At first sight the Koch curve (see Koch Snowflakes) appears to be part of a mathematical fantasy world. Enlarge any part of the Koch curve and the new image is identical to the original - it is self-similar. This property of self-similarity is the key behind all fractals.
The Sierpinsky gasket pictured beside is a typical example of a fractal with self-similarity and arises from repeating the construction infinitely many times. The natural world is surprisingly similar: take the humble cauliflower or a fern frond.
(The fern pattern can be generated in a similar manner to the Sierpinsky gasket.)
Analysis of such mathematical patterns helps us to understand the world around but it cannot lead to predictability, as we shall see.


Original Sierpinsky gasket

Fractals are amazing objects as they are easy to construct but their composition has an infinite structure. To fully understand this, we compare a fractal to a smooth curve. Both are made up of infinitely many points but as we magnify the smooth curve it flattens out and eventually appears almost straight. The position of a point on the smooth curve can be approximately determined by considering the points before and after it. However no amount of magnification of the Koch curve will produce a smooth region. Without prior knowledge of the Koch curve we cannot predict the position of a point from points close by. One way to quantify this difference is to use dimension.
A smooth curve, like a straight line, is 1 dimensional (a plane is 2 dimensional). But, as one might guess from the fact that the perimeter of the Koch curve has infinite length, its dimension is greater than one (log 4/log 3). The dimension of any fractal curve is greater than 1 and less than 2.

With this is mind, Monique Breslin and John Belward in the Department of Mathematics at The University of Queensland have been analysing Queensland's rainfall data.

In the past 1-dimensional smooth curves have been used to model the rainfall. But Monique and John argue that this assumes rainfall patterns on the 1st and 3rd days can be used to predict the rainfall on the 2nd day, which in general is false. Instead they have found that the infinite structure of fractals provides a much closer model for rainfall patterns. In some parts of northern Queensland the dimension of the curve is higher than 1.6 implying unpredictable rainfall patterns. Even in drier parts it is 1.3, and still unpredictable. With this research Monique and John are working with the Department of Primary Industry to improve weather information for Queensland agriculturalists.

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