Fractal Domains is a program for generating fractals on the Power Macintosh. It will only run on MaOS-compatible computers equipped with the PowerPC microprocessor. Fractal Domains takes advantage of the floating point capabilities of the PowerPC to allow rapid zooms into the Mandelbrot set and associated Julia sets. Fractal Domains v1.2 is limited to generating Mandelbrot and Julia fractals, although the next version will be able to generate many other types of fractals as well.
This document gives a short description of the menu commands and dialog boxes used in Fractal Domains. Let's begin by defining some terms used to describe the main features of the program.
Fractal Domains Basics
In order to understand Fractal Domains you need to understand dwells, regions, color maps and the interaction between them.
Dwells
Mathematically, fractals of the type currently displayed by Fractal Domains are defined by an iterative formula defined on the set of complex numbers. The dwell for a particular complex number is defined as the number of times the iterative formula can be recalculated starting at that complex number before the distance of the result from the origin exceeds a pretermined radius. The members of the Mandelbrot set (or of a Julia set) are those complex numbers that have infinite dwell values. In practice, we set a limit on the number of iterations and if we reach that limit without exceeding the radius, we assume the dwell value is infinite.
Interesting images are created generally by assigning a color to each dwell value. In Fractal Domains we call this assignment a color map.
Although technically a fractal consists of those points for which the dwells are infinite, it is usually the non-infinite dwell values near the border of the fractal that give rise to interesting images. In this document, then, we will informally refer to the entire set of points with dwell values assigned to each point as a fractal.
Regions
Fractal Domains allows a fractal to be split up into regions. Each point in a fractal belongs to exactly one region. The main feature of regions is that each region has its own color map. The criteria by which a fractal may be split into regions varies for each fractal. (Right now there are only two types of fractals supported by Fractal Domains, Mandelbrot and Julia, and they happen to have the same set of criteria for region splits, but in the future there will be other fractal types supported by Fractal Domains with some different ways of splitting into regions).
To give an example: the default Mandelbrot fractal has only one region, the exterior (i.e. all points that do not have infinite dwell values). However, using the "Dwells" tab panel of the "Parameters…" dialog box, we can split the exterior into two regions: all points with even dwell values are in one region, all points with odd dwell values are in the other region. We can then apply completely different color maps to each region.
Color Maps
Every region has a color map. Predefined color maps can be assigned to a region from the Palette menu. A random color map can be assigned by choosing Randomize from the Palette menu, or equivalently by pressing Command-R. Any of these color maps may be editied in the “Color Map Editor” window. The structure of a color map is as follows: a color map contains between two and thirty two color breaks. A color break is a dwell value and an associated color. The color breaks are ordered by increasing dwell value.
Points in a region are colored as follows: if the dwell at a point is in the color break list, the point is assigned the corresponding color for that color break. If not, then the dwell generally lies between two color breaks. In this case a linear gradient of colors is generated between the two color breaks and the point is assigned the color corresponding to the dwell's position along the color gradient.
This color map can be modified by a power law if the “mapping exponenet” is different from one. If the mapping exponent is “g” then a mapped dwell is generated according to the equation:
mapped dwell = maxDwell * (dwell/maxDwell)^g
The mapped dwell is then looked up in the color map and the resulting color is assigned to the point.
Tour of the Fractal Domains Windows
About Window
Choosing “About Fractal Domains…” in the Apple menu will bring up the “About…” box. This shows the Fractal Domains version number and also allows you to enter the registration number for the program. More on this later.
Edit Fractal Window
Upon opening an existing fractal parameter file or by choosing New from the File menu to create a new fractal, the user will see the fractal window, containing the fractal image and a palette of some commonly used tools:
The fractal window can be resized; scroll bars will appear if the window is smaller than the total image size. The four tools in the palette at the left side of the window are (in order):
Zoom-in tool — used to zoom deeper into the fractal. The linear scale is magnified by two when this tool is clicked anywhere in the fractal, with the point clicked being used as the center of the new view. If the Command key is held down while clicking the zoom-in tool, the scale will be magnified by four rather than by two.
Zoom-out tool — reverse of the zoom-in tool. The linear scale is expanded by two when this tool is clicked anywhere in the fractal, with the point clicked being used as the center of the new view. If the Command key is held down while clicking the zoom-in tool, the scale will be expanded by four rather than by two.
Julia tool — Used to create a new fractal, a Julia set based on the point clicked with Julia tool. Clicking on a black portion of the set (i.e. part of the Mandelbrot set itself) will create a “solid” Julia set.
