[Marsha] I'd like to hear your explanation. I'd like to hear your explanation.
I'd like to hear your explanation. [Krimel] It is hard to explain the Mandelbrot set in plain text because it makes more sense to see it graphically. There are many good explanations available on the internet. I will comment on this one because it is really fairly easy to understand as its title implies: http://www.tonkoppens.nl/Tutorial01/test.html The equation the produces the pictures you have seen is this: Z = Z^2 + C Doesn't seem possible does it? However, the equation has two important features. First C is a complex number. It is a constant multiplied times "i" which is defined as the square root of -1. This makes it an imaginary number which is a mathematical term for negative square roots. Imaginary numbers were invented because you can't have negative square roots. For example with -1 * -1 = 1. The imaginary number i lets you get the square root of negative numbers. The Mandelbrot set is plotted on a graph that has the real number line as the X axis and the imaginary number line as the Y axis. The other interesting and unusual part of the equation is that it is iterative. This means that you put in a number. Solve the equation. Put the answer back into the equation and solve it again and again and again. What this means is that the equation continues across time. It is dynamic in this sense unlike more familiar equations the produce static answers and then we are done with them. When we put a number into this equation and start solving it over and over again three things can happen. 1. It can yield a 0 and nothing else happens. 2. It will quickly shoot off into infinity. 3. It will oscillate producing answers that vary within a fairly narrow range forever. The answers that oscillate are the points of the set. The way we get the pretty pictures is to take an area on the graph of the XY [Real/Imaginary) axis and plug each point on the graph into the equation, let it run and see what happens. Or think of it this way. Your computer screen is essential a piece of graph paper. It is covered with little dots. It works because the CPU and programs tell each dot on your screen what color to be and how bright to shine. Let's say you are using 800x600 resolution. That means that there are 600 dots on each line across your screen and 800 lines up and down. Thus there are 480,000 points on your screen. To draw a picture from the Mandelbrot set you assign your screen to a particular range of values, creating a kind of window or microscope of the whole set or part of it. Then starting with the first dot in the upper left hand corner you take that number, plug it into our equation and let it run. To make this simple let's say that if the equation produces infinity we color the dot white and if it produces zero we color it black. The Mandelbrot set only occurs in a relatively small area between (0,0) and (2,2) on the number lines. If you plot that area as described you will see the overall set. While this is a fairly small area, remember between any two points on a line there are an infinite number of points. So like using a microscope you can zoom in with infinite precision on the points of the line separating black and white. A couple of notes here. First it should be obvious that while the procedure itself is simple it was impossible to do without computers. You would have to solve that small equation an indefinite number of times for each of the half a million points on a fairly low resolution screen or piece of graph paper. It was conceptually simple but technically impossible. Ok here's how they get all the pretty colors. Some numbers when plugged into the equation reach infinity in one or two iterations; some after several hundred, some after several thousand. The closer you get to the edge of the line the more iterations are needed. So the colors are assigned based on the number of iterations needed to make a decision. In the programs you can get to draw and play with the set you can specify the number of iterations to cap and assign a color. If you told the program to run up to 10 million iterations for each point, it could do it but it would take a really long time. Another limiting factor is the zoom level. You could specify a range between 1.0000000000000000000000000000000000000000000000000000000001 and 1.0000000000000000000000000000000000000000000000000000000002 but this is really hard for your computer to work with for reasons your either already know or probably don't want to know. The point is, the more decimal places you add, the longer it takes to draw the picture. Here is a pretty good web based Mandelbrot explorer. http://users.erols.com/ziring/mandel_applet.html Pick the size images you want to see, small, medium or large. The program will pop-up in a window. It starts with an overview of the whole set. You can use your mouse to draw over a range of point you want to zoom in on and it will draw that range for you. On the top you can see the range of points you are looking at. Ok, that is phase one. There are much better explanations out there than I can give. If this actually interests anyone or any of you are already familiar with this we can talk more about how this relates to the MoQ. Some of that should be obvious. Equations that iterate and persist across time. The dynamic quality of the process itself and the results that become static by flying into infinity or sinking to zero. The set itself that exists on the cutting edge as a dynamic tension between the static alternatives. The self similarity in complexity across levels of scale. The organic Quality of the images. Clear distinctions between polar opposites with fuzzy intricate beautiful edges between them. Outcomes that are purely deterministic but unknowable until we look at them. Oh yeah, and as Case would say, "Zoom in, Zoom out, Refocus." Moq_Discuss mailing list Listinfo, Unsubscribing etc. http://lists.moqtalk.org/listinfo.cgi/moq_discuss-moqtalk.org Archives: http://lists.moqtalk.org/pipermail/moq_discuss-moqtalk.org/ http://moq.org.uk/pipermail/moq_discuss_archive/
