Artificial Forms
Fractal Dimension
Fractal Geometry
More Examples
Why Fractals
The Lorenz Equations
Nonlinear Geoscience

Let's Generate Fractals

A simple artificial self-similar form that has been closely studied for a long time is Koch curve (named after Swedish mathematician Helge von Koch). To generate a Koch curve and others with similar properties is simple; all we need is a construction rule which will then be applied repeatedly (iteratively). The rule is as follows:

Start with a straight line segment of unit length, divide it up into three equal segments of length 1/2, next substitute the middle segment with a V-shaped indentation whose sides are also 1/3. Repeat the construction for all remaining straight line segments.

After applying the rule just a few times, you'll see the orignial straight line segment become a very rugged curve. If we could apply the rule an infinite number of times, no matter how much we enlarge it or zoom into it, its raggedness would always be the same, it would look the same no matter what enlargement we use, and any part would be identical with the whole.

We are now ready to calculate the dimension of the Koch curve.