the model applies as well as what its parameter is.

The one free choice #

The fit needs a centre, and a real shell does not have one marked.

The apex of a shell is a point on the animal, not the mathematical centre of the spiral, and locating the centre is itself an estimate. So the honest statement of a measured growth factor includes how much the answer moves when that estimate moves.

Getting the centre wrong by a quarter of the innermost whorl’s radius changes the fitted factor by 4.56 per cent over two turns and 1.34 over three and a half, and the error falls as the inverse square of the arc measured. This essay used to say “under a per cent”, which is true only above four and a quarter turns and was quoted with no span attached.

That is worth knowing in both directions: it means a careful measurement over enough arc is robust to two significant figures, and it means the measurement cannot be blamed for a discrepancy of a factor of two.

That second point is the one that matters for the nautilus question, where a factor of 2.14 has to be accounted for and the centre cannot account for it.

What the spiral does not say #

The curve is a consequence of proportional growth and nothing more, so it carries very little information about the animal.

It does not say the growth was steady in time. A logarithmic spiral records the geometry of accumulation, not its rate, and a shell that grew in seasonal spurts traces the same curve as one that grew continuously.

It does not say the animal was optimising anything. Self-similar growth is what the result is from the simplest possible rule — add proportionally at the opening — and needs no optimisation to explain.

And it does not distinguish species, families or phyla. Everything that grows this way produces this curve, which is why the growth factor is a useful parameter and the shape is not.

Where the parameter does distinguish #

The factor itself is informative, and that is the reason to measure it rather than to admire the curve.

A tight spiral with a low factor is a shell whose whorls stack closely; a factor of two or three per turn gives the familiar snail. A factor near seven — which is what a golden spiral has — produces a curve that opens so fast that after three turns the aperture is three hundred times its starting width, which is not the shape of any mollusc.

Measuring the factor therefore separates real shells from each other and from the constructions they are said to resemble, and it does so with a number that anyone can check on any drawn spiral.

What self-similarity buys the animal #

The logarithmic spiral is usually presented as a mathematical curiosity that organisms happen to trace. It is better read as the solution to a problem the animal has no way around.

A mollusc’s shell is mineral and, once laid down, unalterable. The animal is inside it, and the animal grows. So the shell has to be extended, and the extension has to be at the opening, and the animal inside has to keep the same proportions because its organs do.

Those three constraints leave one shape. If each increment is a scaled copy of the existing shell, added at the aperture, the accumulated result is a logarithmic spiral, and the animal at any age occupies a chamber geometrically similar to the one it occupied when younger. Nothing has to be remodelled and nothing has to be discarded.

Compare with an arthropod, which solves the same problem by moulting — casting the whole exoskeleton and growing a new one — at the cost of a period of vulnerability at every step. The two strategies are the two available answers to a rigid covering, and the spiral is what the non-moulting answer looks like.

That is the sense in which the curve is explained rather than observed, and it needs no optimisation argument: it is the only shape that satisfies the constraints.

The fit as a test, not just a measurement #

The least-squares fit returns a growth factor, and it returns a residual, and the residual is the part that does the work.

Fitted to a genuine logarithmic spiral, the worst residual in log r is 2 × 10⁻¹⁵ — arithmetic noise. Fitted to a curve that is not logarithmic, it returns some growth factor with a residual that is not noise, and the size of the residual says how badly the model applies.

So the same machinery answers two questions at once: what the parameter is, and whether the parameter means anything. Reporting the first without the second is how a fitted number acquires unearned authority — a growth factor quoted for a curve that is not a logarithmic spiral is a number with no referent.

This is the same discipline the branching exponent fit uses, where the residual at the minimum says whether a power law of that form is the right shape at all.

Where the curve turns up and what that does not mean #

The logarithmic spiral appears in shells, horns, claws, teeth, some seed heads, hurricanes, spiral galaxies and the arrangement of florets on a cauliflower.

That list is often presented as evidence of deep unity. It is evidence of something much weaker and still worth having: all of those things grow or rotate proportionally, and proportional growth has exactly one self-similar shape.

A hurricane is a logarithmic spiral because of the relationship between radial and tangential wind speed, which has nothing to do with a mollusc’s mantle. A galaxy’s arms are a density wave in a differentially rotating disc, which has nothing to do with either. The shared curve reflects a shared mathematical structure and not a shared cause, and treating the list as a unity is the same error as treating Fibonacci counts as a law of plants.

What the curve does supply, in each case, is a parameter — and the parameters are wildly different. A nautilus grows by about 3.2 per turn; a hurricane’s spiral has a completely different pitch; a galaxy’s another. Comparing the parameters is informative and comparing the curves is not, which is the whole of the nautilus question.

Reading a growth factor #

Some values, so the number has a scale attached.

Just above 1 — a very tight coil with many whorls. Foraminifera, some tiny gastropods.

About 3 — the nautilus, and the familiar coil of a planispiral ammonite. Three or four visible whorls in a shell of ordinary proportions.

About 6.85 — a golden spiral, and nothing biological. Three turns take the aperture from one unit to three hundred and twenty.

Tens to hundreds — the shell opens so fast that a single turn dominates. This is the limpet-and-bivalve region, where the “spiral” is barely a spiral at all.

