Phyllotaxis
the spirals are counted, not admired
Shells and growth

The nautilus question

A golden spiral grows by 6.854 per turn. Measured nautilus sections give about 3.2. That is a factor of 2.14 — not a rounding error, not an artefact of where the centre is assumed to be, and not close.
6 min read 5 figures State the claim, state the testOne parameter, one family

The nautilus shell is the standard illustration of the golden ratio in nature. It appears in textbooks, on posters, in design courses, and in a very large number of articles about mathematics being everywhere.

It is not a golden spiral, and the discrepancy is a factor of two.

A golden spiral and a nautilus spiral over 2.4 turns, from the same startAfter 2.4 turns the golden curve is 6× larger. The growth factors are 6.85 and 3.2, a factor of 2.14 apart.golden — 6.85× per turnnautilus — 3.2× per turnsame construction, same start2.14× apart in growth
Fig. 1 Both curves from the same start, over two and a half turns, drawn from their growth factors. After that distance the golden curve is six times larger.

The two numbers

A golden spiral is the one that grows by the golden ratio φ = 1.618 in a quarter turn. That means it grows by φ⁴ = 6.854 in a full turn.

A nautilus grows by about 3.2 per turn. Published measurements cluster between roughly 2.9 and 3.4, varying with the individual and slightly with how the measurement is made.

Those are not close. The ratio is 2.14, and the whole measured range for real shells falls short of the claimed figure by more than the width of the range.

Growth per turn: what is claimed and what is measuredThe measured range is 2.9–3.4; the golden figure is 6.854. There is no overlap and the gap is more than twice the width of the range.3×4×5×6×7×measured nautilus, lowmeasured nautilus, typicalmeasured nautilus, highgolden spiral, φ⁴shaded: the measured rangethe claim sits outside it
Fig. 2 The two on one axis. The shaded band is the measured range; the claim sits outside it, more than twice the band’s width away.

Where the confusion comes from

Three things, and none of them is stupidity.

A nautilus really is a logarithmic spiral. That part of the claim is correct, and it is the part that gets checked. A shell that grows self-similarly traces a logarithmic spiral, and the nautilus does. What does not follow is that it is the particular logarithmic spiral with growth φ⁴.

All logarithmic spirals look alike. Shown one turn of each without a scale, most people cannot tell a 3.2 spiral from a 6.9 spiral. The difference is obvious over two turns and invisible over half of one, and shell photographs are usually cropped to about that.

The golden rectangle construction is suggestive. The nested-squares picture produces a curve that is drawn inside a rectangle whose proportions everyone recognises, and the visual association is strong even though the arc is a sequence of quarter circles rather than a spiral at all.

What the measurement cannot be blamed on

The obvious objection is that the fit depends on where the centre is put, and a shell’s centre is not marked. That objection is testable and it does not survive.

Moving the assumed centre by a quarter of the spiral’s own radius — an error far larger than any careful measurement would make — changes the fitted growth factor by well under one per cent. To turn 3.2 into 6.854 the centre would have to be wrong by an amount that would be obvious in the picture.

How much the answer depends on where the centre is putA shell has no marked centre, so the fit needs one chosen. Getting it wrong by a quarter of the radius moves the fitted factor by 55.9% — nowhere near enough to turn 3.2 into 6.85.02040600510152025assumed centre, as a percentage of the spiral's own radius, away from the trutherror in the fitted growth factor (%)the one free choice in the measurementand it is not free enough to matter
Fig. 3 The sensitivity, measured. The one free choice in the fit is not free enough to account for a factor of two.

Why it survived

A claim this checkable persisting for decades is worth a paragraph, because the mechanism is general.

The check requires two things nobody does casually: measuring a growth factor over more than one turn, and knowing that a golden spiral’s factor per turn is φ⁴ rather than φ. The second is the real barrier — φ is 1.618, the shell obviously grows by more than that per turn, and 3.2 sounds like it could be φ-something to anyone who has not worked out which power applies.

