Is the Nautilus Shell a Golden Ratio Spiral? Myth Checked

What is a golden spiral?

The golden ratio, written with the Greek letter phi (φ), is about 1.618. Two quantities are in the golden ratio when the larger divided by the smaller equals the sum of both divided by the larger. A rectangle with sides in this ratio is called a golden rectangle.

A golden spiral is a special logarithmic spiral that gets wider by a factor of φ every quarter turn. It fits neatly inside a golden rectangle that is split again and again into squares. After one full turn, a golden spiral has grown by φ to the fourth power, about 6.85 times.

The popular “Fibonacci spiral” is a close cousin. It is drawn as quarter circles inside squares whose sides follow the Fibonacci numbers (1, 1, 2, 3, 5, 8, 13…). Because the ratio of neighboring Fibonacci numbers approaches φ, this drawing looks almost like a golden spiral, and it is the picture most often laid over a nautilus shell in books and posters.

What kind of spiral is a nautilus shell?

The chambered nautilus builds a logarithmic spiral, also called an equiangular spiral. Its key property is self-similarity: as it grows, it keeps exactly the same shape, getting wider by the same factor with each turn. The Swiss mathematician Jacob Bernoulli was so fond of this curve that he called it spira mirabilis, “the marvelous spiral”.

A logarithmic spiral can open slowly or quickly. The golden spiral is just one of countless possible versions, with one particular growth rate. Saying that the nautilus forms a logarithmic spiral is correct; saying that it is therefore a golden spiral is the mistake at the heart of the myth. The Scottish biologist D’Arcy Thompson used the nautilus as a classic example of the equiangular spiral in his 1917 book On Growth and Form, which explained how such shapes arise from steady growth.

What do measurements of real shells show?

In 1999, the retired mathematician Clement Falbo measured nautilus shells in the collection of the California Academy of Sciences in San Francisco. He found that their spirals fitted inside rectangles with sides in a ratio of about 1.33, not 1.618. Individual shells ranged from about 1.24 to 1.43. He published the results in The College Mathematics Journal in 2005 and concluded that a nautilus within 2 percent of the golden ratio would be rare, if one exists at all.

The last column is an approximate conversion: in this kind of spiral, the rectangle ratio is roughly the growth per quarter turn, so the growth per full turn is about that ratio multiplied by itself four times. Others, including the writer John Sharp in a 2002 article, made similar measurements and reached the same conclusion: when a golden spiral is drawn over a real nautilus, the two simply do not match. Claims that each new chamber is a golden ratio larger than the one before it are not supported by measurements either.

How did the myth spread?

The idea is attractive because both the nautilus and the golden ratio stand for natural beauty and harmony. The nautilus shell is also a perfect logarithmic spiral to the eye, and the golden spiral is the best-known logarithmic spiral, so the two were easy to connect. Illustrations that place a Fibonacci spiral over a nautilus photo often line up only at one point, or the shell is scaled and rotated until it seems to fit.

Once the claim appeared in popular books on mathematics, art and design, it was repeated in websites, school materials and logos. Mathematicians have tried to correct it for decades, but the picture of a nautilus with a golden rectangle on top remains one of the most widely shared science myths.

How does a nautilus build its chambers?

The logarithmic shape comes from the way the nautilus grows. The animal adds new shell only at the open edge, and each new section is a slightly larger copy of the one before. Growth without any change of shape is exactly what produces a logarithmic spiral, whatever its growth rate.

Inside, the soft body sits in the last and largest chamber. From time to time the animal moves forward and the rear of its body lays down a new curved wall of mother-of-pearl, called a septum, sealing off an empty chamber behind it. A hatchling starts with only a few chambers, and an adult has about 30 or more. A thin tube of living tissue, the siphuncle, runs through all the walls and removes liquid from the sealed chambers so that they fill with gas and keep the animal afloat.

When a nautilus reaches maturity, it stops growing. The last few walls are laid closer together, and the edge of the shell opening thickens. Because of this final crowding, and because every shell grows a little differently, even a single nautilus does not follow one perfect mathematical curve from start to finish.

Where does the golden ratio really show up in nature?

The best-documented examples are in plants rather than shells. In sunflower heads, pine cones and pineapples, the numbers of spirals running in each direction are often consecutive Fibonacci numbers, such as 34 and 55. This pattern arises because each new seed or scale forms at an angle of about 137.5 degrees from the previous one, the so-called golden angle.

Coiled shells of snails and the extinct ammonites also follow logarithmic spirals, each at its own growth rate. You can read more about shell growth in how do snails grow their shells?, and about the deep history of the nautilus in is the nautilus a living fossil?