Libris Britannia 4

home *** CD-ROM | disk | FTP | other *** search

/ Libris Britannia 4 / science library(b).zip / science library(b) / GRAPHICS / GRAPHUTL / 1344.ZIP / MANDLE.ARC / MSET12.DOC < prev next >

Wrap

Text File | 1988-10-05 | 18.2 KB | 351 lines

MANUAL VERSION 1.2 MANDELBROT SET GRAPHICS JULIA SET GRAPHICS INTRODUCTION: Mandlebrot Set Graphics Version 1.2(MSET12) draws graphic representations of a function of a complex variable which were first introduced by Benoit Mandelbrot. This is a shareware program and you are encouraged to copy it and pass it on. It is not necessary to know the details of complex variables or their algebra to use or enjoy the fascinating behavior of the Mandelbrot Set. It is helpful, however, to understand what is being plotted to navigate and explore the nearly limitless variety of designs to be found in the 'universe' of the Mandelbrot Set. If you have looked at the directory, you will have noticed that there is also a file titled JSET12. This is a program which will generate graphics based on the Julia Set; a companion to the Mandelbrot Set. This set is named after Gaston Julia, one of the pioneers in the field of mathematical functions which exhibit chaotic behavior. In fact Mandelbrot was studying Julia Sets when he discovered the Mandelbrot Set. This is a more complicated set to understand than the Mandelbrot Set and will be treated later in this documentation. A Short Mathematics Lesson: This is not mandatory reading, but only for those interested in the details of the calculations the computer is making. Think of your computer screen as an X Y plane. To make it a complex plane, we assign a vector Z to any point on the plane with the equation Z = X + iY where i in the square root (sqrt) of -1. Of course, there is no sqrt -1 and by conversion i is called 'imaginary'. Using ordinary algebra 2 2 2 2 Z = (X+iY) = X - Y +2iXY. All the algebra of complex numbers follows the algebra of ordinary numbers where the 'real' terms are collected together and treated as one group and the 'imaginary' terms are collected and treated as another group. This makes the 'real' and 'imaginary' axes independent or orthogonal. Note i * i = -1. The magnitude of Z is defined as the length of the X Y vector 2 2 magnitude(Z) = sqrt(X + Y ). The points in the Mandelbrot Set are calculated in the following way. Define the recursive formula 2 Z = Z + C n+1 n - 1 - where Z is defined as 0 on the first pass and C is a point in the complex plane. This formula is calculated over and over again until its magnitude exceeds 2. This formula has the curious property that as long as the magnitude of the complex vector remains less than 2 the magnitude of the next recursion may also be less than 2. However, once the magnitude exceeds 2 it will never again be less than 2 and reach very large magnitudes over the next few recursions. The fascinating thing about this recursion is that adjacent points which have almost identical coordinates have greatly different magnitude stability, i.e. one point may quickly 'blow up' and the other not. Plotting The Points: Each point on your computer screen represents a point which has a stability associated with it as defined above. Each point on your computer screen in the graphics mode has a pixel associated with it. The MSET12 program calculates the stability of each point on the screen and assigns a pixel color and brightness as a function of the stability of the point. Some points are infinitely stable, e.g. X = Y = 0. Others blow up immediately, e.g. (X*X+Y*Y) > 4. Between these extremes there are relatively sharp boundaries, and it is at these boundaries that the beauty of the Mandelbrot Set lies. The MSET12 program allows the user to set the location of the central pixel on the screen, the range of values of the pixels calculated over the screen's surface, and the palette of pixel colors and brightness assigned to the pixels as a function of the point's stability. RUNNING MSET12: There are four parameters which are used to control the MSET12 program; - the type of graphic display, - the number of iterations of the recursion executed before the program deems the point 'stable' and moves to the next pixel, - and the palette used to determine the pixel values (colors), - the location of the center pixel on the screen, - the range of pixel values covered by the X and Y axes, The program is started by typing MSET12 and a carriage return. You will then be asked which graphics card is installed in your computer by using the function keys. Next, you can choose the number of iterations desired. Then you select the palette desired. How the points appear on the screen, however, depends on the palette chosen. There are five palettes available. Each palette will change the appearance the array calculated. As a rule, low palette numbers are better for the larger ranges. For example, palette 1 or 2 will produce more attractive displays with