Fractals: concepts, properties, and applications

Fractal, in its completed and perhaps complex form, resembles a fracture or a series of complicated and uncoordinated breaks. Indeed, the word can trace its origins to the Latin fractus, which means fractured or broken. A fractal, as is mathematically understood, is the end product (or a product that is in the process of completion in a recursive manner) of a geometric figure on which a recurring operation is performed. The operation should be necessarily identical at every step, hence the term recursive. Another important term is associated with fractals -- self similarity. This means that if one starts with a geometric construct, then every subsequent recursive operation produces a result that is identical or strongly similar to the starting product. Despite the notion of self-similarity, the end product after several hundred or recursive iterations might occasionally end up not resembling the original or starting geometric figure. Recursivity over several iterations can typically be achieved by using a computer program that creates the fractal. (Barnsley and Rising 1993)

The end product of these endeavors is often a fascinating figure. While it is possible to create a fractal that appears complex, though it might have started from a very simple figure, the end product cannot be described in terms of a simple figure. One can explain this mathematically using the concept of Hausdorff dimension.(Weistsein 2009) Typically, a point can be described as having zero dimension, a line has one dimension and a plane has two dimensions etc., however consider a mountain, which can be reduced to a cone with dimension three, can be considered as a fractal because its dimensionality is non-integral. The dimensions of fractals are non-integral.

One of the simplest examples of a fractal found in nature is a few species of fern. The figure follows

Fern as an example of a fractal in nature

If one imagines a fern it consists of a stem and what might appear as leafs on either side of the stem. But then each leaf itself consists of a main stem to which are attached other fern like structures. Though this is not repeated ad infinitum, since the fern is a finite object, one can see that each small fern- which appears, from a distance, as a leaf, is fairly identical to the main fern. A fern is self-similar. Coastlines of nations are considered to be self-similar. While a line is self-similar, a line recurring upon itself is not considered a fractal because it can be described in Euclidean terms. Similarly, a trapezoid can be constructed out of four similar or identical trapezoids; this is self-similarity, but not a fractal. A snow flake on the other hand can be considered a fractal. (Mandelbrot 1983) Hence all fractals are self similar; but, not all self-similar objects are fractals.

By incorporating mathematical formulas into fractal generating algorithms infinitely many and complex fractals can be generated.

Sierpi-ski Triangle

Sierpi-ski Triangle

One of the first examples of a fractal is the Sierpi-ski triangle, named for Wac-aw Sierpi-ski. This is also called Sierpi-ski gasket or Sierpi-ski sleeve. This fractal starts out with an equilateral triangle. Another triangle is created within this triangle. The vertices of this smaller triangle coincide with the midpoints of the sides of the original equilateral triangle. This results in four equilateral triangles. In each of the three of the four triangles (the one omitted is the middle triangle), additional triangles are created such that their vertices coincide with the midpoints of the triangles in which they are constructed. And this process can be carried out, each time ignoring the central triangle created (of the four). This process can theoretically be carried out ad infinitum. It is possible to create several interesting side-fractals by rotating the fractals and capturing the image following each rotation. At each of this rotation, the features of this fractal are maintained. (Weisstein "Sierpi-ski Sieve" 2009)

Sierpi-ski Carpet and Menger Sponge

Sierpi-ski created another strictly two dimensional fractal called the Sierpi-ski carpet. He used the same idea to create a tetrahedron. In case of the carpet, consider a plane into which are cut nine equivalent squares. The central square is abstracted. Each resulting square is then divided into nine equivalent squares of which the central one is removed, and so on. This is a self-similar square. The effect is that of a complexly woven carpet.

Sierpi-ski Carpet

The Menger sponge is a three-dimensional version of the Sierpi-ski carpet. (Weisstein "Menger Sponge" 2009) It is created by dividing a cube into 27 equivalent cubes. This resembles a Rubik's cube. The central cube is then removed. Each of the remaining cubes are then divided into 27 cubes each of which the central cube is removed. This process is continued to produce a Menger sponge, created by Austrian Mathematician Karl Menger. While 27 cubes result from the first iterative dividing of the starting cube, one can see that the problem becomes quickly insurmountable and needs the use of a computer. On the 6th interation, sixty-four million cubes are produced. The Hausfdorff dimensionality is 2.7268.

The figure shows a Menger sponge after the 4th interation

Dragon Curve

One of the best illustrations of a fractal, especially the ones that produce the complex looking diagrams is called the dragon curve. While there are programs that generate fractals, showing the results, the dragon curve demonstrates this step-by-step. The creation of the fractal begins with a line segment. This segment is folded at 90 degrees and rotated by 45 degrees. This is followed by a further folding of each side of this angle by 90 degrees one segment to the left and the other to the right; this assembly is then rotated by 45 degrees. This is then continued, as illustrated in the figure below. (Weisstein "Dragon Curve" 2009) The Hausdorff dimension for the boundary for this fractal is 1.5236.

Mandelbrot Set

One of the pioneers of the art and mathematics of fractals is Mandelbrot. His famous Mandelbrot Set is another popular fractal. (Alfeld 1998) The mathematical formula for this set is fairly straightforward. This is a function involving two numbers z and c, such that f (n) = zn +c. The resulting figure consists of a sphere to which attached several smaller spheres (which appear as circles in a 2-dimensional representation. It is important to note that z and c are complex numbers. The starting point of a Mandelbrot set (Hausdorff dimenion = 2) appears as follows.

The next two figures show how the recursive creation of a fractal proceeds with n=2 and n=20. As one can see as n reaches a very high number the Mandelbrot set tends to appear as a disc.

n= 3

n=20

Koch Snowflake (Star)

This too is one of the earliest examples of a fractal. It was created by the Swedish mathematician Helge von Koch. It starts out with a line segment whose two endpoints extend into an equilateral triangle. The base of this triangle is then abstracted. For simplicity sake, one can start out with an equilateral triangle. For each side, create an equilateral triangle with the original segment as the base. This base is then removed. After the first iteration, the Koch snowflake looks like the Star of David. The figure illustrates the four recursive steps in the creation of the Kock snowflake. This fractal has a Hausdorff dimension of 1.2619.