
The Mandelbrot Set, named after its discoverer French Mathematician Benoit Mandelbrot (1924- ), is a fractal. Fractals are objects which exhibit “self-similarity” at various scales. Optically magnifying a fractal will reveal smaller scale details similar to the gross characteristics. Although the Mandelbrot Set is self-similar at magnified scales, the fine details are not always identical to the gross forms. The Mandelbrot set is said to be “infinitely complex”, because the number of repeated “designs” rises toward infinity, along with the number of iterations. For all the mathematical, philosophical and theological mysticism which surrounds the Mandelbrot Set, the process of generating it is based on an extremely simple equation involving complex numbers.
Understanding Complex Numbers
The Mandelbrot Set is a mathematical set, a collection of numbers — a collection of complex numbers, to be precise. Complex numbers have a real part plus an imaginary part. The real part is an ordinary number, for example, -2. The imaginary part is a real number multiplied by a special number called i, or some multiple of i, for example, 3i. An example of a complex number would be -2 + 3i.
The number i was invented because no real number can be squared (multiplied by itself) and result in a negative number. This means that you cannot take the square root of a negative number and get a real number. The number i was created in order to be able to express numbers which when squared result in negative numbers. The number i is defined as the square root of -1. This means that i squared is equal to -1. So when you square an imaginary number you can get a negative number. For example, 3i squared is -9.
The complex number planeReal numbers can be represented on a one dimensional line called the real number line. Negative numbers like -2 are plotted to the left of zero and positive numbers like 2 are plotted to the right of zero. Any real number can be graphed on the real number line.
Since complex numbers have two parts, a real one and an imaginary one, we need a second dimension to graph them. We simply add a vertical dimension to the real number line for the imaginary part. Since our graph is now two-dimensional, it is a plane, the complex number plane. We can graph any complex number on this plane. The colored dots on this graph represent the following complex numbers [2 + 1i], [-1.5 + 0.5i], [2 - 2i], [-0.5 - 0.5i], [0 + 1i], and [2 + 0i].
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