Fractals

Author: Greg Paperin.

Fractals are, roughly speaking, objects that exhibit self-similarity on all scales. Self-similarity means that an object consists of smaller copies if itself, which in turn, consist of smaller copies, and so on. The copies do not need to be exact, but the same kind of structures must appear on all scales.

Commonly, fractals are mathematical objects, but many natural objects that demonstrate self-similarity on power-law scales are also considered fractals. This includes surfaces of clouds, mountains and other landscape entities, surfaces of plant structures and various other phenomena. Thus, the study of mathematical fractals of many kinds provides enlightening insights into the dynamics of many natural complex systems.

Fractal dimensions

Intuitively, a fractal is a very rough or winded object – so rough that its surface “breaks out” of its dimension. As an example, consider a normal 2-dimensional square. If we double the length of its base side (i.e. lengthen by a factor 2), its area will be quadrupled (i.e. increased factor 4 = 22). In fact, this is exactly why we call it 2-dimensional. Now consider a 3-dimensional cube. If we double its base side, its volume will increase by a factor of 8 = 23. If we apply the same logic to a 1-dimensional object (e.g. a line segment) its 1-dimensional equivalent of volume (i.e. its length) will increase by a factor of 2 = 21. Generally speaking, if we take an N-dimensional equivalent of a cube and double the length of its base side, the volume of the object will increase by a factor of 2N.

The above process seems intuitive for all objects. But fractals are different and that is one of the most fascinating things about them. Consider the Sierpinski triangle, a very famous fractal. It is obtained by taking a triangular shape and cutting out the middle triangle from it. From each of the 3 remaining triangles, the middle bit is removed again. This process is repeated ad infinitum and the Sierpinski triangle is the set of points that remain.

Construction of the Sierpinski triangle

Figure: Construction of the Sierpinski triangle [source: a Wikipedia article]

The resulting set of points exhibits some fascinating properties typical of fractals:

For instance, we have removed 25% of the remaining area of the structure at each step. It can be mathematically shown that the area of the Sierpinski triangle is in fact zero. At the same time, the vertices of each triangle we have removed remained as a part of the structure and since we have removed infinitely many triangles, infinitely many points do remain! Such counter-intuitive properties are one of the reasons for the great fascination caused by fractals. So what is the dimension of this weird structure?

The dimension of the initial triangle was clearly 2. But the resulting structure has no area. At the same time, the structure cannot be mapped to a line, and thus a dimension of 1 is not really appropriate either. It turns out that the Sierpinski triangle, like many other fractals, exists “between” our “normal” dimensions. It has a “broken” dimension that can be shown to be approximately 1.585 (the technical term for this is Hausdorf dimension). I.e. if we increase the length of a base side of a Sierpinski triangle by a factor of 2, the “contents” of the structure will grow not by a factor of 21 = 2 (as they would do for lines), and not by a factor of 22 = 4 (as they would for a normal triangle), but by a factor of approximately 21.585.

Fractal images

Besides theoretical insights about natural, formal and artificial complex systems that are brought about by mathematical studies of fractals, fractals are also used by artists for generating aesthetically pleasing objects. The fractal nature of landscapes has lead to generation of fractal landscapes – computer images produced through photo-realistic rendering of mathematically generated fractals. Abstract imagery is one of several other areas of fractal art. The renderers available on this site provide an opportunity to explore some very different kinds of fractals in their full complexity. Take some time to learn about and to explore some of the most famous fractals!

Fractals and Scale tutorial

More information and in-depth explanations are available in our fractals and scale tutorial.

Simulations