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Mandelbrot/Julia Set Generator Operating and Reference Manual Version 5.0 Installation The Mandelbrot/Julia Set Generator program requires an IBM compatible computer with at least 512K of memory, a VGA display and a Microsoft compatible mouse. The installation process is quite easy. First, make a backup copy of your Mandelbrot/Julia Set Generator program disk. If necessary, consult your PC-DOS/MS-DOS manual for a description of the Diskcopy command. Save the original program disk in a safe place and use the copy as the working program disk. Second, while it is possible to use the Mandelbrot/Julia Set Generator on a floppy disk system, a hard disk system is a necessity if you wish to store a number of image files. To install the Mandelbrot/Julia Set Generator on a hard disk use the following steps: 1. Insert the floppy disk in your computer in drive A. (or B if necessary) 2. Type A: (or B:) 3. Type INSTALL A C Any hard drive letters can be used, for example INSTALL B D will install the program from floppy drive B to hard drive D. The installation will create a directory called MAND5 on your hard drive and you will need to enter a CD\MAND5 to change to the Mandelbrot/Julia Set Generator directory before running the program. Once installed just type MAN to start the program. All the files for Mandelbrot/Julia Set Generator need to be in the same directory for the program to operate successfully. Quick Start for Impatient New Users Type MAN to start the program. After the mouse cursor appears click it on the Load Image button at the left. When the window appears with the list of image file names, simply clicking on one of them will display the image using the current color mask. If the image file contains a specific color mask filename it will be automatically loaded prior to displaying the image. Most commands can be interrupted by a simple mouse click. The zoom window feature is started by double clicking on the displayed image. Once the zoom window appears, with its crossed center lines, it can be moved by holding down the left mouse button, while the cursor is inside the zoom window, and positioning the window. The zoom window size can be increased or reduced by holding down the left mouse button and moving the mouse cursor horizontally while it is outside the zoom window. Once correctly positioned the mouse cursor should be clicked on the right gray panel, which will store the changed dimensions. The zoom window can be abandoned by clicking the mouse cursor on the left gray panel. Be careful not to drag the mouse cursor onto the gray panels while resizing. The Set Values button should be clicked on next, and the image file name changed. If this is not done the original image file will be erased. Clicking on the Make Image button will start the generation of the new zoomed image. The Command Buttons and Their Function Set Values The Set Values command allows the user to set the initial parameters that will be used by the Mandelbrot/Julia Set Generator to begin generating a new image. These values are also available for inspection when an image has been displayed. The values and their range are: Item Range ----------------------------------------------------- X center value -10 to 10 Y center value -10 to 10 Magnification >0 A value (if a Julia image) -10 to 10 B value (if a Julia image) -10 to 10 Dwell 1 to 8191 Image width in pixels 10 to 4800 Image type [M J] M or J Full/Partial image [F P] F or P Default color mask file xxxxxxxx.MSK Display type [0 1] For future use 256 color palette number For future use Image file name xxxxxxxx.MAN or xxxxxxxx.MAR To change a value simply click inside the rectangle where the value is displayed and then key in a new value or file name. File name extensions must be .MSK for color masks, .MAN for regular images and .MAR for those that are recursive. The A and B values are only displayed with Julia images. If the recursive image generator is used the image width must be a member of the 2^n set, ie. 16, 32, 64, 128 etc. The program maintains the Full/Partial image status and these values cannot be changed by the user. Color Masks When the Color Masks command is chosen a popup window presents the four options: Create/Display color mask Select color mask Select palette QUIT Clicking on the Create/Display color mask option allows the user to create, edit and save color masks. Clicking on the Color mask file name box allows you to type in a file name. The file name must have the extension .MSK or you will not be able to select it later. Ranges of dwell values should be typed into the squares on the left. Just click on the square, type in a dwell value and <Enter>. The colors are selected by clicking on the desired color of the color wheel in the upper right