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The above picture is of a Mandelbrot fractal, like the one on the cover of my book, Philosophy at the Edge of Chaos. A fractal is a mathematically generated shape with unique characteristics. In particular, a fractal has a feature commonly referred to as self-similarity, meaning that as you zoom in on a fractal, the zoomed in portion is similar to the larger whole of which it was a part. At the same time, however, a fractal is more than the sum of its parts, and this is true precisely because of both self-similarity and the possibility for indefinitely zooming in. When one looks at a Koch curve (pictured below), for example, one can see that the process of adding triangles can occur indefinitely. But by adding the triangular shape one also adds length to the line, thus any given fractal will increase in length as the scale of measurement becomes smaller. The example of a coastline is often given to clarify this. From a distance, one measures a coastline and comes up with a given length. As one measures the coastline from a closer perspective, this length increases, and, as in the case of a fractal, it increases indefinitely if the division of the lines occurs indefinitely.

Koch curve:

This is why a fractal is called a fractal. If a line of a given length were divided into five pieces the sum of the five pieces would be equal to the original line. For a fractal, however, the sum of the parts will be greater than the original whole (i.e., as scale decreases, length increases). A fractal is therefore neither a one-dimensional line, nor is it a two-dimensional figure (i.e., it is still a measurable length and not an area). It is a fractional dimension between 1 and 2, and hence the name fractal.

The relationship between fractals and my current work and research should become apparent as one reads my work in more detail, but put briefly a key point in my argument is that becoming (i.e., the dynamical processes whereby identifiable things come-into-being) is much like a fractal. Both becoming and fractals are, from this perspective, indeterminately determinable, meaning that as becoming comes to be identified and determinate, the possibilities for further identifications and determinations increases, and as these identifications come-into-being even further possibilities arise, and so on. Consequently, just as a the length of a fractal increases as it becomes increasingly (i.e., more closely) measured, so too does becoming, as it becomes determined, become indeterminately determinable.

For more on Fractals, check out the following links:

• Introduction to Fractal Geometry is a general overview of fractals and yet goes into much more depth than I did here.
• These Fractal Zoom movies by Eric Bigas give a good visual demonstration of the abstract points made above. They're cool too.