Krimel,
I'm really sorry to have put you to so much trouble. Although it was interesting, your explanation was a bit over my head and not what I was expecting. I imagined you were going to relate the Mandelbrot whatevers to randomness within evolution or the Tao or something like that. But thanks anyway, I had no idea of what Mandelbrot sets were about, and now I have some idea. Many thanks, Marsha At 11:06 AM 4/28/2008, you wrote: >[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/ Shoot for the moon. Even if you miss, you'll land among the stars... 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