Origins of The Chaos Game Algorithm
In his handbook, Barnsley used the following example to illustrate how the algorithm works. Note that he used a triangle (n = 3), r = 1/2, and recommended a large N value, actually in the millions. He also used a numbering system to classify points and vertices.
"Suppose now that you have a die which only shows the numbers 1, 2, and 3 on its faces. Here is how the Chaos Game is played: Mark the three fixed points on a piece of paper, and label them 1, 2, and 3. Choose one additional point wherever you like on the paper. Call it (x0, y0). Roll the die. Mark the midpoint between the point labelled with the same number as shows on the die and (x0, y0), and label this new point (x1, y1). What you have done is to apply the transformation selected by the roll of the die to the point (x0, y0). Again roll the die. Mark the midpoint between the point labelled with the same number as shows on the die and (x1, y1), and label the new point (x2, y2). Repeat this process over and over again to obtain a long sequence of points, preferably millions of them..."
The result of playing the game in this way is the Sierpinski Triangle, one of the most famous fractals. Feel free to reproduce this pattern with our tool. The tool will compute the corresponding pattern and its fractal dimension D where
D = log(n)/log(1/r).
Origins of the Game Name
At this point you might wonder where the name of the game came from. Barnsley wrote in his handbook:
"The Chaos Game Algorithm is the name we use in the Desktop Fractal Design System for the Random Iteration Algorithm, as defined in Fractals Everywhere. How do we know that this algorithm will produce the same image over and over again, independent of the particular sequence of random choices that are made? This remarkable result was first suggested by computergraphical mathematics experiments and later given a rigorous foundation by Iterated Systems Inc.'s mathematician John Elton."
Applications to DNA Sequences
The Chaos Game has found practical applications as a method for representing gene strutures and DNA sequences (Jeffrey, 1990), and as a tool for graphical representation of English text and genetic sequences. The game provides a visual image of a DNA sequence that is different from the traditional linear arrangement of nucleotides (Aswathi, 2009).
Recently, Yin (2017) proposed a novel method based on a variant of the chaos game for encoding a DNA sequence into three integers, containing all sequence information, and with possible applications to DNA sequence compressions, encryption, and steganography.
Ongoing Research
We are currently testing some modifications of the algorithm. We expect to document the results online so third parties might be interested in reproducing them.