and then clicking on the odd and even boxes at the right. The selected color is displayed between the circular menu and the color wheel. If the first line of the color mask reads: 0 9 [blue box] [white box] then dwell values from 0 to 9 will be colored blue if odd and white if even. If a solid color is desired the color boxes should be filled with the same color. The end of a color mask should be designated with a negative value entered into the first dwell box. The default color mask on startup is M1.MSK and its values are: Dwell Range Odd color Even color ----------------------------------------------- 0 9 [blue box] [white box] 10 19 [red box] [red box] 20 510 [yellow box] [yellow box] 511 511 [black box] [black box] -1 In the case of M1.MSK, any dwell values larger than 511 will be colored black (color 0 in the default palette). The circular menu at upper left has four options. Clicking on the up or down arrow jumps to the previous or next 16 color mask entries. A total of 256 entries can be placed in one color mask. The SAVE option stores the color mask currently displayed under the name specified in the Color mask file name box. The new color mask becomes the currently selected color mask. Clicking on the Select color mask option presents the user with a large window and the names of the color masks that have been stored. Clicking on a color mask name will select and load that color mask. It then can be viewed by selecting the Create/Display color mask option. Clicking on the Select palette option opens a window that displays the current VGA color palette of 16 colors. Clicking on the Default box will load the default VGA color palette. Clicking on the arrows will select other prestored color palettes, up to number 57. The QUIT option returns the user to the main menu. Make Image Selecting the Make Image command generates a Mandelbrot or Julia image based upon the parameters entered in the Set Values command. A warning is issued before the generation begins to allow the user to change the file name, as any existing file of this name will be erased. A very simple way to generate images is first to use Load Image to display a previously generated image. Double clicking on the image will produce a zoom window overlaid on the display. Clicking and holding down the left mouse button allows the zoom window to be dragged about the image to an interesting portion of the image. The zoom window can be resized by dragging the mouse pointer to the left and right outside the zoom window. Once the zoom window has been positioned and sized, clicking on the gray panel at right will automatically store the new zoomed values into the Set Values area. Be careful not to drag the mouse cursor onto the gray panels while resizing. The user will probably wish to enter a new image file name using the Set Values command (this will prevent the original image file from being erased), and then generate a new image of the area defined by the zoom window with the Make Image command. While the zoom window is present the procedure can be cancelled by clicking on the right gray panel around the command buttons. The zoom window will only work on images 480 pixels wide, or less. Load Image The Load Image command presents the user with a list of image file names that have been produced with the .MAN extension. Clicking on a file name will display the image with the current color mask if the selected image has no default color mask file name. A brief double tone is sounded if there is no default color mask file name. If a color mask name was included when the image was generated, this color mask will be loaded before the image is displayed. Partially generated images will automatically continue generation when displayed with this command. Once an image is displayed double clicking on the image will produce a zoom window as described under the Make Image command. Make R Image The Make R Image command functions similarly to the Make Image command except a recursive procedure is used in place of the normal line by line generation. The image file should be given the .MAR extension so that it will be properly handled when using the Load R Image command. In some cases this recursive procedure will generate images faster that the normal method. Partially generated images cannot be displayed with generation automatically continuing as is the case with the normal Load Image command. Image files are generally larger with the recursive procedure. Load R Image The Load R Image command displays a recursive image previously generated with a .MAR extension in the file name. A list of such files is presented and the selected image is clicked on. Partially generated images will not be automatically continued as with the Load Image command. 