Benoit Mandelbrot coined the word “fractal” to describe his new object and those like it. He argued that the edge of the set was more than a line (one dimension) and less than a plane (two dimensions). He claimed it had a dimension somewhere between the two, a fractional dimension. #fractegrity #fractalsalltheway #asabovesobelow #fractalart
The Mandelbrot and Julia fractals are drawn in the complex plane (real numbers along the x-axis, imaginary numbers along the y-axis). The value for each coordinate is determined by iterating a mathematical function over and over. There are two possible outcomes for the series of calculations. The answers may diverge, and grow farther and farther away from the origin. Otherwise, the values of the answers may stay near the origin forever. Those points whose series stay near the origin forever are inside the set. Those points whose series diverge are outside the set. To make fractal images more elablorate and interesting, color is added to them. Rather than simply plotting a white point if it escapes, the point is assigned a color relative to how quickly it escaped.
Points are either inside or outside of the Mandelbrot set. It can be shown that if any answer from the calculations is greater than 2, then all successive values will continue to increase. Thus, we can assign a value for each beginning point of calculation equal to the number of iterations it takes that point’s series to exceed the magnitude 2 limit. Points whose series exceed the magnitude 2 limit have a finite value and are outside the set. By convention, points inside the set are colored black; points outside the set are colored according to how quickly they escaped the magnitude 2 limit. This is what provides the patterns of color bands.
To avoid calculating to infinity, an iteration limit is set so that a stopping point for the calculation occurs. Points whose series stay within the magnitude 2 limit up to the iteration limit are colored black. The higher the iteration limit, the more accurately the black pixels represent the actual set.
The black pixels are all connected, similar to the network of nerves of blood vessel in the human body. One can zoom in to no end. Maybe another way of saying this is that the Mandelbrot set contains an infinite potential within a finite space. The complex plane must be aptly named.
The basic technique of these fractals can actually be explained without resorting to confusing mathematical equations and jargon. First, every point on the computer screen is given a unique number. Now take that number and stick it into a formula; you’ll get a result from the formula. Take that result and stick it back into the formula. Keep doing this and watch what happens to the numbers you get. The boundary between numbers that explode and numbers that home in is complicated and twisted – it is the shape of the fractal.
Fractals are, to me, an access to understanding the relationship of chaos to order. With fractals, what appears to be chaos at one scale of magnification can appear to be organized in some recognizable order at another scale of magnification. Alan Watts gave the example of a rope; the observing of a rope shows an organized series of fibers aligned in a particular way. Ifone were to look under a microscope, the appearance of the fibers would seem quite chaotic. Maybe our perception of order and chaos has more to do with how close we are to a situation, and from different perspectives one could see order in the chaos and chaos in the order.
One of the aspects I love about fractals is the feeling of “play” around them. The complex math involved includes an “imaginary” number. An imaginary number is a number that gives a negative result when squared. The name “imaginary number” was originally coined in the 17th century as a derogatory term as such numbers were regarded by some as fictitious or useless, but today they have essential, concrete applications in a variety of scientific and related areas. The rise of computers opened up the possibility of playing more fully with mathematics. With fractals, the calculations involved are repetitive, boring and number in the millions. To produce the Mandelbrot Set on a single screen takes more than 6,000,000 calculations,even more if the iteration limit is set high to increase detail. No human could endure the boredom, but a computer will. Computers are particularly good at mindless repetition. The computer is our telescope, our microscope and our art gallery.
The black pixels are all connected, similar to the network of nerves of blood vessel in the human body. One can zoom in to no end, and still the black pixels of the Mandelbrot set are connected. This reminds me of the term Ubuntu. Archbishop Desmond Tutu explains Ubuntu: “One of the sayings in our country is Ubuntu – the essence of being human. Ubuntu speaks particularly about the fact that you can’t exist as a human being in isolation. It speaks about our interconnectedness. You can’t be human all by yourself, and when you have this quality – Ubuntu – you are known for your generosity. We think of ourselves far too frequently as just individuals, separated from one another, whereas you are connected and what you do affects the whole world. When you do well, it spreads out; it is for the whole of humanity.”