Fractal dimensions

Fractal landscape

A fractal landscape is a surface generated using a stochastic algorithm designed to produce fractal behaviour which mimics the appearance of natural terrain. In other words, the result of the procedure is not a deterministic fractal surface, but rather a random surface which exhibits fractal behaviour. Because the intended result of the process is to produce a landscape, rather than a mathematical functions, processes are frequently applied to such landscapes which may affect the stationarity and even the overall fractal behaviour of such a surface, in the interests of producing a more convincing landscape.

Behaviour of natural landscapes

Whether or not natural landscapes behave in a generally fractal matter has been the subject of some research. Technically speaking, any surface in three-dimensional space has a topological dimension of 2, and therefore any fractal surface in three-dimensional space has a Hausdorff dimension between 2 and 3. Real landscapes however, have varying behaviour at different scales. This means that an attempt to calculate the 'overall' fractal dimension of a real landscape can result in measures of negative fractal dimension, or of fractal dimension above 3. In particular, many studies of natural phenomena, even those commonly thought to exhibit fractal behaviour, do not in fact do so over more than a few orders of magnitude. For instance, Richardson's examination of the western coastline of Britain showed fractal behaviour of the coastline over only two orders of magnitude. In general, there is no reason to suppose that the geological processes that shape terrain on large scales (for example plate tectonics) will exhibit the same mathematical behaviour as those which shape terrain on smaller scales (for instance soil creep).

Real landscapes also have varying statistical behaviour from place to place, so for example sandy beaches don't exhibit the same fractal properties as mountain ranges. A fractal function, however, is statistically stationary, meaning that its bulk statistical properties are the same everywhere. Thus, any real approach to modeling landscapes requires the ability to modulate fractal behaviour spatially. Additionally real landscapes have very few natural minima (most of these are lakes), whereas a fractal function has as many minima as maxima, on average. Real landscapes also have features originating with the flow of water and ice over their surface, which simple fractals cannot model.

It is because of these considerations that the simple fractal functions are often inappropriate for modeling landscapes. More sophisticated techniques (known as 'multifractal' techniques) use different fractal dimensions for different scales, and thus can better model the frequency spectrum behaviour of real landscapes

Generation of Fractal Landscapes

A way to make such a landscape is to employ the random midpoint displacement algorithm, in which a square is subdivided into four smaller equal squares and the center point is vertically offset by some random amount. The process is repeated on the four new squares, and so on, until the desired level of detail is reached. There are many fractal procedures (such as Perlin noise) capable of creating terrain data, however, the term "fractal landscape" has become more generic.

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