3-D Image The 3-D Image command displays an image generated with the Make Image command in a pseudo 3-D style. The display algorithm is a simple one, but very slow. VGA displays have limitations when displaying 3-D Mandelbrot images. Best results occur with color masks that contain multiple colors and have the dwell ranges broken into many small steps. Large values for the maximum dwell may result in the top of the image being lost. Partially generated images will not be automatically continued as with the Load Image command. Plot Dwell The Plot Dwell command reads all the dwell values of an image stored with the .MAN extension and sums them. The sums are then plotted with the current color mask used for each dwell value plotted. Only dwell values of 2,400 or less will be plotted. These plots give an indication of how many points in the image have the various dwell values and can be useful in constructing a color mask that will display the image to best advantage. Make PCX The Make PCX command allows the user to select an image file stored with the .MAN extension and create a PCX image file. A 16 color PCX file using the default VGA color palette can be chosen or several 256 color PCX formats are available. Click on one of the small boxes to select what type of PCX file you desire. The color sequence of each of the 256 color formats is displayed. The first example has magenta blending into red for dwell values from 0 to 64, from red to yellow for dwells from 64 to 128, etc. The PCX image file format allows users to import Mandelbrot and Julia image files into other software such as desktop publishing programs and paint programs. PCX files can also be used for Windows wallpaper. Print Image The Print Image command presents the user with nine different printer types that are supported, or the command can be quit. Epson 9 pin Epson 24 pin IBM 9 pin IBM 24 pin LaserJet DeskJet 500 B/W Epson DM Color DeskJet 500 C PaintJet QUIT The 9 pin printers will output at 120x144 dpi, the 24 pin at 180x180 dpi, the Laserjet, Deskjet 500 B/W and Color at 150x150 dpi and the Epson DM Color and Paintjet at 90x90 dpi. Be patient, the print drivers do take time in exchange for attractive output. Color is the slowest. Black and white images will be dithered. QUIT returns the user to the main menu. The Print Image command is basically for quick hardcopy. If you wish to print museum quality prints try a DeskJet 500 series printer. Create your image and then make a PCX file using one of the 256 color formats. Next, load this PCX file into the Paintbrush program that comes with Windows. This is usually found in the Accessories window. Next print the image from Paintbrush. You will need a 256 color display to do this and the Windows print driver that came with the DeskJet printer. Most IBM PC's and clones being sold today come with a 256 color display. The DeskJet will print your image with a superb color balance at just under 100 dpi. Try an image width of about 750 pixels to fill out the 8-1/2 inch page. I've used this method with a Hewlett-Packard DesignJet 650C and 36 inch wide paper with images 3300 pixels wide to produce colored output that is truly magnificent. If your printer is not supported this method can also be used to print your images. The only thing you will need is the Windows print driver that came with your printer. Remember, images my be created which are much wider than your screen. The upper left corner of your image will be the only area visible. To see the entire image, create a 256 color PCX file and use any paint program that can read 256 color images. These can be very attractive. Help File The Help File command displays the file you are currently reading. Clicking on the arrows to the right displays the next or previous page. Quit MAND51 The Quit MAND51 command returns the user to the DOS prompt. Image File Structure Each image file created by the Mandelbrot/Julia Set Generator begins with a 150 byte header. Byte Item Size Description ------------------------------------------------------ 0 x 8 byte double x center point 8 y 8 byte double y center point 16 mag 8 byte double magnification 24 a 8 byte double a for Julia sets 32 b 8 byte double b for Julia sets 40 maxdwell unsigned int maximum dwell 42 width unsigned int image width in pixels 44 mj[2] char M/J, image type 46 partial[2] char F/P, full/partial 48 mask[32] char color mask file name 80 display integer display (not used) 82 pal integer palette (not used) 84 name[50] char signature 134 fill[16] char filler All char strings are terminated with a hexadecimal 00 byte. The dwell data follows the header. It should be noted that this is not a true image file, rather the dwell values themselves are stored. This allows users to color the image with a large variety of color masks. Storing an image file might be simpler but for every