The Fibonacci Spiral and the Nautilus (Shallow Thoughts)

The Fibonacci Spiral and the Nautilus

I've been working on a short talk on Fibonacci numbers for a friend's math class.

Back when I was in high school, I did a research project on Fibonacci numbers (their use in planning the growth of a city's power stations), and for a while I had to explain the project endlessly, so I thought I remembered pretty well what sorts of visuals I'd need -- some pine cones, maybe some flower petals or branching plants, graphics of the golden ratio and the Fibonacci/ Golden Spiral, and some nice visuals of natural wonders like the chambered nautilus and how that all fits in with the Fibonacci sequence.

I collected my pine cones, took some pictures and made some slides, then it was time to get to work on the golden spirals. I wrote a little GIMP script-fu to generate a Fibonacci spiral and set of boxes, then I went looking for a Chambered Nautilus image on which I could superimpose the spiral, and found a pretty good one by Chris 73 at Wikipedia. I pasted it into GIMP, then pasted my golden spiral on top of it, activated the Scale tool (Keep Aspect Ratio) and started scaling.

And I just couldn't get them to match!

No matter how I scaled or translated the spiral, it just didn't expand at the same rate as the nautilus shell.

So I called up Google Images and tried a few different nautilus images -- with exactly the same result. I just couldn't get my Fibonacci spiral to come close.

Well, this Science News article entitled Sea Shell Spirals says I'm not the only one. In 1999, retired mathematician Clement Falbo measured a series of nautilus shells at San Francisco's California Academy of Sciences, and he found that while they were indeed logarithmic spirals (like the golden spiral), their ratios ranged from about 1.24 to 1.43, with an average ratio of about 1.33 to 1, not even close to the 1.618... ratio of the Golden Spiral. In 2002,John Sharp noticed the same problem (that link doesn't work for me, but maybe you'll have better luck).

As the Science News article points out,

Nonetheless, many accounts still insist that a cross section of nautilus shell shows a growth pattern of chambers governed by the golden ratio.

No kidding! Google on fibonacci nautilus and you'll get a boatload of pages using the chambered nautilus as an illustration of the Fibonacci (or Golden) spiral in nature. It's not just the web, though -- I've been reading about nautili as Fibonacci examples for decades in books and magazines. All these writers just pass on what they've read elsewhere ... just like I did for all those years, never actually measuring a nautilus shell or trying to inscribe a golden spiral on one.

Now do a Google image search for the same terms, and you'll get lots of beautiful pictures of sectioned nautilus shells. You'll also get quite a few pictures of fibonacci spirals. But none of those beautiful pictures will actually have both the nautilus and the spiral in the same image.

And now I know why -- because they don't match!

(Happily, this actually may be a better subject for my talk than the nautilus illustration I'd originally planned. "Don't believe everything you read" is always a good lesson for high schoolers ... and it's just as relevant for us adults as well.)

(Slides from the talk I wrote start here: The Rabbit, the Nautilus and the Pine Cone.)