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MandelMountains V2.0 by Mathias Ortmann Discover the Mandelbrot and Julia Sets From a Completely New Point of View! MandelMountains gives you the ability to render wonderful three-dimensional images of close-ups of the Mandelbrot and - in V2.0! - also Julia sets. The well-known color strips of the usual Mandelbrot images become at once mountainsides that smoothly climb to high plateaus, leaving deep valleys between them. You may have already seen images of this type (e.g. on the covers of the books "The Beauty of Fractals - Images of Complex Dynamical Systems" by H.-O. Peitgen and P.H. Richter or "The Science of Fractal Images", edited by H.-O. Peitgen and D. Saupe) - here and now you have the tool to create them on your own! MandelMountains allows you to produce high-quality non-interlaced or interlaced (and even overscan) images of arbitrary areas of the Mandelbrot/Julia sets. You can easily define magnification windows to zoom deeper and deeper into this fascinating world. Since the development of this program took a lot of time and work, I release it as shareware. This means that if you like and use this program, you should become a registered MandelMountains user by sending me a little contribution of about $10. This will make it possible to develop subsequent versions of this program. Suggestions, comments and bug reports are welcome, too. This is my address: Mathias Ortmann Strindbergstr. 5 D-8000 Munich 60 WEST GERMANY IMPORTANT! MandelMountains V2.0 requires the following libraries to be in your libs:-Directory: - mathtrans.library - mathieeedoubbas.library - mathieeedoubtrans.library This documentation does NOT cover the theory of fractal geometry, complex numbers and Mandelbrot/Julia sets. If you never heard anything about these things, I recommend to read the books mentioned below. 1. The Rendering Method The image is rendered from front to back. A virtual horizon line prevents hidden areas from being displayed. The brightness of the surface is determined by the angle the light falls on it. If the number of iterations exceeds a certain (user-defineable) value, the pixel is set in color instead of gray, thus remains of the usual color strips are visible on the high plateaus, a fact which greatly increases the plasticity of the image. The iterations are effected in pure, speed-optimized assembler code, using the Motorola Fast Floating Point format (FFP), while all other calculations occur in double-precision IEEE standard format. This yields acceptable calculation times (around 20 minutes to 4 hours) without precision problems, even at magnification factors up to 10000. MandelMountains V2.0 now also supports double precision for the entire calculation process, including the main iteration loop. Using this feature will almost triple computation times, but there are practically no restrictions any more concerning the magnification factor. The 1.3 mathieeedoubbas/doubtrans libraries support 68881/68882 math coprocessors. Owners of these expensive (but useful!) devices thus should always use double-precision mode (since I don't belong to this privileged class of people, I couldn't test it, so I can't say if it works, how it works and how fast it works, but presumeably it should be even faster than FFP single precision mode!). 2. The Display Format You can choose between three image sizes: Small for quick test calculations, Normal for the usual screen size (320x200/320x256) and Full for overscan format (352x240/352x282), which I recommend as ideal size. MandelMountains supports NTSC and PAL Amigas and recognizes by itself on which type of machine it is running. All images are generated in 32 color mode: 16 colors for the gray tones and 16 colors for the surface color range. Optionally you can enable the interlace mode, which will double the number of available colors: You now have 32 gray and 32 color tones, which will result in much smoother color ranges. Computation time is not affected by using interlace or non-interlace mode. 3. The File Format MandelMountains writes standard IFF files with an additional "MMD1" chunk containing all parameters of the image, so you can load previously generated images and make further magnifications. You can load MandelMountains files with all available graphics software, but note that if the image is saved again, the MMD1 chunk will be destroyed, and you cannot load it with MandelMountains any more. MandelMountains V2.0 is fully backwards compatible with V1.1, so you can load your previously generated images without problems. 