Juan Luis Martínez
T h i r d . A p e x . t o . F r a c t o v i a

This article explains some basic principles of fractal geometry, from its "discovery" to its application in science and art. Click here to download an Acrobat Reader version of this article for off-line reading.

A fractal is an object that displays self-similarity at various scales. In other words, if we zoom in any portion of a fractal object, we will notice the smaller section is actually a scaled-down version of the big one.

Figure 1: Julia fractal.

Another very important aspect of a fractal is that it has a fractional dimension. That is, instead of being one, two or three-dimensional (as is the case with more familiar objects), most fractals have dimensions that do not fit into that conception. Moreover, their dimension can rarely be expressed as an integer value. This is precisely what lies behind their name.

Regularly, fractals adhere easily to the previous definition. Sometimes they do not, and new and different features will show up in what otherwise would have been a persisting pattern. That depends on the type of fractal we are examining and in the equation that produced the figure in the first place.

Figure 2: Cantor dust.

Possibly the first pure fractal object in history, the Cantor dust was described by the German mathematician Georg Cantor-inventor of set theory-around 1872. It is a very simple figure, yet it contains all the attributes discussed so far: it depicts self-similarity at all scales and has fractional dimension, with value around 0.630929753571457437099527114! (more appropriately, log 2/log 3). We can also use it to introduce another general characteristic of fractals: it is produced by iteration.

Iteration can be described as a feedback process that repeats an n number of times. It refers to the act of performing the calculation of a certain function and then picking the result, or output, as the starting value, or input, for the next calculation of the same function. The operation repeats on and on (even infinitely), becoming an iteration. Any such process will produce a fractal.

For the Cantor dust example, we start with a large segment (the initiator), divide it in three equal smaller segments, and take out the middle one. This process (the generator) repeats indefinitely, producing the cantor dust.

Figure 3: Iteration of the Cantor dust.

In the same iterating fashion, we can produce a Sierpinski triangle, a figure invented by the Polish mathematician Waclaw Sierpinski in 1915:

Figure 4: Sierpinski triangle.

We start with an equilateral triangle. On its inside, we draw a small equilateral triangle with corners at the midpoints of the sides of the large one. Then, we remove the new triangle, such that three similar equilateral triangles remain inside the big one. Repeat the same process (iteration) for each small triangle, and a Sierpinski triangle, or gasket, will result.

    
    
Figure 5: Iteration of a Sierpinski triangle.

Note that when we say, "remove the new triangle", we do not mean to simply take it out, but that the points contained in the area of that particular triangle do not belong to the set of points comprising the Sierpinski triangle. Hence, that part does not belong to the set.

Although fractals were known since the late 19th century (when they were regarded as mathematical curiosities), their kinship was unmistakable unveiled in the 1960s and '70s through the studies of Benoît Mandelbrot and other prolific scientists.