Fractals = Math + Art + Nature

Outcrops at Tidbits look a bit like smaller versions of the larger rock formations.

I hope you all have had a chance to see Nova’s 2008 episode on fractals that was repeated last week. If not, click here to see “Hunting the Hidden Dimension.” It is an excellent overview of fractals, where and why they are found in nature and natural systems, and how scientists are applying fractal geometry to a wide variety of applications.

My computer dictionary gives a decent definition of fractals:

fractal |ˈfraktəl| n.

(Mathematics) a curve or geometric figure, each part of which has the same statistical character as the whole. Fractals are useful in modeling structures (such as eroded coastlines or snowflakes) in which similar patterns recur at progressively smaller scales, and in describing partly random or chaotic phenomena such as crystal growth, fluid turbulence, and galaxy formation.

Organic-looking fractal image generated with Xaos software

Before the “discovery” of fractals, most people thought of most of nature as being irregular and chaotic, because much of it could not be described by traditional Euclidean geometry. Benoît Mandelbrot, considered the father of fractal geometry, was studying iterative equations. These are equations where you plug a number into a function and then take the result and plug it back in over and over again. The resulting patterns show what is referred to as “self-similarity”. Put simply, the details look like smaller images of  the whole.

Mountain shield fern (Dryopteris expansa). Ferns frequently show strong self-similarity, the pinnae and pinnules looking like smaller versions of the fronds.

In my opinion, the brilliance of Mandelbrot wasn’t so much in his mathematical abilities but in his powers of observation. He recognized that the images he was creating resembled what was happening in nature. Branching patterns in trees look similar whether you are looking at the entire tree or small branches. Rock outcrops resemble the mountains they are part of. The study of fractals opened up a whole new way to view and explain natural patterns and to quantify things previously thought to be unquantifiable. It also gave computer artists a simple way to generate more natural imitations of nature. We owe the amazing computer-generated landscapes of science fiction movies to fractal geometry. To see some great examples of natural fractals, check out Miquel.com, or just google fractals. There is a ton of wonderful information out there on the web. To study it in more depth, there is an excellent online course at the Fractal Foundation.

Self-similarity can be seen in both the thrice-dissected leaves and compound flower umbels (larger umbel made up of smaller umbellets) of fern-leaved lomatium (Lomatium dissectum).

I was a math major in college in the late ’70s when fractal geometry was first developing. Somehow I don’t remember hearing about it then (why didn’t my math professors jump on this?). I did, however, have a copy of Peter Stevens’ 1975 book, Patterns in Nature, which I found fascinating. It really opened my eyes to the underlying math in nature. While he didn’t mention fractals—Mandelbrot’s first book on fractals, Fractals: Form, Chance and Dimension, wasn’t published in English until 1977—he did show how few distinct patterns there are in nature—branching, spirals, explosions (like dandelion seeds), meanders and a few others— and why each is used. Nature needs to be efficient, and for the most part, efficiency seems to be the driving force that defines what pattern is used for a particular function. Patterns in nature has always been a subject of great interest to me. It is the coming together of my three favorite subjects: math, art, and nature.

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