Self-similarity and Fractals in Geoscience

Careful geologists always include a scale or scale reference (a coin, a hammer, a camera lens cap or a human) when taking a picture of geologic interest. The reason is that if they didn't, the actual size of the object pictured could not be determined. This is because many natural forms, such as coastlines, fault and joint systems, folds, layering, topographic features, turbulent water flows, drainage patterns, clouds, trees, etc. look alike on many scales.

Look at the dry river channels of a typical drainage network pictured below. Select any small tributary and look carefully at its form. You will see a main stream that splits into two, three or more rivulets, each of which splits again, into two or three smaller ones, and so on. Now look at the entire drainage basin structure, and you will see the same complex design again, starting with the larger stream, you see it split into smaller tributaries, each of which splits again and so on in a seemingly organized manner.

It appears as if the underlying forces that produce the network of rivers, creeks, streams and rivulets are the same at all scales, which results in the smaller parts and the larger parts looking alike, and these looking like the whole. This property is called self-similarity, or scale invariance, and is prevalent in geoscience, where we usually see that small faults look like large ones, short segments of coastline look like the entire coast, small folds one meter long look exactly like folds several kilometers in extent, boulders look like mountains, etc. Self-similarity is also found in time signals, such as the time records of river flows, or in the occurrence of earthquakes, or even the seemingly chancy fluctuations of prices on the stock market.

What is the meaning of the scale invariance, or self-similarity? Why does nature disregard the geometrical scales of things? One way to understand what is going on is to generate artificial self-similar forms which we shall do next.