Mandeldot

Mandeldot is a program that visualizes the Mandelbrot set, a special collection of complex numbers.
There are many such programs. This is the one I wrote finally, after having committed to doing so some three decades ago. ☺ The linked article goes into much more depth about what the Mandelbrot set is and how it’s computed. You may also be interested in the article about fractals, an important concept in understanding the set.

You can download the program by using the link below. The program requires Windows running .NET 4.5. If you don’t have .NET 4.5 installed, the installer might prompt you to download and install it, but you can also download it from Microsoft by clicking here.

The program is meant to be easy to use. You can use the mouse to drag and select a new region to examine more closely (“zoom” in). You can change the “drag mode” to simply pan instead of zooming (use the “Drag mode” option in the “View” menu, or simply hold down the Shift key while dragging). For additional information on how to use the program, see the tips and documentation below.

Tips for using the program (detailed documentation below):

Like similar programs, Mandeldot draws the actual set using black pixels, and uses colors for the rest of the pixels surrounding the set, where the color used corresponds to how fast a pixel not in the set is “escaping” the set. Colors are assigned from a bright “rainbow” palette. This color palette has the advantage that it’s easy to define, but it does lack contrast. If your image seems to be lacking in detail, try adjusting the color offset in the options to find a range of colors that shows contrast better.

One interesting feature of the Mandelbrot set is that it’s a “connected space”. Over-simplifying quite a lot, a connected space is made of points where between any two points, there exists a continuous path of other points in the same set. This means that in theory, all of the points in the set drawn by Mandeldot should touch, but in fact the connection isn’t always visible due to the non-infinite screen resolution used by your computer (never mind the non-infinite resolution of your eyes). You can often make the connections easier to see in the program by reducing the maximum iteration count in the options. Doing so has the effect of erroneously considering points to be in the set when in fact they are simply “close”, making it easier to see the paths of connected points that are in the set.

Another side-effect of the non-infinite nature of your computer is that while the set itself, being a fractal, has an infinite degree of detail (i.e. no matter how closely you look at it, there is the same amount of detail), your computer will eventually reach the limit of being able to distinguish between one complex number and another. When you reach this point, the image will show larger groups of pixels all the same color. If this happens, you’ve gone too far. Zoom back out a bit and things will go back to normal.

Especially as you look at areas close to the set, or where the set wraps around itself, you may find that the image gets kind of “noisy”, in the sense that the color changes dramatically from one pixel to the next. A future version of the program will support alternate coloring methods that will have less of a problem with this. In the meantime, one effective way of smoothing the colors out is to set the image size to something larger than you actually want to look at, and then adjust the view zoom to an appropriate percentage to bring the image back down to the right size. E.g. instead of rendering the set in a 512 by 512 image, use 1024 by 1024 and view it at 50% zoom factor. In scaling the image down to size, the pixels will be blended together and evened out. Yes, it will take longer to render the image this way, but it’s often worth the effort. Try different combinations to see which you prefer: triple the size then view at 33%, or quadruple the size and view at 25%, etc.

Of course, it’s faster and more fun to navigate around the set when the rendering is faster. To speed things up, reduce the image size (I find 512 by 512 pixels works well) and lower the maximum iteration count. Note though, if you zoom in far enough and all you get is black for your image, you may simply need to increase the maximum iteration count again. (Obviously, there are some places where you can’t get anything but black, but it’s usually pretty easy to avoid zooming straight to those areas).

Play. Explore. Have fun!

Documentation:

File menu:

• Render — Begin rendering the currently selected area of the set. Normally not needed, since rendering starts automatically when the options are changed, but is useful if you’ve cancelled rendering and want to restart, or you just want to see it do it again.
• Cancel rendering — Interrupt the rendering in progress. Rendering will automatically be interrupted and restarted if you change the rendering options while it’s already in progress. But you can use this if you know you don’t want to render the current options and want to save some electricity.
• Save image as… — Save the currently shown image.

  Note: this always saves the full resolution of the image. Even if you are viewing the image at some zoom factor of other than 100%, the saved image will have the pixel resolution specified in the image options.

• Exit — This is self-explanatory, right?

Image menu:

• Previous zoom — Return to the previously selected area of the image.

  Note: this restores only the selected area, not other settings such as maximum iteration count or color offset.

• Zoom in 2x — Reduces the dimensions of the area selection by half, effectively “zooming” into the image by a factor of two.
• Zoom out 2x — Doubles the dimensions of the area selection, effectively “zooming” out from the image by a factor of two.
• Options… — Displays a dialog allowing you to adjust the options used to render the image.

View menu:

• Drag mode — Changes the default behavior for dragging the mouse in the image.

  Note: in both “selection” modes, the selection always has the same aspect ratio as configured in the current image options.

  • From center — Creates a new selection, centered at the initial mouse click. The Alt key temporarily selects this mode.
  • From corner — Creates a new selection, with one corner anchored at the initial mouse click. The Control key temporarily selects this mode.
  • Pan — Drags the image to allow selection of a new area without changing the dimensions of the area selection. The Shift key temporarily selects this mode.
• Zoom — Changes the zoom factor at which the image is displayed. This simply adjusts the visual display, and not the options used to render the image.

