CITS4403 Computational Modelling Fractals

A fractal is a mathematical set that typically displays self-similar patterns. Fractals may be exactly the same at every scale, or they may be nearly the same at different scales.

Where fractals come from… Fractals are closely linked to complexity theory. Self similarity and patterns that are independent of scale have been studied since the 17 th century. More recent studies of complex systems have found that simple equations for describing weather, cellular automata, agent interactions has a similarly fractal nature.

Complex Systems and Fractals Complex systems and fractals have some common properties. They can be simple to specify, but hard to predict. They are studied via computer simulations. They frequently have the scale free property. Treating complex systems as fractal allows us to use the Fractal Dimension to “quantify” complexity.

Fractal Dimension To understand fractals, we have to start with dimensions. The dimension of a space is the number of coordinates we need to specify a point in a space. A number line takes one coordinate, a Euclidean plane takes 2, a solid takes 3, etc. A fractal, such as the Koch Snowflake is approximated by a line, but at its limit it has a fuzzy, infinite border, more like an area….

Fractal Dimension We can describe non-integer dimensions by examining what happens to the shape under scaling. Image we switch from Metric to Imperial. Going from centimeters to inches, we multiply every length by 2.5, every area by 6.25 (2.5*2.5) and every volume by and so on. By examining the properties of fractals under scaling we can assign them a fractal dimension. The Koch Snowflake has infinite length, but we can examine what happens when it is approximated at a scale:

Box Counting Complexity To compute the box-counting dimension, we divide the space into a grid where the size of each cell is ε. Then we count N(ε), the number of cells that contain at least one element of S. As ε gets smaller, N(ε) gets bigger. For many objects the relationship has the form: Taking the log of both sides and rearranging yields:

Fractal Dimension of CAs To investigate the behavior of fractal dimension, we’ll apply it to cellular automata. Box-counting for CAs is simple; we just count the number of “on” cells in each time step and add them up.

Fractal Dimension of CAs In some cases (Type 1) the CA is very predicable, and in these cases has an integer dimension, but in others (Type 2,3,4) we see fractional dimensions:

Other Complex Systems In 1990 Bak, Chen and Tang proposed a cellular automaton that is an abstract model of a forest fire. Each cell is in one of three states: empty, occupied by a tree, or on fire. The rules of the CA are: 1.An empty cell becomes occupied with probability p. 2.A cell with a tree burns if any of its neighbors is on fire. 3.A cell with a tree spontaneously burns, with probability f, even if none of its neighbors is on fire. 4.A cell with a burning tree becomes an empty cell in the next time step. Percolation is a process in which a fluid flows through a semi-porous material. Examples include oil in rock formations, water in paper, and hydrogen gas in micropores. Percolation models are also used to study systems that are not literally percolation, including epidemics and networks of electrical resistors. Seehttp://en.wikipedia.org/wiki/Percolation_theory. Percolation processes often exhibit a phase change; that is, an abrupt transition from one behavior (low flow) to another (high flow) with a small change in a continuous parameter (like the porosity of the material). This transition is sometimes called a “tipping point.”

Mandlebrot and Julia Sets These classic fractals are the result of applying iterated complex functions, and examining the boundary of the convergent set:

Philosophy of Science?? Does fractal dimension give us a measure of complexity? Discuss this claim… Does fractal complexity allow non- determinsim and free will to exist within a deterministic universe?