Rules of the game:

- A point (x,y) is selected at random and drawn. Then the following process starts:
- A four-sided die is rolled in order to choose one of the four transformations given below.
- This randomly chosen transformation is applied to the point most recently obtained and the new point is born and drawn.
This Markov process produces the Apollonian Window with a great accuracy.

The

four transformationsF_{i}: (x, y) --> (x', y') are defined here:

F_{1}:x' = x

y' = - yF_{2}:x' = x / (16x ^{2}+ 16y^{2}- 8y + 1)

y' = (4x^{2}+ 4y^{2}- y) / (16x^{2}+ 16y^{2}- 8y + 1)F_{3}:x' = (x ^{2}+ y^{2}- x - 2y + 1) / ((x-1)^{2}+ (y-1)^{2})

y' = (x^{2}+ y^{2}- 2x - y + 1) / ((x-1)^{2}+ (y-1)^{2})F_{4}:x' = - (x ^{2}+ y^{2}+ x - 2y + 1) / ((x+1)^{2}+ (y-1)^{2})

y' = (x^{2}+ y^{2}+ 2x - y+1) / ((x+1)^{2}+(y-1)^{2})Can you guess what they represent?

Remarks:The Markov chain of transformations that allows one to obtain Sierpinski Gasket as the orbit of a randomly chosen point is a well known example of a so-called "Chaos Game". Here is how you do it:

- Choose tree points in a plane, say
A,B, andC(vertices of a triangle).- Now, pick a random point
Pin the plane.- Pick at random one of the vertices and draw a midpoint
P_{1}betweenPand this vertex.- Pick at random one of the vertices and draw a midpoint
P_{2}betweenP_{1}and this vertex.- Iterate this process, getting a sequence of points (
P_{i}).In a short while the points, except the first few, will form a recognizable, almost perfect, shape of the Sierpinski Gasket.

I do not know who is the author of this idea (let me know if you do). Writing a program that plots these points is however a standard exercise for students of Computer Science, as the writing a rather short script is rewarded by a very aesthetical graphical result.

As for the rules for the Apollonian Window that I gave above, I believe that no simpler formulas would do the job.