\grayb1000\cb1\pard\tx960\tx1920\tx2880\tx3840\tx4800\tx5760\tx6720\tx7680\tx8640\tx9600\f0\b\i0\ulnone\fs28\gray0\fc2\cf2 GENERAL
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This application computes the Mandelbrot Set in a separate thread using distributed objects for communication and displays the result on the screen.\
While this application can generate beautiful images on a monochrome display, it is at its best when run on a color system.\
\b THEORY
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The Mandelbrot Set can be defined in many ways; e.g. in terms of Julia Sets, iterative maps, chaotic electrostatics, and so on. A purely algebraic description is the following. Let c be a point in the complex plane, and set z = 0. Then iterate the non-linear relation:\
z := z*z + c\
until | z |, the magnitude of z, exceeds 2. For some extremely
some would say unthinkably
pathological set of initial points c the iteration goes on forever, with | z | < 2 always. This set is the Mandelbrot Set. Since computers do not have infinite speed, we approximate by finding those points c for which the iteration goes at least Depth iterations, where Depth is some cut-off parameter. As most Mandelbrot programs do, this one colors each point c according to how many iterations arise from that point. You can assign colors as a function of iteration count using the Color Map (q.v.). By default, points within the Mandelbrot set are colored black.\
\b PARAMETERS
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Parameters are set via the five TextFields at bottom center of the window. The meanings of the parameters are\
X Center, Y Center: the coordinates in the complex plane of the current center of the display.\
X Scale, Y Scale: the side of the display square. Very small scale means very high magnification. NOTE that allowing the DSP to calculate out of bounds regions will cause integer overflow effects, such as replicated patterns. For sufficiently small values of X Scale and Y Scale, you will see the effects of limited precision of the DSP (24 bits).\
Depth: the maximum Mandelbrot iteration per pixel.\
NAVIGATING THROUGH THE MANDELBROT SET\
When the program is first started, the parameters are set up to display the entire Mandelbrot set. Press the RUN button to compute and display the entire set. \
There will always be worlds within the Mandelbrot set that have never been visited. You can go to a new place on the current display by dragging out a square with the mouse. This new display box is active when you next hit the Run button. Remember that if you go too deep you will see the effects of the limited precision of the processor. \
The Places menu defines a number of
pretty
places within the Mandelbrot set which you can go visit any time you wish. Simply click on the desired name and hit the Run button to see it.\
\b THE COLOR MAP
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The Mandelbrot iteration algorithm terminates when | z | exceeds 2 or after Depth iterations, whichever is less. The Color Map lets you assign a color for each point according to the number of iterations before the algorithm terminated. \
The Color Map is implemented as a sum of Cosines, one for each of Red, Green, and Blue. For each color component, you have the following controls:\
\b Frequency
\b0 : This controls the number of cycles for that color between 0 and Depth. A frequency of 1.0 will complete one full light-dark-light cycle between 0 and Depth.\
\b Phase
\b0 : This controls the initial phase of the cosine for that color. A phase of 0.0 means that color will start out fully on (this is a cosine). A phase of 0.5 means the color will start out fully off (as in a sine wave).\
\b Contrast
\b0 : At a value of .5, the color will alternate evenly between light and dark. As you increase the contrast, the color will tends towards darkness with sharp bands of light. At a contrast of 1.0, the sharp bands of light will become infinitely narrow, and the entire color will be dark. The situation is reversed as you decrease the contrast below .5: the color will tend towards lightness with sharp bands of darkness.\
\b Colors
\b0 menu item defines some colors for you. But don't be fooled
the visual effect of any given color map depends highly on what part of the Mandelbrot set you are visiting and the setting of the Depth parameter. For example, try the
\b Storms
\b0 color map while visiting
\b Home
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\b Black Hole
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\b HINTS ON USING THE COLOR MAP:
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A color with an integer frequency and a phase of .5 will always start and end in darkness (corresponding to iterations ranging from 0 to Depth). This tends to look good, especially if all three colors start and end in darkness.\
A color with a frequency of N+.5 (for integer N) and a phase of 0.0 will start in lightness and end in darkness. This is another good starting point.\
You may wish to turn off two of the colors while you play with a third. To turn off a color, you can set its Frequency to 0.0 and phase to 0.5. A simpler (though less intuitive) approach is to set its Contrast to 1.0.\
If all three color components are identical, you will get gray scales. This may be especially useful if you are working on a monochrome display. See the predefined
Zebra
color map for an example of this.\
\b REMOTE EXECUTION:
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By default, the calculations will take place on the local host. You can execute the calculations remotely by specifying a remote host in the "Host" field. You might want to do this if you have a faster machine on your network than your local host. Note that you must have a Mandelbrot server already running on the remote host for this to work. The Mandelbrot application will not run it for you. You must run it by hand either by running the Mandelbrot application on the remote host or if you have Portable Distributed objects running the separate program "MandelServer" on the remote host instead.\
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