ranges larger than 0.1 while palettes 3 and 4 will be better for ranges less than 0.1. This is not a hard rule and they all will produce interesting displays provided the location and range define an interesting - 2 - area. And then there is the matter of aesthetics. The user is encouraged to experiment. Palette 5 is recommended for use with the Hercules monochrome adapter card. Finally, the program will then prompt you for the X and Y values of the center pixel and the range of the pixel values for the X axis. For example, if the X value entered is 0 and the range value entered is 1, the range of the X axis will be from -0.5 to +0.5. The range of the Y axis will be automatically set so that the image is not distorted. The smaller the range, the greater the magnification. MSET12 now has all the parameters needed to calculate the stability of a large array of points on the computer screen. It takes a long time to complete a Mandelbrot Set. Just how long depends on the display selected, the computer clock rate, the microprocessor (8088 or 80286), whether there is a coprocessor present, and the nature of the area selected (stable areas take longer to calculate than unstable areas). The CGA display requires 64,000points, EGA requires 128,000 points, extended EGA requires 243,200 points, and the Hercules requires 250,560 points. The coarse Hercules mode plots only 62,000 large pixels and is four times faster than the normal Hercules mode. If you have an EGA display, chances are that it is CGA compatible so you might wish to search with the CGA mode until you have found an interesting area and then use the EGA mode to display the final product. Rough approximations are that a 10mHz AT with an 80287 coprocessor in the CGA mode and in an unstable area can complete an image in 5 minutes where a Hercules monochrome graphics equipped 4.7mHz XT in the Hercules mode in a stable area can take over 18 hours. Patience is required, but the results are often worth it. Extreme patience is required if you do not have a coprocessor. If you run out of patience, you can stop the process by typing 'S' for stop and then any key to clear the screen. When the image is complete, touching any key will clear the screen and return to the command line. Tips A good way to begin is to view the entire region of interest in the Mandelbrot Set. Enter the location X = -.5, Y = 0 and a Range = 3. The dark areas near the center of the screen are very stable areas. They are sometimes called 'lagoons'. The borders to the lagoons have the most intricate patterns and are the most interesting to investigate. The dark areas near the edges of the screen are very unstable and of no interest. This is a slow image to complete because there are many stable points in the image. Images with many stable points proceed much faster if a low number of iterations are selected. Next, it is instructional to locate an interesting area and to zoom in on it with successively smaller ranges (larger magnifications). One of the most fascinating characteristics of the Mandelbrot Set is that no matter how high the magnification of the Set, the detail is never lost. Note also that the patterns repeat over and over again. The characteristic of repetitive patterns at finer and finer detail is called 'fractal', a word coined by Benoit Mandelbrot. The MSET12 uses single precision - 3 - floating point in its calculations for speed. This is the only limit to the detail in the patterns and is not a problem to even the serious user. Try the following series: X Y Range Palette -0.8 0.25 .5 1 -0.75 0.15 .1 2 -0.75 0.113 .02 3 Note that the magnification for each of the series has been increased by a factor of 5 over the previous one. Now that you have the idea, you are on your own. Continue this series, or try something new. It is very useful to keep a notebook or log of the points you have tried or are trying. Believe me, having your computer grinding away for hours on a Set which proves to be beautiful and not remembering the location and range is very frustrating! JULIA SETS: When you run the JSET12 program you will notice that the menus are almost the same as the MSET12 program. That is because the equation solved to generate Julia Set is the same as the equation used for the Mandelbrot. Much of what we have said about the Mandelbrot Set could also be said of the Julia Set. Another Mathematics Lesson The mathematics lesson above was not mandatory. If you wish to master the use of the JSET12 program, however, it is necessary to have a basic understanding of the math involved. Nothing very deep, mind you, but at least a little understanding. If you have skipped the mathematics tutorial above, read it now before you proceed. To generate the Mandelbrot Set, we set Z = 0 on the initial pass through the equation 0 2 Z = Z + C n+1 n and expanded the equation in the complex plane of C. Notice that this equation has two complex terms; C and Z. Each of