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Measuring the Coast of England; images from wikipedia.com |
Fractals part 1 |
The term 'fractal' was coined by Mandelbrot, who used it to describe objects that have self-similarity and a fractional dimension. Mandelbrot explained these ideas in a 1967 paper entitled "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension."
The ideas of one dimension (represented by a line), two dimensions (a plane) and three dimensions (the world in which we live) are easy to grasp. At least, they make sense to us, who live in a three-dimensional world. To people who live with less dimensions, such as characters from Edwin Abbott Abbott's Flatland, the higher dimensions are nearly impossible to understand. Similarly, we have difficulty visualizing dimensions higher than three because we cannot experience them. Mandelbrot's definition of fractals depends on their having a fractional dimension, which can also be a difficult concept to grasp. The basic idea, however, is simple. A fractal, because it iterates infinitely, is not able to be classified with the lines as being one dimensional. However, because it does not fill the plane-- there is empty space, a fractal cannot be two dimensional. |
In his paper, Mandelbrot used the coast of Britain to show how an object might have a longer length the smaller the increment of measurement. He described how Britain could be measured with a long ruler to give a rough approximation of the length of the coast. Then make that ruler half as long, and the approximation will be more similar to the true object. You can continue this process many, many times and never level off; one could never know exactly how long the coast of Britain is (if it were a true fractal). In his paper, Mandelbrot used the work of English meteorologist Lewis Richardson, who discovered that the length of a coastline grows the smaller the unit used to measure it.
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Measuring the Coast of England; images from wikipedia.com |
Britain, however, is not a fractal, merely an example: it acts like a fractal only over a certain range. However, it allowed Mandelbrot to show that length of an object was a function of the scale used to measure it.
Length (L) as a function of the measurement scale (G) is equal to a constant (M) multiplied by the measurement scale raised to 1 minus the dimension: L(G) = M*G^(1-D). This formula can be solved for D to give an estimate of the fractional dimension of a fractal. The fractional dimension of the west coast of Britain, for example, is 1.25. |
A simple fractal can be used to describe self-similarity: the Koch Snowflake. This is created from equilateral triangles. Begin with an equilateral triangle, and at the center of each side put the base of another equilateral triangle, with sides 1/3 the length of the original. At the center of each new side, put the bases of new equilateral triangles, sides 1/3 the size of the previous iteration's triangles (and 1/9 the size of the original triangle). Continue this process infinitely, and you end up with the Von Koch curve, as shown below. As you can see, the curve is self-similar: if you magnify it you cannot tell if you are 10x or 100x, because each part looks the same.
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Image from wikipedia.com |
The image below shows the first 4 iterations of the formation of several fractals. If iterated infinitely, each would be self-similar at any level and each would have a fractional dimension which could be determined by the formula discussed above.
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The Computational Beauty of Nature, a wonderful book on complex systems by Gary William Flake, has a series of online java applets, one of which allows the building of fractals. Choose an applet size, then pick L-systems from the drop-down menu. The commands for this are simple: F draws forward, G moves forward without drawing, + turns to the right, - turns to the left, [ saves a position, ] returns to the last saved state.
If you want to draw a Koch snowflake, the angle is 60 (an equilateral triangle), the axiom is F + + F + + F (forward, turn 120 degrees, forward, turn 120 degrees, forward to the first position), and the rules are F = F - F + + F - F. Alternatively, you can pull down Koch-Island from the drop-down menu and hit 'restart' to watch it go. |
Continue reading about fractals
Katie Dektar, September 2008