Reading the number rather than the shape is the whole point of measuring it. A photograph of one turn of a shell is compatible with almost any factor; two turns pins it down; and the factor, unlike the curve, distinguishes one animal from another.

What the fit needs and what it does not #

To be explicit about the inputs, since the measurement gets used to settle an argument.

It needs the coordinates of points along the curve and an assumed centre. That is all. It does not need to know which end is older, what the shell is, how many turns are present, or what factor to expect.

The angle has to be unwrapped so that it increases monotonically along the curve, which is a bookkeeping step and the one place an implementation can quietly go wrong — a fit that lets the angle wrap will report a slope averaged over a discontinuity and give a plausible wrong answer.

And the sensitivity to the assumed centre is measured rather than assumed, because it is the only free choice and it is the obvious objection. A quarter-radius error in the centre moves the factor by 4.56 per cent over two turns and needs 4.25 turns to fall under one, which is still far too little to matter for the comparison that follows.

The name, and a small history #

The curve has been called the spira mirabilis since Jacob Bernoulli, who was sufficiently taken with its self-similarity to ask for one on his gravestone with the motto eadem mutata resurgo — though changed, the same rises again.

The stonemason carved an Archimedean spiral instead, which grows by a fixed amount per turn rather than a fixed factor, and is not self-similar at all. It is still there in Basel.

That is a pleasant story and it is also the essay’s point in miniature. The two curves look alike at a glance and are different in the property that matters, and telling them apart requires measuring rather than looking — which is precisely the difficulty the nautilus claim lives in.

An Archimedean spiral, incidentally, is what a rolled carpet or a coiled rope makes: a fixed thickness added per turn. A logarithmic one is what a growing organism makes: a fixed fraction added per turn. The difference between adding a constant and multiplying by one is the difference between the two shapes, and it is the difference between accumulation and growth.

The self-similarity, stated as an equation #

The property everything else follows from is worth writing once.

A logarithmic spiral is $r = a\,e^{b\theta}$. Rotate it by an angle α and the radius at every point is multiplied by $e^{b\alpha}$ — a constant. So a rotated copy is a scaled copy, and the curve is indistinguishable from itself at any magnification once rotated to match.

Two consequences matter for shells.

The angle between the curve and the radius is constant, at arctan(1/b). This is why the curve is also called equiangular, and why a shell can be built by an animal that adds material at a fixed angle to the direction it is growing — a local rule requiring no measurement of the whole.

The growth factor per turn is $e^{2\pi b}$, which is the quantity everything on this site is fitted to. It is a full description: two spirals with the same factor differ only by rotation and scale, which is what makes it a legitimate one-number summary of a shell’s coiling and what makes the nautilus comparison a comparison of like with like.

An organism growing this way never has to know how big it is. Adding a fixed proportion at the aperture, at a fixed angle, produces the whole curve — which is presumably why the shape is so common, and is a better answer to “why this curve” than any appeal to efficiency.

The one number, and the habit behind it #

Reducing a shell’s coiling to a growth factor is the same move this site makes everywhere, and the reason for it is worth stating once more in this context.

A shape cannot be argued about. Two people looking at the same spiral will disagree about whether it is “golden-looking” and neither will be able to settle it. A number can be argued about, because it can be measured twice.

So the nautilus claim becomes a comparison of 3.2 against 6.854; Raup’s three parameters turn a shell into a point; Murray’s exponent turns a branching network into a scalar; and a divergence angle turns a seed head into one.

In each case the parameterisation is doing the same job: it makes disagreement resolvable. That is a lower bar than explanation and a much more reliable one.

What the curve is used for on this site #

The logarithmic spiral appears in three places here, doing three different jobs, and distinguishing them is worth a closing paragraph.

As a model of shell coiling, where the growth factor is a real parameter of a real animal, recoverable by least squares and comparable between individuals. That is Raup’s W, and it is the load-bearing use.

As the subject of a false claim, where the assertion that a particular shell’s factor is φ⁴ turns out to be wrong by a factor of 2.14. The curve is fine; the number attached to it is not.

And as a counterexample to reading a shape, since two spirals with wildly different growth factors look alike over a single turn and are told apart only by measuring more than one — which is the same difficulty that makes a spiral count depend on where one counts and a packing claim depend on head size.

The third is the one that generalises. A curve family with a free parameter will always contain a member that matches any given picture, and the picture is therefore never the evidence.

Two recoveries, and what makes one exact #

This essay’s growth factor comes back to fifteen digits, which is the strongest closure the founding essays managed. Expansion built a second exact recovery, on a cylindrical lattice, and comparing the two says what makes such a thing possible.

Both are solves rather than fits. A logarithmic spiral’s growth factor appears linearly in the log of the radius against angle, so a straight line through those coordinates recovers it with no objective to choose. A cylindrical lattice’s two parameters appear in two hop-length equations, so bracketing a sign change recovers them the same way.

What both have in common is that the measurement pins the unknowns exactly. One parameter, one linear relation; two parameters, two equations. Nothing is left over for a tolerance to absorb.

Compare the divergence angle recovered from spiral counts on a disc, which returns an interval, because a pair of integers does not determine an angle — a range of angles makes the same two offsets shortest.

The pattern across all three: exactness is a property of what was measured, not of how carefully. A recovery that returns fifteen digits and one that returns an interval can be equally well done.