4. The Parameters An image is defined by several parameters. You can see and modify all of them in the window MandelMountains opens on the Workbench screen. The cx/cy values are used only when working in Julia set mode. They are the real/imaginary part of the fixed parameter c in the term z = z²+c. See below for a detailed description of how to generate images of Julia sets. In the next two lines, there are the xmin/xmax/ymin/ymax values. They determine the rectangular part of the Mandelbrot set that is to be shown in the image (xmin/xmax represent the range of the real part, ymin/ymax of the imaginary part of c [in Julia set mode: z] in the term z = z²+c). The Depth value limits the number of iterations. If then the value of z has not exceeded a certain maximum, the point will be drawn in black. Increasing this value will result in a more detailed rendering of the border between color and black, but it will also increase computation time if there are larger areas of black. Normally, a Depth of 400 to 1000 is sufficient. Linear/Nonlinear/Super-Nonlinear Transformation: If your mountainsides look extremely steep, you should switch to Nonlinear Transformation (especially useful for magnifications of the Seahorse Valley!). For extreme cases, I added the Super-Nonlinear transformation, which is especially useful for magnifications of the western side of the Seahorse Valley. ColorMin: If you wish to have the surface of the plateaus colored (and you surely will!), you set ColorMin to the number of iterations from which on a pixel is to be drawn in color. Increasing this value will make the colored areas smaller. The range of ColorMin is normally from 20 to 300 (set it to 0 if you wish no coloring). ColorDiv: This value determines the "step rate" for the surface colors. For example: You have a ColorMin of 100 and a ColorDiv of 50. The number of iterations for a point is 300. The color of the point is now (300-100)/50, i.e. 4. Usually ColorDiv is around 50 to 150. It is not very critical, but if you choose it far too low, the colored areas on your surface will look rather fragmented. If you choose it too high, they will all have more or less the same color. HZoom: This very important value is the decimal logarithm of the factor all heights are multiplied with. If you choose this value too low, the whole surface will be flat like a sheet of paper; if you choose it too high, you will not see more than some vertical walls. This value is probably the most critical and must be chosen carefully. It depends very much on the magnification factor (increase it after each magnification) and can range from 2 (initial picture) to 30 (blow-ups of details in the Seahorse Valley, for example). Remember that for nonlinear and super-nonlinear transformation, HZoom must be considerably higher than for linear transformation. HSmooth: Sometimes it may occur that the border of a plateau looks rather fragmented. In this case, simply increase the HSmooth value. It can range from 0 up to more than 200, depending on the HZoom value you are using, but is not very critical. Set it to 50 for normal images. It must be said that you will have to experiment a little to get perfect results, but soon you'll get a feeling for these things (look at the sample pictures and their parameters). The 'Parameters' menu of MandelMountains V2.0 contains the submenu 'Suggest' which allows you to precalculate two of the above parameters. See below for a further description of this feature. 5. The Menus Project Menu: Choosing the Load Image or Save Image option will bring up a file requester (thanks Justin!) that allows you to choose a file name for the image to load/save. Images are compressed before saving. Start Rendering: This option clears the current screen, brings it to front and starts the computation. Mode: You can select between Mandelbrot and Julia Mode (see below for details about Julia sets). (NEW in Version 2.0) Precision: This submenu allows you to choose either single precision (FFP) or double precision (IEEE) mode. (NEW in Version 2.0) Parameters Menu: Zoom In brings the current screen to front and creates a mouse-directable magnification window. Pressing the left mouse button while moving the mouse to the right/left changes the size of the window. The right mouse button confirms the magnification. Restore Aspect is useful when you have entered xmin/xmax/ymin/ymax values by hand. By restoring the aspect of width and height you ensure that the image does not look squeezed. Suggest is a new feature of MandelMountains V2.0, which I implemented to give you a basis for your experiments by generating suggestions for the parameters Depth and HZoom. After some initial problems, I found a quite adequate way to precalculate the HZoom value. The results of the algorithm are surprisingly good - in most cases they need not be altered at all and thus will save you a lot of experimenting time! The algorithm needs some time to compute its result. The percentage of completion is shown by a horizontal bar. You can cancel the computation by pressing the Escape key, or terminate it at any time by pressing <Return>, accepting the current state of evaluation, which in most cases is not too far away from the final result. Screen allows you to choose one of the three standard screen sizes and to toggle interlace mode. Color Range Menu: This menu controls range of surface colors. Different color ranges may greatly change the impression of an image, so experiment a little. Available as from/to colors are: Black, Blue, Red, Magenta, Green, Cyan, Yellow and White. Additionally, you can cycle the colors by pressing F1/F2 at any time. (NEW in Version 2.0) 6. Julia Sets Julia sets are generated by varying the initial value of z in the iteration formula z = z²+c, leaving c constant for each pixel (Mandelbrot set: The parameter c is varied instead of z, and he iteration always starts with z = 0). There exists an infinite number of Julia sets, all determined by the fixed parameter c (displayed in the top line of the parameter block, cx is the real, cy the imaginary part of c). You can either enter a c value directly by hand, or choose it from a position out of the Mandelbrot set. This is done as follows: 1. Load an arbitrary MandelMountains image of the area you are interested in. 2. Select "Zoom In" from the Parameters Menu. 3. Mark the desired point with the TOP LEFT edge of the zoom window and press the right mouse button. 