The Julia tool is only enabled when viewing part of the Mandelbrot set; it is disabled when viewing a Julia set.
Selection tool — Used to select a portion of the fractal. When a selection is made with this tool, the three items in the Fractal menu:
Extract
Magnify
Extract & Magnify
will become enabled. Extract will create a new window containing the selection without changing the scale. Magnify zooms the current window to contain the selection without changing the image area (the aspect ratio of the image will change to match the aspect ratio of the selection, however). Extract & Magnify has the same effect as Magnify but opens a new window for the magnified selection.
Parameters Dialog
The Parameters… dialog box can be used to change the parameters that define the image being viewed. The Fractal Parameters dialog box contains three tab panels for a Mandelbrot set image: the Area, Dwells, and Orbit panels. A Julia set image has these three panels and an additional one titled Julia.
At the bottom of the Parameters… dialog box are three buttons labelled OK, Apply and Cancel. The OK and Cancel buttons will both dismiss the dialog box. If OK is clicked, any parameters you changed will be applied to the frontmost fractal window; if Cancel is clicked the fractal will be unchanged.
If Apply is clicked, any parameters that were changed will be applied to the fractal, but the dialog box will not be dismissed. This allows you to see the effect of your changes and modify them without having to continually reopen the dialog box. This is a moveable modal dialog box, so if the box is obscuring the fractal you want to modify you can drag the dialog box to another part of the screen.
If you have clicked Apply and are satisfied with your changes, and you have not modified any parameters since the last time you clicked Apply, you may simply click Cancel to dismiss the dialog box. This prevents the fractal from being redrawn unnecessarily.
Area Panel
The Area panel looks like this:
Pixels — The size of the image in pixels is deterimined by the h and v fields. If either or both is changed, the image is resized if there is memory available for the new size.
Center — determines the center of the image in the complex plane.
Resolution — determines the total area covered by the image in the complex plane; the smaller this number is, the deeper the zoom.
Dwells Panel
The Dwells panel looks like this:
Escape Criteria
Dwell Limit — the total number of iterations allowed for each point in the fractal. If this limit is reached without exceeding the escape radius, the point is considered to be in the Mandelbrot set.
Escape Radius — in the process of iterating the fractal formula, if the absolute value of the iterated value exceeds this number, iteration is halted and the number of iterations is recorded as the “dwell value” for the starting point of the iteration.
Exterior Region Split
By Iteration (Even/Odd) — Checking this splits the Exterior region into two regions, based on whether the dwell value is even or odd. This produces a characterisic "zebra" effect.
Skip by — only effective if “Split by Iteration” is checked. This causes the even and odd bands to be wider than one dwell. It is sometimes useful for zooms where the dwell values change so rapidly that simply splitting according to even or add dwells does not produce a coherent pattern. The “Skip by” value determines the width of the bands.
Dwell Method — The numerical value assigned to each point in the fractal image is called the "dwell" but in fact Fractal Domains provides several ways of computing the value at each point.
Escape Time — This is the conventional way of assigning a value. The number of iterations required before the orbit exceeds the escape radius is called the Escape Time.
Continuous Potential — This is a number given by a formula containing the number of iterations required to move past the escape radius and the actual position in the plane at that point. It is closely related to the escape time when the escape radius is very large (256 is a good value for the escape radius when using this method) but it is a continuous function of position and therefore yields smooth gradients rather than bands as the escape time does.
Argument (Decomposition) — A number is assigned based on the angle the orbit point makes with the real axis (called the "argument" of that point) when it moves past the escape radius. Again, the results are senstive to the exact value of the escape radius.
Distance Estimator — This method uses a formula to estimate the distance of each point from the Mandelbrot set (the method can also be used for Julia sets). The closer the distance, the smaller the dwell value. With an appropriate color map, this method can be used to reveal the fine structure of the fractal which may not be apparent when using the escape time method.
Orbit Panel
The Orbit panel is rather complicated, but it can be used to create a wide variety of effects. It looks like this:
The contents of this window will possibly be different for fractal types supported by future versions of Fractal Domains. For the current version this is the window you will see for Mandelbrot and Julia fractals.
Orbit Trap — Turn this on to use the orbit trap options. All other controls in this panel will be disabled unless this checkbox is on. Causes the Exterior region to be split into Exterior and Trap regions.
A point is considered to be part of a trap region if, during the iterative process, one of the intermediate points generated in the iteration process (the set of all these points is called the orbit) passes within a trap area centered at the origin of the complex plane. The shape and size of the trap area is defined by the controls in this panel.