different color mask a new image file would have to be created. The dwell data is stored as a series of two byte unsigned integers. Each unsigned integer contains the dwell value and a run length corresponding to a string of identical dwell values. The number of bits required to hold the maximum dwell is first obtained. If the maximum dwell is 511, then 9 bits are required, 1023 would require 10 bits, etc. Using 1023 for the maximum dwell as an example, the right most 10 bits of the 16 bit integer represents the dwell value and the 6 left most bits contain the run length. As a run length of zero is not very useful, this value is always incremented by one such that a run length of zero equals 1, 1 is 2, etc. Given a maximum dwell of 1023 the following 16 bit unsigned integer represents a dwell of 1000 and a run length of 32. 011111 1111101000 7FE8 hex When an image is being displayed and the unsigned integer above is read, a line of 32 pixels will be drawn using the appropriate color from the active color mask for dwell value 1000. Each line of a display is encoded with no wraparound. This means that each line will end with the display of an encoded unsigned integer and no extra pixels of the same dwell will be added for the beginning of the next line even if there is room in the run length. It should be noted that the maximum run length that can be stored varies with the maximum dwell chosen. Files with a maximum dwell of 1023 will have a maximum run length of 64, those with maximum dwells of 8191 will only store 16. This does not limit a run length because if it exceeds the space available in a single unsigned integer it simply creates additional ones until the run of dwells has been stored. For this reason images with high maximum dwell values are often large in size. This method of file compression strikes a good balance between file size and speed when displaying an image. The Mathematics of the Mandelbrot Set The Mandelbrot set is computed by operating on a fairly simple equation that contains complex numbers of the form x + yi where i = sqrt(-1) The Mandelbrot equation is z <- z^2 + c where z = x + yi and c = a + bi substituting these values into z^2 + c we have (x + yi)^2 + a + bi x^2 + 2xyi - y^2 + a + bi separating the real and imaginary parts of z gives x <- x^2 - y^2 + a y <- 2xy + b To determine whether a point (a,b) in the complex plane is a member of the Mandelbrot set, the real and imaginary parts of the equation are iterated. The x and y values are first initialized to zero. The constants a and b, the point in the plane, are then substituted into the equations giving x <- a and y <- b for the first iteration. The two new values for x and y, along with the constants a and b, are now substituted into the equations again. This procedure (iteration) continues until the absolute value of x + yi > 2, ie. sqrt(x^2 + ^y2) > 2. For those cases where this value never exceeds 2, the maximum number of iterations is preset. A value of about 500 is usually adequate, although this value is raised to several thousand when smaller details at high magnification are examined. The number of times the equations are iterated before the value of sqrt(x^2 + y^2) > 2 is called the dwell. Those initial points (a,b) where the dwell is infinite, or for more practical purposes attains the preset maximum, are members of the Mandelbrot set. Another way to describe this is to say that for points within the Mandelbrot set, the sequence of points produced by this iteration procedure is bounded inside a circle of radius 2, where points outside the set are unbounded and continue to grow and escape the circle. The Mandelbrot set exists entirely within the area defined by -2 <= a <= 2 and -2 <= b <= 2 in the complex plane. A Mandelbrot image is produced by taking this area of the complex plane and dividing it into a array of 1200 x 1200 points. Each one of these points becomes the constant (a,b). The iteration procedure previously described is used on each of the 1.44 million points, coloring each point in the Mandelbrot set black and all others white. The algorithm is: maxcount <- 1000 for b <- 2 to -2 stepdown 1/300 for a <- -2 to 2 step 1/300 x <- 0 y <- 0 count <- 0 while sqrt(x^2 + y^2) < 2 and count < maxcount x <- x^2 - y^2 + a y <- 2*x*y + b count <- count + 1 end while if count = maxcount plot(a,b,BLACK) else plot(a,b,WHITE) end for a end for b While the algorithm is not that complex, the amount of computation is enormous. Depending on programming language and style, the inner loop has at least four multiplications and a square root. For a point in the Mandelbrot set this loop is executed 1000 times and there are over a million points to check! It is not surprising that the Mandelbrot set was not discovered until the age of computers. In the