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For instructions on the applet and information on the fractals in it, scroll down.
You can zoom in on a fractal by clicking where you want to zoom in. To zoom out, hold Shift and click. If you click Start, it will zoom back to the original position.
Currently, the easiest way I have for you to invent your own fractal is fractal words, followed by patterns and quilts. I designed this program to be idiot-proof, so feel free to play with any of the controls on screen even if you don't know what you're doing to see what happens. In other words, don't worry about the consequences -- just have fun.
The fern is probably the most popular IFS fractal, mainly for its beauty. Notice that each frond is a picture of the whole fern.
The twist and the screw are the only fractals here that I invented. I got the maps (numbers) for the Fern from Michael Barnsley's book Fractals Everywhere. The others are ones that I have seen pictures of before ( mostly in the same book) and wanted to add to my collection. They are mostly standard fractals that can be seen in just about any book or web site dealing with IFS fractals. To add them, I had to toy with numbers until I got the image I wanted. Click Stop to make it stop drawing the picture and leave it as it currently is.
You can stop, as well as restart, the drawing of any fractal by click the start or stop buttons. Most fractals look better if you stop the drawing soon after you start it.
The angle scrollbar or text field will change the rotation of the fractal about the vertical axis. This works with 2D or 3D fractals, although the effect on 2D fractals can be weird at times.
The Keep Proportions check box does exactly that. If you remove the check and click Start, it will stretch the fractal to completely fill the frame.
The fractals created by this applet have up to four maps corresponding to four smaller square images inside a larger square. To select one of the four smaller squares, click in the box that has the multicolored boxes. A white box shows which area of the fractal you are editing. You can rotate that section clockwise or flip it vertically by clicking Rotate or Flip, respectively. Note that changing any area of the fractal changes the whole fractal. This is because the entire fractal is made of images of itself.
The drop-down list box controls how brightly a section of the fractal appears or whether it appears at all. If set to "deleted," that area will not be part of the fractal. It is possible to create a pattern with only two fractal maps. (two sections not set to deleted) Most of the interesting patterns use three.
If all four sections are included in the fractal at equal brightness, a dull filled square will appear. However, if the sections are set to different brightnesses, interesting patterns known as fractal quilts will appear. If you allow the applet to draw one of these quilts for too long, it will turn into a solid square. This is why the Start button automatically turns into a Stop button when you click it.
The idea for symmetric fractals was presented in the book Symmetry in Chaos by Michael Field and Martin Golubitsky. One of the many symmetric fractals presented in the book was the Circular Saw. The other three fractals shown below are standard fractals. To view one, click the link and then go back to the Java applet, where the numbers will be entered. Then click Start.
The Twin Dragon has an interesting property -- it can be tiled. The top and bottom borders are the same, and so are the left and right. It is the background of this web page, and two versions of it are available on my fractal wallpaper page.
The inside border of the Snowflake fractal is a famous fractal coastline known as the Koch curve. It can be made by starting with an equilateral triangle and replacing the middle 1/3 of each segment with two sides of a smaller equilateral triangle. Replace the middle 1/3 of each segment of the new figure with two sides of even smaller equilateral triangles, and continue the process infinitely. The result we be an infinitely long coastline that surrounds a finite area.
The Sierpinski Triangle shows how dull it can be to zoom in on a self-similar image. Click in the middle of the upper-right triangle. You will have the same image, except it will draw more slowly.
The idea for this panel was inspired by the popular "coke" fractal. I originally had the coke fractal in my collection, and once I had applied the idea to create my name as a fractal. When I started adding ways for users of this applet to invent their own fractals, I thought about fractal words. It's the easiest way I could invent for anybody to invent their own fractals. Just type a word and click Draw Word. Use only letters and spaces. Three to eight letters are recommended.
Economists have had to deal with the fact that standard statistical models do not fit the economy very well. The stock market jumps around more radically than the standard model predicts it should. It has more large jumps and less small jumps than what would fit the bell curve. Mandelbrot (a pioneer in the field -- he invented the word "fractal") offered fractal models as a substitute for the standard statistical models. These graphs have proved to be better at modeling real-world data in areas like economics than the more commonly used bell curve.
To create a fractal graph, first specify a set of points that make up the basic structure of the graph. There must be at least three points, they cannot be in a straight line, and no two points may have the same x value. Enter the x and y values of the first point, click Add Point, enter the x and y values of the second point, etc. Make sure you click Add Point after every point. When you are done adding points to the graph, click Done. You will then be able to modify the properties of each segment of the graph or display it. The drop-down list box allows you to select a segment of the graph to modify. The h variable controls how chaotic each section of the graph is. It must have a value from -1 to 1. Zero will make that section a straight line. If h is negative, that section of the graph will be an upside-down image of the whole graph. If backwards is checked, that section of the graph will be a backwards image of the whole graph. To view the graph, click Draw Graph. To start over with a new set of points, click New Graph.
If you want to slightly modify one of the two colors you have already made, click the box that it is in and the color will be set on the scrollbars.