Friday Fractal XXIV

What's so special about a spiral? Why does it catch our eye, inspiring our art and architecture? Why is it even there? This week, I reviewed a program about the emergence of order, showing how organized patterns appear in nature and society, without the guidance of a leader. Spiral forms are an excellent example for this subject... did some creator or guide put these elegant forms in nature? Or do they appear almost by accident, a result of simple, repeated processes? Sounds familiar. Before comparing today's fractal with an image from nature, compare these two definitions:

A self-similar structure whose geometrical and topographical features are recapitulated in miniature on finer and finer scales.

Has a radius which varies in length in a exponential relationship with the angle of revolution.

Before you run your mouse over (or click on) the links, try to guess: which defines a fractal, and which defines a spiral? Well, perhaps they aren't quite that indistinguishable, but they do seem to fit together naturally. In fact, in many fractal sets, such as the following Carlo Julia, spirals are quite common:

This familiar shape once dominated the oceans:

Eutrephoceras fossil (a Cretaceous-era Nautiloid)

Both the living species of Nautilus and extinct nautili like this Eutrephoceras fossil, took advantage of their spiral shape, using the osmotic pressure of gas and fluids passing between chambers to navigate the waters. The shapely shell didn't begin as a specialized swimming tool, however, but emerged from a simple anomaly: a basic shape, repeated almost exactly, but increasing in relative size.

Imagine, you live in a shell. (I know, it's a stretch, unless you're an apartment dweller, in which case you're right with me.) You have one cavity inside the shell to move around in. Now, imagine that by some chance, you could have multiple cavities, but had to keep the same basic shape. What's the best way to keep these multiple cavities together? Line them up, side by side, smallest to biggest. Done neatly, your shell becomes a logarithmic spiral.

These fascinating, simple shapes have emerged all over the place. I'll leave you with a few to consider:

To see more spirals emerge, just look at the world around you.

Image notes: Nautilus fossil photo by the author. Images in spiral picture: (Clockwise from the top left) rose by comma? via Flickr, Van Gogh's Starry Night via the WebMuseum, Paris, fiddlehead fern by Anderw Howard via photo.net, slinky study by Dave April via his Digital Art & Photo Gallery, hurricane Isabel via CalculateMe, staircase in the Au Sable Point Lighthouse by Dave Wobster, snail via freebird, spiral galaxy NGC 4414 via the NSSDC photo gallery. All fractals made by the author using ChaosPro.