Fractal GeometryFractal Geometry
Fractal geometry is a branch of mathematics having to do with fractals. Fractals are geometric figures, just like rectangles, circles and squares, but fractals have special properties that those figures do not have. In geometry two figures are similar if their corresponding angles are congruent in measure. Fractals are self-similar meaning that at every level the fractal image repeats itself. An example of self-similarity would be a triangle made up of triangles that are the same shape or are similar to the whole. Another important property of fractals is fractional dimensions. While in Euclidean geometry figures are either zero dimensional points, one dimensional lines, two dimensional planes, or three dimensional solids, in fractal geometry figures can have dimensions falling between these whole numbers, that is being made up of fractions. For example a fractal curve would have a dimension between one and two depending on how much space it takes up as it twists and curves. The more a flat fractal fills a plane the closer it is to being two-dimensional. As few things have basic shapes, fractal geometry provides for the complexities of these shapes and allows the study of them better then Euclidean geometry which is only successful in accommodating the needs of regular shapes.
Fractals are formed by iterative formation, meaning one would take a simple figure and operate on it in order to make it more complex, then take the resulting figure and repeat the same operation on it, making it even further complex. Algebraically fractals are the result of repetitions of nonlinear-equations. Using the dependent variable for the next independent variable a set of points is produced. When these points are graphed a complex image appears.
One does not have to try very hard in order to experience fractals first hand in the real world as they are ever present in nature. For example in the instance of a river and it’s tributaries, each tributary has it’s own tributaries so that it’s structure is similar to that of the entire river. Many of these things would seem irregular, but in fractal geometry they each have a simple organizing principle. This idea of trying to find underlying theories in what seem to be random variations is called the chaos theory. This theory is applied in order to study weather patterns, the stock market, and population dynamics. Fractals can also be used in order to create computer graphics. It was found that the information in a natural scene can be concentrated by identifying it’s basic set of fractals and their rules of construction. When the fractals are reconstructed on a computer screen a close resemblance of the original scene can be produced.
The first person to study fractals was Gaston Maurice Julia, who wrote a paper about the iteration of a rational function. This work was essentially forgotten until Benoit Mandelbrot brought it back into the light in the 1970’s. Mandelbrot, who now works at IBM’s Watson Research Center, wrote The Fractal Geometry of Nature that demonstrated the potential application of fractals to nature and mathematics. Through his computer experiments Mandelbrot also developed the idea of reconstructing natural scenes on computer screens using fractals.
In conclusion fractals are irregular geometric objects made of parts that are in some way similar to the whole. These figures and the study of them, Fractal geometry, allow the connection between math and nature.
M. Barnsley, Fractals Everywhere, 2d ed, 1992
T. Vicsek, Fractal Growth Phenomena, 1992