Furthermore, the dwell value assigned to a point in the stalk region is not its actual dwell value, but a value based on the distance of its orbit from real or imaginary axis, in the case of the cross (stalks) or square traps, or the distance from the origin in the case of the circle trap.
Skip by — Only applies if “Split by Orbit (Even/Odd)” or “Split by Dwell (Even/Odd)” is selected. Increases the width of the bands generated by these option in the same manner as the “Skip by” parameter for the Exterior region.
Trap Extent/Outer and Inner — Determines the inner and outer dimensions of the orbit trap. You may have to change this by many orders of magnitude to get a good picture depending on the orbit trap type and the fractal area itself. Circle and square traps call for much larger values than the cross and axis traps. Try entereing different values and hitting "Apply" until you see results of the size you desire.
Setting "Inner" to a non-zero value creates a ring-shaped trap.
Exclude Iterations/Less Than and Greater Than — The orbit of a point is tested for intersection with a trap only for points in the orbit with iteration values within these bounds. Points that intersect the trap at low iteration values tend to contribute to large features in the final fractal image, so if you want to remove the larger features, set the "Less Than" value to a higher number. Similarly, if you want to exclude very small features, set the "Greater Than" value to a smaller number.
Trap Interior Points — Subjects points interior to the Mandelbrot or Julia set to the same trap criterion, so that points in the set can be mapped to trap regions also. The default is for this to be on. It is here for compatibility with FracPPC, which always excluded set interior points from the trap region.
Scale by Iteration Limit — If this is on, values in the trap region are scaled so that the maximum value is equal to the iteration limit. If this is off, the maximum value is close to 2^15. This option is off by default, it is available solely for compatibility with FracPPC, which always scaled values by the iteration limit.
Region Split — Further splits the Trap region. This menu offers the following choices:
None — Does not further split the Stalk region.
Split by Axis (Real/Imag) — Splits trap region into two separate regions based on whether the point where the orbit passed through the trap was closer to the real or the imaginary axis.
Split by Orbit (Even/Odd) — Splits trap region into two separate regions based on whether the number of iterations where the orbit passed through the trap is even or odd.
Split by Dwell (Even/Odd) — Splits trap region into two separate regions based on whether the actual dwell value is even or odd.
Julia Panel
For Julia fractals, there is an additional tab panel labelled "Julia." There are two fields to set in this window, for the real and imaginary parts of the Julia set constant C used in the iterative operation z^2 + C -> z that defines the Julia set.
Image Dialog
Choosing the menu item Image… from the Fractal menu brings up a tab dialog with two panels labelled Map and Color. It has OK, Apply and Cancel buttons that work as the buttons in the Parameters dialog work.
Map Panel
The Map panel looks like this:
This panel controls the way in which color maps from the Palette menu or the Color Map Editors are applied to the regions of a fractal, and also controls how random color maps are generated.
Random Color Limit — determines the maximum number of color breaks in a random color map. The number of color breaks in a random map will vary randomly between two and this number.
Map Limits — this item is intended to control how the maximum and minimum dwell limits mapped vary with a zoom into the fractal.
Randomize Color Breaks — If this is not checked, color breaks in a random color map will be spaced uniformly between the minimum and maximum dwell. When checked, the spacing in random color maps will also be randomized.
Apply Same Map To All Regions — Causes a map applied to any region to be applied to all other regions automatically.
Randomize All Regions — If not checked, random maps are generated only for the current regions (see the section on the color map editor to see how to change the current region). If checked, commanding "Randomize" generates a different random map for each existing region. However, this option is overriden by “Apply Same Map To All Regions.”
Color Panel
The Color panel is shown below:
Macs with 24 bit displays will automatically generate 24 bit color fractals. This dialog box allows Macs with displays of lower depth generate higher depth fractals offscreen. The resulting map will then be copied to the screen with optional dithering.
The value set in this window also affect the Print… and Save As PICT… commands. The picture sent to the printer or PICT file will be generated at the depth currently selected in this window. You can change the depth at any time, even after the dwells have been generated.
Render Image Dialog
Whenever a fractal editing window is visible on the screen, the "Render Image…" command is enabled in the Fractal menu. Selecting it brings up the following dialog box:
This dialog enables you to produce a higher-quality image of the fractal and/or spool large images to disk.
Image Size/ h and v — These fields are initially set to the same values as the fractal window the image will be derived from, but you can set them to whatever you want. Generally this is used to generate an image that is much larger than the image in the fractal window.
