## 14 September 2008

### Nautilus shells aren't golden spirals

You know how everybody says that the nautilus shell grows in a golden spiral, i. e. a logarithmic spiral that grows at a rate of φ = 1.618... per quarter turn? (Formally, it's the graph of r = φ in polar coordinates.)

Turns out it's not true. (This is also why I'm not worrying about saying who "everybody" is.)

I'm not surprised; a lot of the places where φ shows up in nature seem to be related to the fact that it's the "most irrational number" (i. e. the one that's hardest to approximate accurately by rational numbers, which follows from the fact that its continued fraction expansion consists entirely of 1s). This is useful for, say, growing leaves on a tree; if the angle between successive leaves is φ-1 revolutions then leaves end up not on top of each other, and therefore not blocking the sun. I can't see any reason why a shell should grow like that, though I could be suffering from a failure of imagination, and of course I am not a biologist. A logarithmic spiral with any growth rate is self-similar, though, and self-similarity is everywhere in biology.

Of course, technically I should go out and find my own shells and do my own measurements, and not just trust Some Guy On The Internet. But it's Sunday morning, and I really shouldn't even be awake yet, so cut me some slack.

Ivars Peterson at Science News wrote about this a few years ago, it seems. A more reputable source seems to be The Golden Ratio: A Contrary Viewpoint, Clement Falbo, The College Mathematics Journal, March 2005, pp. 123-134; among other things Falbo measured some actual shells, and the growth rate for a typical shell appears to be about 1.33 per quarter-turn, although there's a pretty wide range, but the growth rate never seems to get as large as φ. The mathematical study of mollusk shells from the AMS web site might also be of interest

CarlBrannen said...

The human brain is tuned to recognize symmetry even in places where it doesn't exist, even to the amount of ignoring evidence to the contrary. So the golden spiral belief is similar to the belief that planets orbit perfect circles and that string theory defines elementary particles.

Anonymous said...

I am not surprised either, I have learned to be sceptical when mathematicians speak of applications of mathematics. It is not unusual for the claims to be exagerated.

Pat's Blog said...

If you accept that your 1.33 is about 5% high, and that the truth is closer to 1.2688, then you might consider that the cross sectional area is growing by something closer to Phi which is how I always thought it was...

Anonymous said...

Minor nitpick: For your formula in polar coordinates to be right, it seems you would need to measure the angle in turns rather than radians.

But yeah, the claims concerning the golden ratio are surely exaggerated. I am not sure it's right to lay the blame at the feet of mathematicians, though.

Anonymous said...

I tend to be skeptical about any claims involving the golden ratio. Long have I seen images where rectangles where drawn on top supposedly proving the golden ratio connection, when said rectangles could easily be tweaked to represent many different ratios.