Clever mathematicians over the years have devised some interesting and visually stunning examples of fractal figures which help them to better understand this concept.

A fractal - which is described by mathematicians as being both continuous and non-differentiable - can best be described as a figure which intrinsically contains ever smaller versions of itself. It is a figure which can be zoomed in to finer and finer detail, and continue to show more and more of the same form as the larger (a common analogy is that of a coastline, which contains more and more curvature the closer one looks at it).

While many forms of fractals exist in nature, and have proved invaluable for mathematicians who have tried to understand the complexity of the universe around them, others have been developed within the minds and on the papers of mathematicians, who continue to develop ever-more-clever means of understanding these truly "infinite" figures.

Here are a few examples of such fractal figures:

The Mandelbrot Set

Perhaps the most famous example of a fractal figure, the Mandelbrot set was developed in the 1980's by the mathematician Benoit Mandelbrot (called by many "the father of fractal geometry"), who had been exploring the graphical features of certain quadratic polynomials.

What Mandelbrot discovered was that certain complex quadratic polynomials when graphed created images (which are surprisingly varied, though many of them, as a simple online search will show, look something like tye-die designs) which contained the very unique feature of lacking any semblence of simplicity, no matter how closely one looked at it.

In other words, for a Mandelbrot set, the closer one looks at it, the more detailed and complex it becomes.

It was the work of Mandelbrot (including his coining of the term "fractal" itself) which opened the doors for other forms of fractals, some more complex, and some more simple.

Serpinski Gasket

The Serpinski Gasket, developed in 1915 by the Polish Mathematician Waclaw Serpinski (though it wasn't called fractal at that point - it was more of a mathematical curiosity), is a geometrically simple, self-similar triangle.

This is a good example of the type of fractal that is fairly easy for a person to understand. It consists of a triangle which is divided into three other triangles (see image below). These three triangles are then divided the same way into three more triangles, which are themselves divided, and so on. The result of all this is that at any magnification, any one of these triangles will look identical to the original, being composed of ever decreasing triads of triangles.

Variations of the Serpinski gasket are almost unlimited, and have been extended to include squares, three dimensional constructs and rotational translation.

The importance of the Serpinski gasket lies in both the mathematics that have been used to describe its phenomenon, and the fact that it is a rather simple example of what is now known as a fractal figure.

The Koch Curve

The Koch curve, named after Swedish mathematician Helge von Koch who first developed it in a 1904 paper, is another rather simple example of an easily-constructed fractal.

A description of how to make a Koch Curve goes like this:

Draw a straight, horizontal line. In the middle of this line, draw an isoscelese triangle using the middle third of the original line as a base. Now, using every resulting line segment (the two remaining portions of the original line and the two sides of the triangle), perform the same action - drawing smaller triangles using the middle third of the lines as bases. Then repeat the process again.

While this process, in theory, should continue ad infinitum, there will soon be a point where the resulting triangles are much too small to actually draw.

Now, the fractal aspect of this figure should be obvious, for like all the others, the closer one looks at any section of this figure, the more detailed it will become, due to its self-similarity.

Another remarkable aspect of this figure, like other similar fractals, is the counter-intuitive fact that the length of the resulting perimeter is, by definition, infinite. As new lengths in the form of triangles will continually be forming, there is no absolute way of measuring this never-ending perimeter. To make matters even more difficult to comprehend still, any single line segment, no matter how small, would also be infinitely long, for the same reasons.

Fractals in Nature, and the Future of Mathematics

This "infinite" nature of fractals is one of the more important (and mathematically challenging) aspects of them, while at the same time being absolutely necessary for their existence. It is for this reason that all fractals may be considered to lie firmly in the realm of non-Euclidean geometry - for any standard, Euclidean geometry must by definition possess the ability to be measured and studied in traditional ways.

It is also this feature which shows that naturally occurring examples of fractals are very rarely "true" fractals, as they almost always possess some sort of "end" to how far they can be magnified and remain self similar (even if this limit lies at the atomic level). These mathematical, man-made examples of fractals, however, remain important examples of the concept in a general sense, and will surely continue to play ever-more-interesting roles in many forms of science, mathematics, and art.