We all know that absolutely regular patterns are not very common in the natural world around us. The exceptions are unusual enough attract attention, for example, the regular hexagons of wax forming the brood cells in a beehive. (There is a good mathematical explanation for this particular regularity: it uses the smallest amount of wax to form the maximum number of cells: so bees evolved over time to optimise their use of resources in producing the maximum number of young.) Similarly, natural crystals arise from the underlying regularity of the way atoms pack together at the microscopic level.
Every tree, however, looks a bit different to its neighbour - even if that neighbour is a genetic clone. Yet, the shape of an oak tree is recognisably different to the shape of an ash or a maple. An artist can draw a tree from his imagination and it will look like a tree (and, if he or she is an observant artist, it may look specifically like an oak tree or specifically like an ash tree). There is, in fact, something about such natural forms that remains characteristic even when every natural example is a distinctly different randomly structured individual.
The concepts of fractal geometry help to capture some of these characteristics.
Let us look at the tree again, and measure the total length of all the branches and twigs. From a distance we see only the thickest branches and our measurement will reflect the detail we can see. As we move closer we see thinner branches and the length we measure increases. When we are close, we now see fine twigs and the length we measure increases again.
It is the same with coastlines: from space we measure one value for the length of the UK coastline, from a plane a somewhat longer length because we can see the wiggles on a finer scale, and walking the coastline we see even more convolutions and increase our length estimate again. If we had a microscope we would see even more normally invisible wiggles. It has (we would now say) a "fractal" structure.
Mandelbrot captured this phenomenon by introducing the idea of a "fractional dimension" (he also called it the "art of roughness"). If we double the size of a square the length of the perimeter would increase by a factor of two (and the area by a factor of four). However, if we take the UK coast line and halve the size of our measuring ruler then the estimate of its length would change by a factor somewhere between one and two. That value - the fractional dimension - measures something very characteristic about the visual quality of the convoluted coast line. Mandelbrot called fractal geometry the geometry of nature.
Fractal Clouds over a Fractal Landscape?
It also means that we can use fractal geometry to generate images that have some characteristics of shapes we see in nature. It appears to be the case that many people viewing such images find them more pleasing than either highly regular patterns, or images with a high degree of irregularity. I suspect we respond to the characteristics which resonate with natural structures, even though we may not recognise them explicitly. (It may even be that our brains are wired to recognise fractal characteristics at some level, having evolved as a means of efficient recognition of the irregular patterns of the natural world.)
Most people have probably encountered images based on calculations of the Mandelbrot Set, but this is only one example of a fractal structure. There are many algorithms and variations that can be used in the production of generative art. I aim to explore some the possibilities.
My first exploration is, of course, the Mandelbrot Set itself. Why? Because it is easy to generate highly decorative results: small effort and high reward.