Anti-Alias — Turn this on to produce an anti-aliased version of your fractal image. Several samples are generated for each pixel in the image, and an anti-aliasing filter is applied. Currently, a Lanczos-windowed truncated sinc filter is used. Future versions of Fractal Domains may allow a choice of filters.
Filter Parameters
Support — This is used to determine where the sinc function is truncated. If you don't know what this means, you don't have to worry about it. Generally, this doesn't need to be changed.
Sampling Ratio — Determines how many samples (in both dimensions) should be generated for each pixel in the original image. This is important because it controls both the quality of the anti-aliasing and the amount of time required for generation.
It takes longer to generate an anti-alised image than an image generated without anti-aliasing. The time factor involved depends mainly on the sampling ratio. The table below shows the approximate extra computation time:
If Sampling Ratio = 2, image takes four (4) times as long to generate.
If Sampling Ratio = 3, image takes nine (9) times as long to generate.
If Sampling Ratio = 4, image takes sixteen (16) times as long to generate.
Generally, sampling ratios above 4 don't yield noticable improvement in quality. A sampling ratio of 3 is often a good compromise. Try all three values on some small fractal images and see if you think the improvement in quality is worth the extra time.
Create Image On Disk — If this button is pressed, the image will be spooled to a disk file with the image size and anti-alias options that you specified. You will be prompted for the name of the spool file.
Facts about spool files
The spool file window can be used to view the image, but the entire image is never in memory at one time. The size of the image in the spool file is not limited by the amount of memory allocated to Fractal Domains.
The spool file itself is not an image file, that is it is not in any standard graphics file format. However, it contains all of the image information and can be used by Fractal Domains to generate a PICT file of the image.
If spooling to file is interrupted by closing the spool window or quitting Fractal Domains, generation of the image can be resumed at any time by opening the file with Fractal Domains. Thus, you can generate very large images without having to insure that the computer is continuously up and running for long periods of time. In fact, if a crash occurs during generation of a spool file there is a good chance that the image generation can still be resumed.
When a spool file is first created, Fractal Domains does not check to see if there is enough free space on the volume for the complete spool file. This is actually a good thing, as you can start a spool file and clear the space for the whole file later. If there is not enough space when you start, make sure you clear more space before the disk is full, or stop the image generation before the disk is full, move the spool file to another volume, and resume image generation.
The space required by the complete spool file can be simply calculated. If the height and width of the image are h and v, then the spool file will require (approximately) 3*h*v bytes of storage.
The elapsed time is stored in the spool file; thus, if you resume image generation at a later time, the estimated time to completion can still be calculated.
Disk accesses usually happen whenever the spool file window needs to be updated. If this becomes inconvenient, the spool file window can be collapsed by clicking in the collapse box in the upper right hand corner of the window if you are using MacOS 8, or by triple-clicking the title bar if you are using System 7.x and have WindowShade enabled. Collapsing the window does not interfere with the image generation process.
Create Image In Memory — An image will be rendered in memory rather than on disk. A window will be opened that shows the generated image, but you will only be able to save as PICT and will not be able to resume a partially generated image once you close it. This is mainly useful for quickly checking the effects of anti-aliasing on small images.
Preferences Dialog
Selecting the Preferences item in the Fractal menu brings up the following dialog box:
Fractal Generation Time - The time in milliseconds that is spent calculating dwell values before allowing other process to have CPU time. Increasing this value speeds up fractal generation somewhat, although very large values will make the computer less responsive.
Save Dwells - If checked, the dwell values are saved when a fractal is saved to a file, so they don't have to be regenerated when the file is reopened. Since this option makes the file very large and many fractals can be regenerated fairly quickly, you probably won't want to use this option very often.
Center (If Possible) When Printing - If the total area of your fractal image is greater than the printable area provided by the printer and Page Setup options you have selected, then this option has no effects; the fractal will be printed on multiple pages, as many as it takes to print the image.
However, if your fractal image is smaller than the printable area of one page, then checking this option instructs Fractal Domains to center the image on the page. If this option is not checked, the image will be printed justified to the top left corner of the printable area of the page.
Open Window On Startup - These options allow you to determine which windows will be opened initially when Fractal Domains is started.
• Statistics - Statistics floating window is visible on startup
• Cursor - Cursor floating window is visible on startup
• Color Map Editor - Color Map Editor window is visible on startup
• New Fractal Document - a new fractal is opened on startup
Floating Windows
There are three floating windows that show information about the fractal in the frontmost fractal window.
The cursor window shows the region, dwell value, and complex coordinates of the point where the cursor is currently located.
