The existence of Mandelbrot and Julia set fractals depends on complex numbers. But if we are going to talk about the latter, we have to introduce imaginary numbers first. Two Italian mathematicians, Girolamo Cardano and Raffaele Bombelli, proposed both types of numbers in the sixteenth century.

As we know, negative numbers have no square roots that could be expressed in real numbers. Nevertheless, mathematicians have given them an imaginary value i defined as the square root of -1 (hence their name).

or     i^2 = -1

Complex numbers are those composed of a real part and an imaginary part. The real part is a real number-for example, -2, 1, 1/2, 0.2154-, while the imaginary part is a real number plus the special number "i", as in 3i. An example of a complex number would be 2 + 3i.

Not all fractals are made by the iteration of mathematical expressions with complex numbers. Iterating elementary geometric figures also produce some of them. The Sierpinski Carpet, for instance, is produced from a square.

Figure 21: From left: second, third and forth iterations of the Sierpinski carpet.

An equation is defined as a statement which shows that two mathematical expressions are equal, such as in x + 1 = 3 - x^2.

A function is defined as an association between two or more variables, in which to every value of each of the independent variables, or arguments, corresponds exactly one value of the dependent variable in a specified set (called the domain of the function). Simply put, in a function such as f(y) = x + 1, the value of the variable y depends on and varies with that of x. In that expression y is the dependent variable, while x is the independent variable.

A formula, on the other hand (and in our case), expresses a mathematical fact. As an example, the formula for calculating the area of a triangle is a = ^bh/₂, where b is the base, h is the height, and a the area of the triangle.

When we talk about the Mandelbrot set f(z) = z^2 + c, it would be proper to talk about functions. While they are equations, since we are expressing that both parts are equal, they are functions, since their values are limited to a given set of numbers.

For several reasons fractals have been associated with chaos. However, there are many fractal objects that have nothing to do with chaos. As we have seen, many of the early simple mathematical fractal constructions date from the late 19th century, long before chaos theory came forth in the 1960s. Nevertheless, thanks to the technological advances in computer science, chaos theory has generated some new types of fractals. One of the pioneers of chaos theory is Dr. Edward Lorentz of the Massachusetts Institute of Technology (MIT)—even though Jules Henri Pointcaré was talking about the "Butterfly Effect" as early as the 1830's.

Figure 22: Lorentz attractor.

Strictly speaking, chaos theory is the study of nonlinear systems, for which the rate of change is not constant. They are characterized by unpredictability. The weather and population growth are good examples of non-linear systems, both of which are also fractals.

In nonlinear systems, each state of the system is determined by its previous state (iteration), and a small change in the initial input values will have dramatic effects on the final outcome of the system.

Thanks to the discoveries of chaos theory and fractal geometry, scientists have been able to understand how systems once thought to be completely chaotic actually have predictable patterns. One of the most significant contributions of fractal geometry is its "capability" to model natural phenomena, such as plants, clouds, geological formations and atmospheric phenomena. Fractal theory has also contributed in such diverse fields as linguistics, psychology, image compression technology, superconductivity, circuitry and other electronic applications.

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