Self Similarity

Fractals usually possess what is called self similarity across scales. That is, as one zooms in or out the geometry/image has a similar (sometimes exact) appearance. The following will illustrate various types of self similarity as well as present some real world examples.

The self similarity may be exact, this normally only occurs in mathematically defined fractals where the realities/constraints on structures by the physical world don't apply. The following example is the well known Koch snowflake curve created by starting with a single line segment and on each iteration replacing each line segment by four others shaped as follows . As one successively zooms in the resulting shape is exactly the same no matter how far in the zoom is applied.

Approximate self similarity

A far more common type of self similarity is an approximate one, that is, as one looks at the object at different scales one sees structures that are recognisably similar but not exactly so. An example of this in a mathematically defined system can be readily demonstrated by almost all the patterns seen in the Mandelbrot set. The following show three successive zooms and at each level a structure similar but not exactly the same as the whole Mandelbrot set can be found.

Note that in the above example the self similar structure occurs at discrete levels of scale while other systems like the Koch curve the self similarity occurs at all scales.

Sometimes the self similarity is isn't visually obvious but there may be numerical or statistical measures that are preserved across scales. One obvious measure might be the fractal dimension, in the example below of 1/f noise. the fractal dimension is constant as one zooms in.

Example 1 - Terrain

Consider the following image, is it on the scale of a large piece of rugged terrain photographed from an aeroplane, or the side of a mountain, or a patch of dirt on the scale of a few meters, or a magnification of the surface of a rough rock? Whichever it is, it could also easily be imagined to be any one of the others. So one could start at the large scale view from the air and apply successive zooms down to a microscopic scale, the surface maintains self similarity across those scales.

As with most fractal structures found in nature, the self similarity only occurs over a range of scales. In the above example, there is no self similarity as we zoom out to see the whole planet, or zoom in to microscopic scales.

In the case of the self similarity found in the fern, not only is there a limit to the range of scales at which the self similarity occurs but it also occurs at only at a few discrete scales.

Example 3 - Tree branching

Branching structures such as roots, capillaries, river networks often exhibit self similarity across scales. In some cases, a young tree may be self similar in time, namely, as it grows. A Bonsai tree is a good example (if a little artificial) of this. The trees below, besides the fact that you can see them in thier context and thus scale, could easily be imagined to be anywhere in size from a small shrub of 10cm, as they are at about 1m tall, and even as small trees of 10 to 20m in height, or large trees of 100m. They would all pass for possible trees across that large range of scales.

Bonsai trees in Chinese gardens, Singapore