(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
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.
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.
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.
There are many fractals in nature and for whom precise mathematical formalisms have been developed. Mathematicians have also noted that whether the fractal is deterministic or stochastic (with a random element accompanying the determinism) that it is possible that a random or chaotic event or structure is really at its basic level the result of a relatively simple structure that keeps repeating, and for which a mathematical notion has not been developed. One example is Brownian motion. (Mandelbrot 1983) This is the random motion of particles in a fluid. It is possible to develop a fractal that can trace the random instances as part of a not yet determined non-chaotic whole. Fractal-like behavior has also been observed in the breaking of fabricated structure and scientists have begun to see patterns in the earths crust following an earthquake as a system of fractals.
Note: All figures have been taken from the one of the following web sources.