Symmetric Icons

Symmetric Chaos

Choose a color scheme:
Regular: Blue Cyan Green Yellow Red Magenta	White: Blue Cyan Green Yellow Red Magenta

To use this applet, click on links to change the colors and see other pictures, click to zoom in, and shift-click to zoom out. You can play with the numbers to see if you can invent your own picture -- just enter valid numbers (read on to find out what kind of numbers to use) and click View Icon to see what you've invented. If you create a neat picture that you think I would like to add to this page, see the form at the bottom of this page.

The information I have on symmetric icons, as well as all of the icons listed above, came from one book, Symmetry in Chaos by Michael Field and Martin Golvbitsky. If you like these pictures and want to learn more about the math behind them, I strongly recommend this book.

The formula behind these pictures is z' = z * (lambda + alpha * z * conjugate(z) + beta * real(z ^ n) + omega * i) + gamma * conjugate(z) ^ (n - 1). In this formula, exponents are shown with ^'s. n is the degree of symmetry, z is a complex number corresponding to a point in the image, and i is the square root of -1. The conjugate of a complex number is a complex number with the same real part, but opposite imaginary part. For example, the conjugate of (3 + 2i) is (3 - 2i). To create these pictures, z is set to any initial value. The formula gives a new value for z. The point corresponding to z is hit, and the new value for z is used in the formula. The colors are determined by how frequently a screen pixel is hit.

The formula for these images was invented strictly for the purpose of producing chaotic, yet symmetric, images. When searching for a formula to produce beautiful images, the authors of this formula started with the logistic map formula. They added several variables to fill the complex plane, create or destroy symmetry, and add variety. A brief description of how to use the variables is below:

lambda -- This value must be between -3 and 3. This number is very important in creating chaos and should not be too close to zero.
alpha -- This value should have the opposite sign of lambda. It is also necessary that this value not be too close to zero.
beta -- This number was added to the formula for variety; it is unnecessary. You can leave it as zero or play around with it.
gamma -- This can be any value from -1 to 1. I do not recommend values close to zero.
omega -- This number was added to destroy bilateral symmetry. If this number is set equal to zero, the resulting image will have bilateral symmetry -- that is, if the image is flipped, it will look the same. It doesn't take a large number to have an effect -- I recommend using numbers between -1 and 1.
symmetry -- This can be any whole number greater than two. It gives the rotational symmetry of the image -- it's the number of ways the resulting image can be rotated to get the same image.

If you think I would like the icon you have created, fill out the form below and click Send. This form will send me a copy of the parameters and everything you write in the form. If I like it, I will add it to the web page.

All fields are optional.