This program draws escape-time fractals, as described by Clifford Pickover in his article ╥Automatic Generation of General Quadratic Map Basins╙, in Computers and Graphics, Mar/Apr 1995.
How do I use it?
Here╒s a description of the various menus. Self-evident items have been omitted.
File Menu:
Ñ Save Picture As╔ : Saves the currently-displayed image as a SimpleText picture. Any color paint program can open those for editing.
Edit Menu:
Ñ Undo: Allows you to undo the last change you made. You may go back to the previous formula, or ╥zoom back╙ to the previous view.
Ñ Copy formula: Copies the current formula onto the Clipboard. You can then paste it in a text editor, or the Note Pad, or whatever.
Draw Menu:
Ñ New Formula: Allows you to enter a new formula. Formulas are strings of 12 letters, A through Y. Other characters are invalid and will be replaced by either 'A' or 'Y'. You may use lower-case letters if you want.
Ñ Random Formula: Determines a new formula at random. This formula is first tested to verify that it produces at least a marginally interesting picture (if you try to input random letters yourself, you╒re much more likely to get nothing but a tiny fleck of color). Note: since this menu item is chosen so frequently, pressing the spacebar will also activate it (this involves less stress on the fingers than constantly typing command-R).
Ñ Set View: Allows you to determine numerically which area of the plane you want to view. You may want to select this item to note where exactly a given region of a fractal is, so you can get back to it with precision at later time. See ╥Mathematical underpinnings╙ for details.
Ñ Reset View: Resets the view to the region (-5,-5); (+5,+5). See ╥Mathematical underpinnings╙ for details.
Ñ Draw it!: Redraws the fractal. In case the image gets messed up (which isn╒t supposed to happen).
Ñ Zoom in: Look ╥closer in╙ at the current image.
Ñ Zoom out: ╥Pull back╙ from the current image.
Ñ Select Palette: Use this hierarchical menu to select one of the different color palettes available. Different palettes give different ways of looking at the same fractal, and produce sometimes markedly different pictures.
Window Size Menu:
Allows you to select one of five standard sizes for the window. You may also use ╥Other╙ to set a non-standard size. The window must be between 32 and 512 pixels on a side╤it will always be a square.
Formulas, More and Still More Menus:
Choosing one of the fancy names in those menus selects one of 54 predefined formulas. These produce some pretty neat pictures. In some cases, it may be necessary to zoom deep into one of the areas of the fractal in order to see the really nice part (the image that appears when the program starts is a tiny part of ╥Fire Swirls╙). The formula for ╥Pickover╒s Spiral╙ is taken from Mr. Pickover╒s original article.
Zooming in on an Image
To zoom in on an image, hold the cursor over the window; it changes to a cross. Click and drag, to draw a square enclosing a region of the image. This region will be recomputed to fill the window. Note that the first click always selects the top left corner of the square. If you don╒t drag to the right and down, no zooming in will take place. You may of course use the ╥Zoom in╙ menu item instead, but this always zooms in on the center of the current image.
To see more of an image, either use the ╥Zoom out╙ menu item (to pull back a little), or use ╥Reset View╙ to get back to the original view.
Mathematical Underpinnings
What this program actually does is compute and display the behavior of iterated functions. The window ╥looks at╙ a portion of the x,y plane (initially, this portion goes from (-5,-5) in the top left corner to (+5,+5) in the bottom-right corner). To each pixel of the image there correspond initial values for the variables x and y. Given these initial values, we compute new values using the formulas:
new x = a + bx + cx^2 + dxy + ey + fy^2
new y = g + hx + ix^2 + jxy + ky + ly^2
where a, b, ╔, l are parameters. This process is repeated until the current values of x and y are outside a circle of radius 1000 from the origin (0,0). The number of iterations needed to go outside the circle is noted and will be used to determine the color of the pixel. (This value has a maximum of 1,024 for purposes of this program.)
Values for the parameters are indicated by a letter code, where
A = -1.2 B = -1.0 C = -0.8 ╔ X = +1.0 Y = +1.2
Letter-codes for the values of each of the 12 parameters produce a ╥formula╙.
The color palettes used in the program map the number of iterations to a set of up to 256 different colors. In most palettes, colors are reused frequently to produce a ╥banding╙ effect and to always get a nice color range. The palettes ╥Rainbow 3╙ and ╥Shades o' Grey╙ use a full 256-color stretch, which for most fractals produces large areas of nearly-identical color. Some fractals, however, show up very nicely with those palettes. Experiment!
Finding Nice Pictures
Although ╥Random Formula╙ will almost always produce a marginally interesting picture, most fractals will still look like pixel puddles with funny-shaped holes (try the program and see what I mean). Only about 5-10% of randomly-generated fractals will show pretty structures. Among those, you will often see variants of the same type, so truly striking pictures aren╒t so easy to come by. The text file ╥Other nice formulas╙ gives a list of formulas for interesting fractals above and beyond the 54 in the menus.
Who did this?
EscapeFractals is copyright ⌐Yves Meynard, 1996. It was compiled for the PowerPC using MetroWerks CodeWarrior. It is freeware and may be freely distributed, but not altered.
Version History
1.0 Initial Release
1.1 Now handles drawing correctly when in 256 colors.
1.2 (April 97) Draws a nicer zoom square. Changed e-mail address in ╥ReadMe╙ file.
1.3 (August 97) Fixed a long-standing (and embarrassing) bug with the ╥Save as picture╙ routine. Improved ╥Set view╙ interface; it now uses a center and a radius, instead of top, bottom, left and right.
