Fractals, as defined by Hans Lauwereier, are geometrical figures 'in which an identical motif repeats itself on an ever diminishing scale'. The term for such repetition is iteration, and these iterations can be used to make models of any natural object, be it a fern or a galaxy.
Inherent in these fractals is the similarity of the original shape in such an ever-diminishing scale. The common example of such a concept is a tree with a trunk which divides into two branches, which divide into two branches, and so on. Note, however, the word similarity, for none of the iterations of the original shape are ever precisely the same. The end result of the concept is of a tree fractal with an infinite number of branches, which can be regarded as whole trees in themselves.
Fractals can now be applied to many areas where complex calculations that rely on chaos theory are needed. Areas such as weather forecasting, image enhancing, summarising data, and, of course, making beautiful pictures, benefit from the application of fractal equations.
Fractal mountains can be one of the most beautiful computer-generated applications of fractals. The equations used to produce the mountains create startlingly life-like(!) representations of mountains, with jags, peaks, and many other features.
There are undoubtedly many ways to construct fractal mountains, but the method we chose is explained below.
The fractal mountains are created by beginning with a single large triangle, then subdividing that into smaller triangles so you now have 4 small triangles, then subdividing those triangles, ad infinitum.
To make these subdivisions, we created an array of points from all the points where lines cross, and created a new point from the middle of each line. We then drew a line from each point to the next point above and to the right, to the next point directly to the right, and to the next point below and to the right, thus creating new triangles. Obviously, exceptions were made for points on the edges of the triangle.
The method above creates 'flat' mountains. To create mountains which attract more attention to the eye, vary the y co-ordinates of each point up or down by a random amount. A large random range will create a very ragged mountain. The random range must be halved every subdivision so as not to create undue raggedness, otherwise the result would look rather 'interesting' (i.e.: not look at all like a mountain!)