Behaviour of Natural Landscapes
Whether or not natural landscapes behave in a generally fractal manner 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) exhibit the same mathematical behaviour as those that 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
Read more about this topic: Fractal Landscape
Famous quotes containing the words behaviour of, behaviour and/or natural:
“The methodological advice to interpret in a way that optimizes agreement should not be conceived as resting on a charitable assumption about human intelligence that might turn out to be false. If we cannot find a way to interpret the utterances and other behaviour of a creature as revealing a set of beliefs largely consistent and true by our standards, we have no reason to count that creature as rational, as having beliefs, or as saying anything.”
—Donald Davidson (b. 1917)
“I look on it as no trifling effort of female strength to withstand the artful and ardent solicitations of a man that is thoroughly master of our hearts. Should we in the conflict come off victorious, it hardly pays us for the pain we suffer from the experiment ... and I still persist in it that such a behaviour in any man I love would rob me of that most pleasing thought, namely, the obligation I have to him for not making such a trial.”
—Sarah Fielding (1710–1768)
“The common notions that we find in credit around us and infused into our souls by our fathers’ seed, these seem to be the universal and natural ones. Whence it comes to pass that what is off the hinges of custom, people believe to be off the hinges of reason.”
—Michel de Montaigne (1533–1592)