The mathematics of nautilus shells

Just in case it’s not immediately apparent from the title, this post is a bit of a geeky diversion into the mathematics behind spirals.  Skip now if the word “logarithm” sends you into a cold sweat!

Essentially, a nautilus shell is a logarithmic spiral, also known as an equiangular spiral or Bernoulli spiral. If you imagine a number of equally spaced lines radiating from a central point, an equiangular spiral will hit each of these radials at the same angle.

Image from Wolfram Research
Image from xahlee.org

What this means is that each layer of the spiral gets bigger as it grows, unlike an Archimedean spiral, where the width of the layers remains constant.

Image from spiralzoom.com

Nature seems to like logarithmic spirals – as well as mollusc shells you can see them in the shape of galaxies, like the spiral arms of the Milky Way, and in hurricane and cyclone formations.

Will any of this help me construct a nautilus shell in felt? Probably not, but it’s been an interesting diversion.

Normal service to be resumed next time.