Fractals - An Introduction

Real fern leaf (left) and a fractal fern leaf (right), miracle in nature and a reality in fractal world

While the classical Euclidean geometry works with objects which exist in integer dimensions, fractal geometry deals with objects in non-integer dimensions. Euclidean geometry is a description lines, ellipses, circles, etc. Fractal geometry, however, is described in algorithms -- a set of instructions on how to create a fractal.

The world as we know it is made up of objects which exist in integer dimensions, single dimensional points, one dimensional lines and curves, two dimension plane figures like circles and squares, and three dimensional solid objects such as spheres and cubes.

Fractal Dimension

To explain the concept of fractal dimension, it is necessary to understand what we mean by dimension in the first place. Obviously, a line has dimension 1, a plane dimension 2, and a cube dimension 3. But how is this happens to be ???

We may break a line segment into 4 self-similar intervals, each with the same length, and ecah of which can be magnified by a factor of 4 to yield the original segment. We can also break a line segment into 7 self-similar pieces, each with magnification factor 7, or 20 self-similar pieces with magnification factor 20. In general, we can break a line segment into N self-similar pieces, each with magnification factor N

A square is different. We can decompose a square into 4 self-similar sub-squares, and the magnification factor here is 2. Alternatively, we can break the square into 9 self-similar pieces with magnification factor 3, or 25 self-similar pieces with magnification factor 5. Clearly, the square may be broken into N^2 self-similar copies of itself, each of which must be magnified by a factor of N to yield the original figure


A square may be broken into N^2 self-similar pieces, each with magnification factor N


Finally, we can decompose a cube into N^3 self-similar pieces, each of which has magnification factor N.

So we can define Fractal dimension as...




for the square, we have N^2 self-similar pieces, each with magnification factor N. So we can write




However, many things in nature are described better with dimension being part of the way between two whole numbers. While a straight line has a dimension of exactly one, a fractal curve will have a dimension between one and two, depending on how much space it takes up as it curves and twists.

As an example let's consider Sierpinski triangle



Dimension


The more a fractal fills up a plane, the closer it approaches two dimensions. In the same manner of thinking, a wavy fractal scene will cover a dimension somewhere between two and three. Hence, a fractal landscape which consists of a hill covered with tiny bumps would be closer to two dimensions, while a landscape composed of a rough surface with many average sized hills would be much closer to the third dimension.



Some Fractal Scenes