Mandelbrot
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].
Graphing the Mandelbrot Set
Since the Mandelbrot set is a set of complex numbers, we graph it on the complex number plane. However, first we have to calculate many numbers which are part of the set. To do this we need a test that will determine if a given number is inside the set or outside the set. The test is based on the equation Z = Z2 + C. C represents a constant number, meaning that it does not change during the testing process. C is the number we are testing. Z starts out as zero, but it changes as we repeatedly iterate this equation. With each iteration we create a new Z that is equal to the old Z squared plus the constant C. So the number Z keeps changing throughout the test.
How magnitude is calculated We’re not really interested in the actual value of Z as it changes, we just look at its magnitude. The magnitude of a number is its distance from zero. For example, the number -9 is a distance of 9 from zero, so its magnitude is 9. The magnitude of a complex number is harder to measure. To calculate it, we add the square of the number’s distance from the x-axis (the horizontal real axis) to the square of the number’s distance from the y-axis (the imaginary vertical axis) and take the square root. In this illustration, a is the distance from the y-axis, b is the distance from the x-axis, and d is the magnitude, the distance from zero.
As we iterate our equation, Z changes and the magnitude of Z also changes. The magnitude of Z will do one of two things. It will either stay equal to or below 2 forever, or it will eventually surpass two. Once the magnitude of Z surpasses 2, it will increase forever. In the first case, where the magnitude of Z stays small, the number we are testing is part of the Mandelbrot Set. If the magnitude of Z eventually surpasses 2, the number is not part of the Mandelbrot set, and is not plotted
Graph of Magnitude of Z
As we test many complex numbers we can graph the ones that are part of the Mandelbrot set on the complex number plane. When you plot tens of thousands of points, an image of the set will appear:
The plotted points will, even after trillions of iterations remain in the basic shape shown above. Color is added to the image such that points inside the set are colored and those outside are black, and this according to how many iterations were required before the magnitude of Z surpassed two. Not only do colors enhance the image aesthetically, they help to highlight parts of the Mandelbrot set that are too small to show up on the graph.
November 4th, 2006 at 6:12 pm
[…] The Mandelbrot Set may offer us a true and terrifying glimpse of “divine intelligence”. It may be a Rosetta of Creation. There is a place just beyond philosophy and outside of mathematics, where only the poet can go. His capacity for awe and sensitivity to beauty cast a light where others find only darkness. Emerson had those eyes. Perhaps you have them too. Can you conceive of the mind for which infinity itself is not a difficult abstraction but a territory of touristic familiarity? The Mandelbrot Set places in our hands a view of the universe which our own science can neither refute nor confirm, for we cannot see the edges of its truth. It suggests that all of the recurring patterns in nature we have so far discerned (the tree, the spiral, radial symmetry, the orbit, etc.) are the consciously duplicated themes of an intelligent and systematic Creation. God’s own fingerprints, if you will. Are these recurring themes merely signs of the Maker’s creative style, or are they the perfect designs which only a perfect designer could have achieved? Faith can accommodate both interpretations. The Mandelbrot Set is quite simply a “little universe”, which humans can inspect from the transcendent perspective of an omniscient god. As we can perceive and marvel over Mandelbrot, so must God see all he has made. And in the same way that there are no insignificant replicas among Mandelbrot’s infinite replications, we are as sovereign in all of existence as we have so often dared to believe. […]