Mandelbrot/Julia Set Generator program some additional refinements are made to standardize the initial parameters used to generate a specific image. Instead of defining the range of (a,b) values used for an area, a center point and a magnification are specified. The center point is simply a chosen (a,b) value. The length of a side which encloses the area of interest is defined as side = 2/magnification The following values can now be defined a_minimum = a_center - side/2 b_maximum = b_center + side/2 gap = side/width where width is defined as the number of points that make up a side (or on a computer screen the number of pixels), and the gap being the distance between each point. The Mandelbrot set is an interesting image, a sort of cardioid with a spiked head attached at the left. The boundary of the set sprouts self similar buds of different sizes. Vastly more interesting images are forthcoming when we examine the boundary of the Mandelbrot set under higher magnification. To obtain higher magnifications we simply divide a smaller area into our array of points. For example, the area defined by the center point (-0.77,0.17) and magnification 20 is located in the upper valley between the head and the cardioid shaped body. If we continue with these magnifications, very different and interesting images can be produced by coloring the dwell values in specific ways. Along with coloring points in the Mandelbrot set black, we can assign different colors to other points based upon their dwell value. For example, we might assign yellow to dwell values in the range 400 to 499, red to 300 to 399, etc. When we do this a great deal more detail begins to appear in the boundary regions. This region of interest exists only in a narrow band just outside the Mandelbrot set. The skill one uses in choosing the various colors for differing dwell values is very important when attempting to produce an attractive image. The Mandelbrot/Julia Set Generator uses a file called a color mask to store the colors used in painting the various dwell values in an image. This technique allows many different coloring schemes for a single image. Consider the following color mask: Dwell Range Odd Color Even Color ---------------------------------------------- 0 9 blue white 10 19 red red 20 510 yellow yellow 511 511 black black -1 Dwell values from 0 to 9 will be colored blue if they are odd numbers and white if they are even. Values from 10 to 19 will be colored red, 20 to 510 yellow and 511 will be colored black. Choosing the maximum dwell value to be in the set 2n - 1 maximizes the file compression method the Mandelbrot/Julia Set Generator program uses. Generating Julia set images is a similar process. The point (a,b) is chosen from one of the interesting boundary areas of the Mandelbrot set. This value is held constant and the (x,y) value is initialized to the various points in the complex plane defined by -2 <= x <= 2 and -2 <= y <= 2 This would be a magnification of 0.5, actually the Julia image can often be enlarged slightly to fill the screen and magnifications from 0.6 to 0.9 are often used. The algorithm for generating a Julia set is maxcount <- 1000 a <- constant b <- constant for y <- 2 to -2 stepdown 1/300 for x <- -2 to 2 step 1/300 count <- 00 while sqrt(x^2 + y^2) < 2 and count < maxcount x <- x^2 - y^2 + a y <- 2*x*y + b count <- count + 1 end while if count = maxcount plot(a,b,BLACK) else plot(a,b,WHITE) end for x end for y Selected References Barnsley, Michael, Fractals Everywhere. San Diego, CA: Academic Press, 1988. Briggs, John and Peat, F. David Turbulent Mirror. New York: Harper & Row, 1989. Devaney, Robert L.Choas, Fractals, and Dynamics. Menlo Park, CA:Addison- Wesley, 1990. Devaney, Robert L. and Keen, Linda, Editors. Chaos and Fractals, The Mathematics Behind the Computer Graphics: Proceedings of Symposia in Applied Mathematics.Providence, RI: American Mathematical Society, 1989. Gleick, James Chaos, Making a New Science. New York: Viking Penguin, Inc., 1987. Mandelbrot, Benoit B. The Fractal Geometry of Nature. New York: W. H. Freeman and Co., 1983. Peitgen, Heinz-Otto and Richter, Peter H. The Beauty of Fractals, Images of Complex Dynamical Systems. Berlin: Springer-Verlag, 1986. Peitgen, Heinz-Otto and Saupe, Dietmar, Editors. The Science of Fractal Images. New York: Springer-Verlag, 1988. Pietgen, Heinz-Otto, Jurgens, Hartmut and Saupe, Dietmar Fractals for the Classroom, (Volumes I & II), New York: Springer-Verlag, 1992. (There is a single volume work entitled Chaos and Fractals, New Frontiers of Science, which is essentially the same work as the two volume set above.) Pickover, Clifford A. Computers Pattern Chaos and Beauty: Graphics from an Unseen World.New York: St. Martin's Press, 1990. Pickover, Clifford A. Computers and the Imagination: Visual