Mandelbrot Set
The Mandelbrot set is the most famous object in the branch of mathematics known as fractals and chaos theory. It was discovered in 1980 by the mathematician Benoit Mandelbrot, who pioneered this relatively new field.
A picture of the Mandelbrot set is reminiscent of a beetle with two main segments: a large circular bulb adjoining the left end of an even larger cartioid shaped central region. While these two segments visually dominate the Mandelbrot set, the geometric intricacies of its border truly define it.
A glance at that border reveals a number of bulbs of varying sizes attached to the two main regions of the set. Each of these bulbs, no matter the size, has a number of antennae protruding from it. Like the main region of the Mandelbrot set, the border of these antennae are decorated with a series of bulbs, which in turn have antennae protruding from them. Zooming in on those antennae will reveal more bulbs, each of which have antennae protruding from them. Zooming in on these antennae again reveals more bulbs, which contain more antennae, which contain more bulbs, and so on and so forth. Magnifying various regions of the Mandelbrot set's border not only reveals an infinitely repeating series of bulbs and antennae. It also reveals copies of the Mandelbrot set itself within the antennae. Each of these 'miniature' Mandelbrot sets contain all of the geometric properties and similarities described above, including additional copies of the Mandelbrot set, which in turn contain all the previously discussed properties of the Mandelbrot set. These patterns, which reveal themselves at every scale of magnification, intrigue not only mathematicians, but scientists and artists as well.
The Mandelbrot set is a graph of an algebraic function, much like a parabola is a graph. Unlike a parabola, which exists on the Cartesian plane, the Mandelbrot set exists on the complex plane. The horizontal axis in this plane is the x-axis, as is true in the Cartesian plane. The vertical axis in the complex plane, however, is known as the i-axis. Each point in the complex plane represents a complex number. It is these complex numbers that produce the Mandelbrot set. In order to understand the geometry of this graph, it is necessary to understand the algebra behind it.
The fundamental principle underlying the Mandelbrot set is iteration, which means to repeat a process over and over again. In mathematical terms, to iterate a function means to substitute a value derived from that function back into the function to derive the next value. This can be repeated indefinitely. This process of iteration is illustrated in the following example using the function f(x) = x2 + 1 and an initial value of x = 0.
f(0) = 02 + 1 = 1
f(1) = 12 + 1 = 2
f(2) = 22 + 1 = 5
f(5) = 52 + 1 = 26
f(26) = 262 + 1 = large
f(large) = large2 + 1 = very large
etc.
As shown in the example above, the orbit, or iteration, of x2 + 1, with an initial value of x = 0, tends to infinity.
Orbits do not always tend to infinity. This is illustrated in the iteration of the function f(x) = x2 + 0 with an initial value of x = 0.
f(0) = 02 + 0 = 0
f(0) = 02 + 0 = 0
f(0) = 02 + 0 = 0
etc.
The orbit of f(x) = x2 + 0, with an initial value of x = 0, is a fixed point.
The above examples illustrate an important dichotomy in iterative functions: Sometimes orbits of a function go to infinity, sometimes they do not. This dichotomy provides a definition of the Mandelbrot set: The Mandelbrot set consists of all c-values in the function x2 + c, where x and c are both complex numbers, for which the orbit of 0 does not go to infinity.
This explains how the graph of the Mandelbrot set is created. A complex number is used as a c-value in the function x2 + c. That function is then iterated using an initial value of x = 0. If this iteration does not tend to infinity, then that c-value is in the Mandelbrot set. Its corresponding point on the complex plane is colored black. If the iteration does tend to infinity, then the c-value is not in the Mandelbrot set. Its corresponding point on the complex plane is represented by a color other than black. This explains why the black region of the graph is the actual Mandelbrot set. It also hints at the extraordinary number of calculations required to create the graph of the Mandelbrot set, as this iteration process is applied to thousands of points on the complex plane.
Though the colored regions in the graph of the Mandelbrot set are not actually part of the Mandelbrot set, they do have meaning. They represent how quickly the orbit of 0 escapes to infinity when iterated in the function x2 + c. The points that directly border the Mandelbrot set are colored violet. When used as c-values in the function x2 + c, these points result in an orbit of 0 that tends to infinity very slowly. It may take 40 to 50 iterations for these orbits to explode to infinity. The points farthest away from the Mandelbrot set are colored red. When used as c-values in the function x2 + c, these points result in an orbit of 0 that escapes to infinity very rapidly. These orbits will explode to infinity after only 5 to 10 iterations.
The Mandelbrot set results from a relatively simple algebraic procedure. Yet this procedure produces an intricate geometry that is full of similarity and patterns in the midst of apparent chaos. These patterns have provided new understanding of many diverse fields, including ecology, economics, and meteorology.
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