these terms represents a two dimensional plane, and the planes are not in the same space. In fact, it is not possible for these planes to have more than one point in common. The Julia Set is generated by setting the value of C as a constant for a whole set and incrementing the real and imaginary parts of Z to define the pixel 0 - 4 - values across the computer screen. The Mandelbrot Set expands the equation in the C plane and the Julia Set expands the same equation in the Z plane. The characteristics of the Julia Set are important 0 to understand if you are to control the images produced. If C values are picked which are in a stable region of the Mandelbrot Set, Then the Julia Set will appear as a connected figure with a boundary which may appear simple. As the value of C is moved near edge of a stable region in the Mandelbrot Set, the boundary of the Julia Set exhibits a more complicated structure. Once the value of C crosses the Mandelbrot boundary into an unstable region, the Julia Set explodes into a set of of small regions which quickly evaporate as the value of C moves further into the unstable region. Because the appearance of the Julia Set depends on the value of C chosen, there are as many Julia Sets as there are values of C. There are many Julia Sets, there is only one Mandelbrot Set. RUNNING JSET12: It is assumed that the reader has already read the MSET12 instructions and run the program. JSET12 is very similar. The major difference is seen when the values controlling the calculation of the Set are entered. You are first asked to enter the complex values Z (X and Y). It is possible to position the center of the JSET12 image on the screen by the selection of Z values. You will then enter the complex values for C. This parameter must be entered with high precision if it is near the boundary of the Mandelbrot Set because it is very sensitive in this region. The boundary of the Julia Set shows the same fractal character as the Mandelbrot Set, even under high magnification. Finally you will enter the range which the value of Z will be varied. The magnitude of the range controls the magnification of the image. Small values lead to a high magnification. Tips Let us start by viewing the Julia Set associated with the Mandelbrot example in 'Tips' above. We now must enter four coordinates as well as the range. When the computer is started on a Julia set using a Hercules display, it may take several minutes for the image to begin to show. Notice how many of the shapes resemble each other in the true fractal fashion. They may remind you of elephants. The second example is very near the edge of the Mandelbrot Set and has begun to evaporate and break up into a set of separate pinwheels. Try some other changes and see what happens. - 5 - X Y real C imag C Range Palette 0 0 -.74543 .11301 3 5 0 0 -.76 .11301 3 5 The palettes used when displaying JSET12 can alter the appearance of the set just as it is in MSET12. The need to change palettes as you go to high magnification is not as critical because you will usually work with low magnification so that you can display a full set to enjoy its symmetry. Palette 5 is always a good choice whether displaying Hercules, CGA, or EGA. The other palettes, however, should not be ignored because they sometimes make pleasing designs. Let's try another series: X Y real C imag C Range Palette 0 0 .25 0 3 choose any 0 0 .255 0 3 but keep it 0 0 .26 0 3 the same for 0 0 .26 .001 3 the whole 0 0 .26 .002 3 series Notice how sensitive the image is to these small changes in the parameters. Now let's zoom in on the last set with 6 times magnification. The center of the screen is repositioned with changes in the X and Y values. X Y real C imag C Range Palette .25 .88 .26 .002 .5 any The beauty of the set's symmetry is lost, but notice how the fine detail remains in true fractal fashion. Julia Sets offer more variety than Mandelbrot Sets and include aesthetically pleasing symmetry. If you would like to see the area of the Mandelbrot Set you are in, run MSET12 and enter: X Y Range Palette .27 0 .05 3 The area is that in the center left of the screen. Now you are on your own to do some exploring. - 6 - CONCLUSION: MSET12 and JSET12 are (c) copyrighted by Image Laboratories. They are for private use only and may not be modified or sold commercially without written permission. These programs may be copied freely for private use. If you like the MSET12 and JSET12 programs, support them by sending $35 to: Image Laboratories 67 Old Stage Road Chelmsford, MA 01824 In return, you will receive the latest version of the MSET and JSET programs which will allow you to save and display the Mandelbrot and Julia Sets you have generated and to color these images with your own palettes instead of the ones provided with the program. This allows you to save your best images for easy, fast display at a future time and for you to apply your own artistic interpretation to them.