Adventures Beyond the Edge. New York: St. Martin's Press, 1991. Pickover, Clifford A. Mazes for the Mind. New York: St. Martins Press, 1992. Schroeder, Manfred Fractals, Chaos, Power Laws, Minutes from an Infinite Paradise.New York: W.H. Freeman and Co., 1991. Stevens, Roger T. Fractal Programing in C. Redwood City, CA: M&T Publishing, Inc., 1989. Stevens, Roger T. Advanced Fractal Programing in C. Redwood City, CA: M&T Publishing, Inc., 1990. Stewart, Ian Does God Play Dice? The Mathematics of Chaos. Oxford: Basil Blackwell, 1989. Stewart, Ian and Golubitsky, Martin Fearful Symmetry, Is God a Geometer? Oxford: Blackwell, 1992. Registration You may freely copy and distribute this shareware Version 5.0 of the Mandelbrot/Julia Set Generator. Shareware users who find the Mandelbrot/Julia Set Generator useful should support the author and register their copy. The form found below should be used for registration. Registered users will receive a copy of Version 5.1 of the Mandelbrot/Julia Set Generator with additional images and a printed manual. Registered users will also receive support, by letter mail or e-mail, for six months from the date of registration. The Mandelbrot/Julia Set Generator is a "shareware program" and is provided at no charge to the user for evaluation. Vendors who distribute shareware programs may charge a small fee for an evaluation copy. Feel free to share this program with your friends, but please do not give it away altered or as part of another system. The essence of "user-supported" software is to provide personal computer users with quality software without high prices, and yet to provide incentive for programmers to continue to develop new products. If you find this program useful and find that you are using the Mandelbrot/Julia Set Generator and continue to use the Mandelbrot/Julia Set Generator after a reasonable trial period, you must make a registration payment of $25. plus $2. shipping to Theron Wierenga. The registration fee will license one copy for use on any one computer at any one time. You must treat this registered software just like a book. An example is that this registered software may be used by any number of people and may be freely moved from one computer location to another, so long as there is no possibility of it being used at one location while it's being used at another. Just as a book cannot be read by two different persons at the same time. The registration fee is $25. ($35. outside the United States.) Please include $2.00 for shipping and handling. A complete listing of the program, which is written in Borland C/C++, is also available for an additional $20.00. All prices are in U.S. dollars. Checks should be made out to: Theron Wierenga, P.O. Box 595, Muskegon, MI 49443 Ombudsman Statement This program is produced by a member of the Association of Shareware Professionals (ASP). The ASP wants to make sure that the shareware principle works for you. If you are unable to resolve a shareware-related problem with an ASP member by contacting the member directly, ASP may be able to help. The ASP Ombudsman can help you resolve a dispute or problem with an ASP member, but does not provide technical support for members' products. Please write to the ASP Ombudsman at 545 Grover Road, Muskegon, MI 49442-9427 USA, FAX 616- 788-2765 or send a CompuServe message via CompuServe Mail to ASP Ombudsman 70007,3536. User Support Registered users will receive support, by letter mail or e-mail, for six months from the date of registration on any problems they encounter. The author is available by e-mail on the internet at mups_wiereng@wmich.edu. Registration Form Mandelbrot/Julia Set Generator, Version 5.0 Name____________________________________________________________ Address_________________________________________________________ City_______________________________________State_____Zip________ Disk size desired: 5 1/4 in._______ 3 1/2 in._______ Registration fee . . . . . . . . . . . . . . . $25.00 __________ Registration fee (Outside the USA) . . . . . . 35.00 __________ Borland C/C++ program code . . . . . . . . . . 20.00 __________ Shipping . . . . . . . . . . . . . . . . . . . ___2.00___ Total enclosed . . . . . . . . . . . . . . . . __________ (All prices are in U.S. dollars.) Method of payment: Check or MO______ MasterCard______ Visa______ Account number____________________________ Expir. date__________ Signature (necessary)___________________________________________ How did you receive your copy of this program?__________________ ________________________________________________________________ Suggested improvements__________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ The Mandelbrot/Julia Set Generator, Version 5.0 is a software product of Theron Wierenga, P.O. Box 595, Muskegon, MI 49443