frac·tal (frāk“tėl) n. A geometric pattern that is repeated at ever smaller scales to produce irregular shapes and surfaces that cannot be represented by classical geometry. Fractals are used especially in computer modeling of irregular patterns and structures in nature. [French, from Latin fractus, past participle of frangere, to break. See FRACTION.]
Fractals are extensions of traditional Euclidean shapes, such as lines, squares, and circles, with two fundamental properties. First, when you view fractals, you can magnify them an infinite number of times, and they contain structure at every magnification level. Second, you can generate fractals using finite and typically small sets of instructions and data. Fractals grew out of the goal of mathematicians to completely describe the world using standard geometrical expressions. IBM mathematician Benoit B. Mandelbrot, PhD, proved and published the theory behind fractals in 1981 and was the first to view computer-generated fractal structures. The well-known Mandelbrot Set is named in his honor. Another famous fractal researcher, with a set also named for him, is French mathematician Gaston Julia.
The standard Mandelbrot fractal equation takes the form z(n+1) = z(n)^2 + c, where c is the complex number x+iy corresponding to any point on the (x,y) coordinate plane. Fractal equations are iterative, in that the result of one calculation of the fractal equation becomes the z input to the next calculation. Over repeated evaluations of a fractal equation, values for each point in the (x,y) coordinate space either converge at single points, move toward the (0,0) origin point, or move toward infinity. The diverse colors in fractal plots reflect the rate of this movement for each point. Discussions of chaos theory frequently use fractals as examples, because slight variations in the fractal equation produce radically different results.
But first, a quick set of definitions:
FORMULAS are the mathematical statements (especially equations) of a fact, rule, principle, or other logical relation, used to perform the basic calculations of fractal images. Many fractal generating applications are capable of using various formulae, whether hardcoded within the program or using a formula parser.
PARAMETERS are the constants in an equation that vary in other equations of the same general form. They are values supplied to, and used by, a formula within the generating application. These are established by the user of the software to adjust various settings, quantities, etc. to achieve varied results in the image itself.
IMAGES are the the pictorial representation as used in computer-aided design, the process by which a computer displays data pictorially.
MAPS, or GRADIENTS, are the color palettes used to cause variations in the way an IMAGE may be colored. (Thousands of these are in the public domain already.)
The PARAMETERS and IMAGES made by an individual are their own property, and are considered to be copyrighted. Any use by another person of said items should require the permission by the creator/author/originator. Whereas FORMULAS are usually considered to be public domain, and virtually can not cause an IMAGE without supplying PARAMETERS.
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