Add that the claim is attractive, appears in reputable places, and is repeated by people who are not making an error so much as passing one along, and it becomes self-sustaining. The correction has been published repeatedly since at least the 1990s and it does not travel, because the claim is more useful to a lecture than the correction is.

What is actually true about it

The nautilus is a genuinely elegant object and it does not need the golden ratio.

It grows self-similarly, which is a real constraint with a real consequence, and it does so with remarkable regularity — the fit residuals on a well-measured section are small, meaning the animal held its growth factor nearly constant over its whole life. That is more impressive than matching a particular constant, and it is the thing the measurement actually supports.

The chambers are also spaced so that each is a scaled copy of the last, which is the same property expressed in the septa, and the buoyancy control that uses them is a genuinely sophisticated mechanism.

None of that requires 6.854. The animal is interesting for what it does, and attaching a number to it that it does not have is a way of finding it interesting for something it is not.

A logarithmic spiral growing by 3.20× per turnFitting log r against angle on the drawn points returns 3.2000, and the model's worst residual is 9e-16 in log r.growth 3.20× per turnrecovered 3.200×
Fig. 4 The machinery the comparison rests on. A spiral generated at a stated factor, with the factor recovered from the drawn points to fifteen digits — so a measured 3.2 is a measurement rather than an impression.

What the claim would need to be true

Worth stating positively, because it is a short list and every item fails.

The shell would have to grow by 6.854 per turn. It grows by about 3.2, and the whole published range falls short.

The measurement would have to be sensitive to the assumed centre. It is not — a gross error in the centre moves the answer by under a percent.

Or the claim would have to be about a different quantity — the ratio of successive chamber widths, say, rather than growth per turn. That version can be checked too, and a growth factor of 3.2 per turn gives successive quarter-turn ratios of 3.2^0.25 = 1.34, not 1.618.

There is no reading of the claim on which the numbers work.

The pattern this belongs to

The nautilus is one of three famous assertions this site measures, and the three sort neatly.

One is true of a branch rather than of plants. One is right, in a sharper form than the version usually told. And this one is simply wrong, by a factor of two, about a quantity anyone can measure in an afternoon.

What they share is that none of them had a test attached. Each was passed along as a fact rather than as a claim, and a fact does not invite checking. Attaching a test to each — state the number, state how it would be measured, report what the measurement gives — is the whole of what this site does, and it is why the one that survives is worth more after the exercise than before it.

A golden spiral and a nautilus spiral, drawn from their growth factorsA golden spiral grows by φ⁴ = 6.854 per turn; measured nautilus sections give about 3.2. That is a factor of 2.14, not a rounding error.golden — 6.85× per turnnautilus — 3.2× per turnsame construction, two factorsa factor of 2.14 apart
Fig. 5 The two growth factors drawn from the same start. After two and a half turns they are a factor of six apart.

Where this goes next

growth 3.20× per turnrecovered 3.200× Growth as a rule A logarithmic spiral is not a shape somebody admired. It is what a thing grows into when it adds material at its opening without changing shape, its one parameter is how much it grows per turn, and that parameter can be recovered from any drawn curve to the last digit. 00.2000.400100120140160divergence angle (°)resistance to rational approximation (higher is more irrational)1/√5 — the bound137.508° — 0.4381500 angles on a 0.05° grid, plus the golden angle exactlythis claim is sharp The claim that survives Of the three famous assertions about this subject, one is out by a factor of two, one is true of a branch rather than of plants, and one is right — in a sharper form than the version usually told, and about arithmetic rather than about packing. golden 137.51°55 and 89FibonacciLucas 99.50°47 and 76Lucas151.14°50 and 81neither77.96°37 and 60neithercounted from the pointsone sequence per branch Fibonacci is a branch, not a law Fibonacci counts come from one branch of the model. The Lucas branch — which the same rule reaches at a different growth rate — gives 47 and 76, and neither is a Fibonacci number. The sequence is a consequence of an angle rather than a property of plants.