  Note: less than the pre-defined values in this menu, there are implicitly defined values at intervals of 5%, and greater than the pre-defined values there are implicitly defined values at intervals of 100%. These implicit values are used, along with the pre-defined values, when using the “Zoom in” and “Zoom out” menu items.

  • 25%, 33%…200% — Select a pre-defined zoom factor.
  • Custom… — Displays a dialog allowing you to enter a specific zoom factor.
• Zoom in — Increases the zoom factor to the next higher value.
• Zoom out — Decreases the zoom factor to the next lower value.
• Fit to window — When this menu item is toggled on, the displayed image will be resized so that it always fills the program window.
• Fit window to image — Resizes the program window so that it’s just large enough to display the image at its current zoom factor. Not a toggle.
• Show status — When this menu item is toggled on, some status information will be displayed at the bottom of the program window: the elapsed time during rendering, and the currently selected number of threads used for rendering.

Options dialog:

• Width — The width, in pixels, of the rendered image
• Height — The height, in pixels, of the rendered image
• Maximum iterations — The maximum iteration count for any given pixels. Pixels are colored according to the iteration count if they are determined to definitely be outside the Mandelbrot set before reaching this limit. Otherwise, they are considered inside the set and drawn as black.
• Thread count — The number of threads used to render the Mandelbrot set. This initially defaults to the number of processors detected on your computer.

  Note: modern Intel processors include a feature called Hyper-threading in which a single processor “core” is made to appear as two processors. The additional “processor” that Hyper-threading enables provides a modest increase in computing power but does not allow for complete concurrency within that single processor core. This means that if your computer supports Hyper-threading, the program will select a number of threads that is double the actual number of processor cores on your computer, but will not get twice the performance that it would get running with half that number of threads. Up to the number of actual cores, multi-threaded execution increases performance nearly linearly (e.g. twice as many threads gives almost double the speed), but past that and up to the number of processors, performance increases only a bit more. You can expect to see 10-20% improvements in speed at that point. Using more threads than there are processors on your computer will not improve rendering speed, and in fact may well make it run more slowly, as the threads fight over the smaller number of processors.

  At the moment, AMD CPUs do not have any equivalent to Hyper-threading, so if you have an AMD-based computer, the performance increase for the number of threads should be nearly linear all the way up to the number of processors detected.

• Color offset — Shifts the color used for the image by the specified amount. There are 1530 colors used, so values between 0 and 1529 have a useful effect (any other value simply wraps around to the useful range)
• Real center — The real component (x-axis value) of the point drawn in the center of the image.
• Minimum real — The real component (x-axis value) of the point drawn in the lower-left corner of the largest square visible in the image.
• Imaginary center — The imaginary component (y-axis value) of the point drawn in the center of the image.
• Minimum imaginary — The imaginary component (y-axis value) of the point drawn in the lower-left corner of the largest square visible in the image.
• Minimum radius — The radius from the center of the image to a nearest edge of the image.
• Minimum side — The distance from one edge of the image to the other, in the shorter direction.
• Relative to center — Enables centered coordinates for specifying the area to draw. If this is unchecked, then the “Real center”, “Imaginary center”, and “Minimum radius” options become “Minimum real”, “Minimum imaginary”, and “Minimum side”, respectively.
• Defaults — Replaces the current render coordinates with their original, default values.
• Defaults (All) — Replaces all of the current options with their original, default values.

Custom Scale dialog:

• Percent scale — The zoom factor at which to view the image.

JPEG Quality dialog:

• Quality — The quality value used for JPEG compression when saving a JPEG file. Specify a value between 0 and 100, inclusive. Better quality results in larger files. Use a lower value for smaller files. (This dialog is shown automatically after confirming the “Save image as…” dialog, if you choose the JPEG file format in that dialog.)

Glossary (of terms that may be unfamiliar, or unique to this page):

aspect ratio — the ratio of the the image’s horizontal dimension to its vertical dimension.

iteration — for each pixel in the image, the program starts with the corresponding complex value (based on what area the image is showing), repeatedly squaring and adding the result to the original complex value; this repetitive computation is the iteration that ultimately determines the pixel’s color.

largest square visible — for non-square images, one can imagine a square in the middle that has all four sides the same length as the shorter dimension of the actual image; this is the largest square visible in the image, and is used to determine how the rendering coordinates are interpreted (i.e. they are interpreted as if the image were this square…that’s why only three values are needed to describe the area being rendered).

render — to draw the image, by calculating for each pixel whether it’s in the set, and if not, assigning a color other than black to it.

selected area — the boundary, expressed as components of a complex number and a dimension, that describes the area of the set being viewed.

thread — a sequence of computer instructions executing within a program; running more than one of these concurrently allows a computer with more than one processor to compute the color for more than one pixel at a time, allowing the whole image to be rendered more quickly.