Fractal, or fractional, dimension is something that will never exist and will never be understood inside the realm of elementary geometry. It is but another field in which at least one of Euclid's postulates-those compiled by the Greek mathematician in the fourth century B.C.-does not hold, and where other mathematical realities emerge. Thus, we can say that there are two main types of geometry: Euclidean and non-Euclidean geometries. In the first group we find plane geometry, solid geometry, trigonometry, descriptive geometry, projective geometry, analytical geometry and differential geometry. In the second category, there are hyperbolic geometry, elliptic geometry and fractal geometry. For a detailed list of geometries, click here.

The word "fractal" comes from the Latin word "fractus", which means, "fragmented", "fractured", or more simply "broken", very appropriate for objects with fractional dimensions. It was Benoît Mandelbrot who coined the term back in 1975. The study of fractal objects is generally referred to as fractal geometry.

Figure 14: A fractal near the border
of the Mandelbrot.

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Fractal images get their "shapes" and colors because we assign a given color range to a series of points depending on their behavior as we resolve the function with the aid of a computer. That's also the only way we have to view them. We have several possibilities for assigning color values:

• If the output goes to 0 (hence, they belong to the set),
• If it escapes to infinity (hence, they do not belong to the set),
• If it oscillates among a number of states,
• If it does not exhibit any discernible pattern.

The first case occurs inside the figure; the second, outside the figure; and the third and fourth cases, near the borders.

If it weren't for those artificial color designations fractals would resemble any other plain "unattractive" looking graph.

Figure 15: Simple Mandelbrot set.