0: Defined

Ok, so what is a fractal?  Merriam-Webster’s Student Dictionary has a definition that’s easy on the eyes:

An irregular shape that looks the same at any scale on which it is examined.

“Irregular” means that fractals tend to be rough and bumpy, rather than smooth and flat.

La Vieja, a natural rock spire in Argentina A larger-than-life traffic cone
A rough and bumpy rock spire. Photo: “Argentina – Frey climbing 52 – climbing la Vieja” copyright (c) 2012 McKay Savage. Made available under an Attribution 2.0 license. A smooth traffic cone. Photo: “Cone Olympic Park Seattle” copyright (c) 2008 amanderson2. Made available under an Attribution 2.0 license.

“Irregular” also means that the roughness is going to have a somewhat random look to it. Like waves on the ocean, the bumpiness of a fractal isn’t going to be perfectly regular. For example, a saw blade and a mountain range are both rough, but only one is irregular.

Olympic Mountain Range - photo by Lassi Kurkijarvi
Top image: Sawblades with regular teeth. Photo: “Sawblades” copyright (c) 2011 Rachel James. Made available under an Attribution-NonCommercial-ShareAlike 2.0 license.Bottom image: Mountain range with irregular peaks. Photo: “Olympic Mountain Range” copyright (c) 2009 Lassi Kurkijarvi. Made available under an Attribution-NonCommercial 2.0 license.

Perhaps the key characteristic of fractals is that they “look the same at any scale”, as the definition says. Basically, this means that if you take a small piece of a fractal, it’s going to look similar to any other piece of the same fractal, or even similar to the whole thing. The following video illustrates this idea using a simple fractal called the Sierpinski gasket.

Note: Adobe Flash is required to view this video.
Video: “Sierpinski Gasket” copyright (c) 2012 Jason Hine. Made available under an Attribution-NonCommercial 2.0 license.

The Sierpinski gasket exhibits self-similarity, and every fractal has this characteristic to some degree. In some cases, like in the Sierpinski gasket, a fractal will be exactly self-similar across a never-ending range of scales. Many mathematically-generated fractals (but not all!) are like this. In nature, fractals are often only approximately self-similar, and the self-similarity is only apparent across a limited range of scales. A good example is a fern frond, where each leaflet looks like a smaller version of the entire frond, but the resemblance isn’t perfect.

Approximate self-similarity in a fern frond. The main stem is highlighted in white; stems of smaller versions of the frond are highlighted in red and yellow.Photo: “Self-Similar Fern Frond” copyright (c) 2012 Jason Hine. Made available under an Attribution-NonCommercial 2.0 license.

Now that you know what to look for, you’re ready to begin seeing the fractals in the world around you. You’ll find mathematical fractals (like the Sierpinski gasket) and natural fractals (like the fern). In every case, you’ll find that fractals are:

  • irregular (rough, somewhat haphazard), and
  • self-similar (pieces, when magnified, look similar to each other)

Suggested further reading:

2 thoughts on “0: Defined

  1. Pingback: What’s a fractal? | See the Fractals

Leave a Reply

Your email address will not be published. Required fields are marked *