The Statistics window shows some facts about the fractal in the frontmost window: the minimum and maximum dwell values, the time elapsed while generating the fractal, an estimated time to completion, and the actual time the CPU required to generate the fractal.
The Color Map Editor is used to modify the color map. It is very crude right now and will definitely be replaced with a better editor in a future revision of Fractal Domains. Right now it at least allows the color map data structure to be edited, albeit in a clumsy fashion.
Region - This popup menu selects the current region. This sets the region not only for the color map editor but also for the commands in the Palette menu.
Mode - This popup menu offers three choices: Auto, Fixed and Wrap. The mode determines how a color map will adjust when it is first mapped to a fractal region.
• Auto - The minimum and maximum dwell (dwell values for the first and last control points) will be scaled to match the minimum and maximum dwell in the fractal. Intervening dwells will be scaled accordingly.
• Fixed - The minimum and maximum dwell do not change unless they are manually adjusted in the color map editor.
• Wrap - The minimum and maximum dwell do not change; however, after the maximum dwell, the color map wraps around. After the maximum dwell, the color map is reproduced, but scaled by the wrap factor.
# of control points - Displays the number of color breaks in the color map. Adding and deleting color breaks changes this number.
Control Point - sets the number of the color break currently being edited
Dwell - sets the dwell value for the current color break (the color breaks are ordered by dwell value, so if changing this number changes its place in the color break list, the control point number changes).
If the Uniform Control Point Distribution option is checked, this field will not be visible, as the dwell values at each control point will be determined by Fractal Domains.
Uniform Control Point Distribution - if this is checked, the control points in the map are distributed uniformly between the maximum and minimum dwell value. All control points are equidistant.
If this option is checked, the "Dwell" field will disappear. When unchecked, the Dwell field will reappear with the dwells distributed uniformly.
Color Control - the color square displays the color assigned to the current color break. Click on this to bring up the Color Picker to change the color.
Add - adds another color break with the same dwell and color as the current color break. You can then modify these values to get a new color break.
Delete - deletes the current color break.
Mapping Exponent - determines the current power law exponent for this color map. This can also be increased or decreased by pressing the right arrow and left arrow keys respectively. The mapping exponent can be reset to one by pressing the up arrow key. These keys have the same effect even if the color map editor window is not currently visible.
Apply - this button causes the changes made in the color map editor to be applied to the current region.
Apply Immediately - when checked, any change in the color map editor is applied immediately to the fractal image.
Registration
Fractal Domains is shareware. Some features are restrictred and are not available until you register your copy. You may use the unregistered copy for an unlimited period to evaluate it. In order to have use of the restricted features, you must register. The current shareware fee is $20. Registration is via the Kagi application. Payment may be made electronically with credit card, or by mail with check, money order or credit card. Further instructions are in the "Read Me" file that accompanied the Fractal Domains distribution.
Restricted features
An unregistered copy of Fractal Domains may generate all available fractals with all available dwell methods and orbit trap types. Saving and printing is restricted, however. You may save or print the parameter file or PICT file of any Mandelbrot or Julia fractal generated with the standard "Escape Time" dwell method and with either no orbit trap or the cross (stalk) orbit trap. Fractals generated with any other dwell method or orbit trap cannot be saved or printed. In addition, any image generated with the "Render Image…" command, whether generated in memory or spooled to disk, cannot be saved as a PICT file with an unregistered copy. However, if you generated a spool file with an unregistered copy of Fractal Domains, you can save the spool file and register Fractal Domains later. When your copy of Fractal Domains is registered, you can generate a PICT file from your saved spool file.
Entering your registration number
After you have paid the registration fee via the Kagi application, you will receive by email a registration number that is keyed to your name. Save a copy of this email in a safe place, in case you ever have to reregister your copy. In order to register your copy, start Fractal Domains and choose "About Fractal Domains…" from the Apple menu. There is a button on the "About" dialog box labelled "Register." Click this button. You will see the following dialog box:
Enter your name and registration number exactly as you received them and click OK. All previously restricted features of Fractal Domains will now be enabled:
Important note about registration
Your registration information is stored in the "Fractal Domains Preferences" file which resides in the "Preferences" folder in your System Folder. If you move your copy of Fractal Domains to another computer, this preferences file will not follow it. Also, you may lose this file if you have a hard disk crash or if you do a clean install of the operating system. For these reasons, please make some sort of backup of your registration information. If you lose it for some reason, contact the author at the email address listed at the end of this document.
More info?
The latest information can be found on the Fractal Domains web page, which is currently:
http://members.aol.com/ddemars/fracppc.html
You can always email me with any questions or bug reports.