4. Select "Julia Set" from the Project/Mode Menu. The parameters xmin/ymin, representing the selected top left edge, will be copied to the cx/cy fields. The other values will be initialized with default values, giving the best possible starting environment for most Julia sets. Experimenting with this feature, you will discover a fascinating resemblance of the Julia sets to the area their parameter c is taken from (e.g. for a c value out of the Seahorse Valley, there will be the same seahorse tails and eyes as in the Mandelbrot set). Thus the Mandelbrot set can be interpreted as a "one-page dictionary" of all existing Julia sets. I recommend to choose the c values mainly from the border area of the Mandelbrot set, because otherwise the resulting Julia sets will look rather boring. Certainly, you can zoom into Julia sets exactly like into the Mandelbrot set. 7. When is a Picture Finished? Normally, you will wait until the screen is filled completely (up to the upper edge). In this case, the program stops the computation by itself. Sometimes, it may be necessary for aesthetical reasons to interrupt the rendering earlier, resulting in a black background of the image. In this case, you simply have to click on the STOP Gadget at the right moment (Initial.NTSC, Julia1.PAL and Julia2.PAL were generated using this method). Important: Clicking on the gadget near the depth gadgets of MandelMountain's workbench window will make the current screen appear (with correct centering if it's an overscan screen), clicking on the screen and then pressing the right mouse button will make the screen disappear again. If you haven't got a true Fast-RAM machine, I recommend to click the screen to back during computation. This will save memory cycles for the CPU and greatly reduce computation time. Special thanks go to Heinz-Otto Peitgen for publishing the formula that smoothes the surface of the Mandelbrot set in such a perfect way! It is called Continuous Potential Method for Mandelbrot Set (CPM/M) and can be found in chapter 4.2.4 of the book Heinz-Otto Peitgen/Dietmar Saupe (editors): The Science of Fractal Images Springer-Verlag New York Berlin Heidelberg 1988 ISBN 0-387-96608-0 Springer-Verlag Berlin Heidelberg New York 1988 ISBN 3-540-96608-0 If you are looking for an excellent introduction into fractal theory, history and mathematical backgrounds of the Mandelbrot and Julia sets, including a couple of wonderful color pictures, you should take a look at H.-O. Peitgen/P.H. Richter: The Beauty of Fractals - Images of Complex Dynamical Systems Springer-Verlag New York Heidelberg Berlin Tokyo 1986 ISBN 0-387-15851-0 Springer-Verlag Berlin Heidelberg New York Tokyo 1986 ISBN 3-540-15851-0 -------------------------------------------------------------- PathMaster File Selector User Documentation 89-06-26. -------------------------------------------------------------- Copyright © 1989 by Justin V. McCormick. All Rights Reserved. The selector is completely dynamic and asynchronous; you may change directories, select files, scroll the file list, etc., while the selector is scanning a directory. The three Sort Gadgets (Alpha, Size, and Date) work like switches. For instance, if you click on the "Alpha" Sort Gadget, the gadget will remain highlighted ("On") when you release the mouse button. The File Selector will now alphabetize directories automatically until you turn the Sort Gadget "off" by clicking it again, or click on a differnt Sort Gadget. Clicking the Menu Button activates the FILE or PATH String Gadget, allowing you to type in a new file or pathname without having to click inside of the string gadgets first. Also, some keyboard shortcuts are available: ESCAPE key - Same as clicking on the "Cancel!" gadget. Cursor-Up key - Same as clicking on the Up Arrow gadget. Cursor-Down key - Same as clicking on the Down Arrow gadget. Shift-RETURN keys - Toggle between FILE and PATH gadget. Note that if the PATH, FILE, or PATTERN String Gadgets are in use, the keyboard shortcuts are disabled. The selector has a software path length limit of 344 characters. You will receive a warning message in the titlebar if you try to exceed this limit. The PathMaster name is a trademark of Justin V. McCormick. PathMaster File Selector source and documentation written by: Justin V. McCormick. Copyright © 1989 by Justin V. McCormick. All Rights Reserved. * --------------------------------------------------------------------- * THIS SOFTWARE IS PROVIDED "AS IS". NO REPRESENTATIONS OR WARRANTIES ARE MADE WITH RESPECT TO THE ACCURACY, RELIABILITY, PERFORMANCE, CURRENTNESS, OR OPERATION OF THIS SOFTWARE, AND ALL USE IS AT YOUR OWN RISK. THE ABOVE AUTHOR ASSUMES NO RESPONSIBILITY OR LIABILITY WHATSOEVER WITH RESPECT TO YOUR USE OF THIS SOFTWARE. * --------------------------------------------------------------------- *