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\(\mathcal{F} = (\mathbb{R}^2,T_1\) ,\(R_1T_2\) ,\(R_2T_1\) ,\(R_3T_2\) ,\(R_4T_1\) ,\(R_5T_2\) )
This is a chaos game implementation. The rules for a chaos game are encoded in iterated function systems, or IFS's for short. An IFS is just a bunch of transformations that map points to other points in the plane.
To start a chaos game, pick a random point in the plane and mark it down. Then pick a transformation from the IFS at random and use it to transform the starting point. Mark down the resulting point. Pick another transformation from the IFS at random and use it to transform the new point. Repeat.
After enough moves, a fractal pattern should appear.
The IFS for this chaos game, (labeled \(\mathcal{F}\)), has six transformations. Each transformation maps a regular hexagon, (centered at the origin), to a smaller regular hexagon with a shared vertex. See the figure below.
The transformations are listed in counter-clockwise order, with the first transformation corresponding to the rightmost hexagon.
Each transformation has the form \(RTX\), (which is executed in the order \(X,T,R\)).
First, a rotation or reflection \(X\) is applied that preserves the regular hexagon centered at the origin, (think symmetry). The value of \(X\) can be selected from a drop-down menu for each transformation. \(R_n\) corresponds to a counterclockwise rotation of \(n * 60\) degrees. \(F\) corresponds to a reflection about the x - axis.
Second \(T\) contracts the hexagon down to an appropriate size, then translates the hexagon to the right until the new hexagon shares a right vertex with the original hexagon. \(T_1\) contracts the hexagon by a factor of 1/2, while \(T_2\) contracts the hexagon by a factor of 1/4.
Finally, \(R\) rotates the new hexagon about the origin until it is in the correct vertex position.
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