### Chaos Game Mode

The Chaos Game rules for a fractal define transformations based on scalings, rotations, and reflections with respect to chosen fixed points rather than with respect to the origin in an underlying coordinate system. The idea is to describe rules on how to move from a point X towards a fixed point P.

Scaling by a factor of s means to move 1-s of the distance from X towards the fixed point P.

A rotation of theta degrees (measured counterclockwise) means to rotate X around P by theta.

If the rule is a reflection, then you either reflect the point across a vertical or horizontal line through P.

While in the Chaos Game Mode, the IFS window will show the rules for each fixed point. The first four columns are used to specify the order of moves (since rotations and reflections do not commute, the order is important.) The scaling and rotation columns will show the values of the s and theta parameters respectively. The last two columns give the coordinates of the fixed point.

The following image shows the IFS window for a chaos game consisting of two moves:
1. Scale by .707 (i.e. move 1-sqrt(2)/2 = 0.293 the distance towards the fixed point at the origin), then rotate by 45 degrees.
2. Scale by .707 (i.e. move 1-sqrt(2)/2 = 0.293 the distance towards the fixed point (0.968, 1.030)), then rotate by -45 degrees. This chaos game will produce the Levy Dragon. The coordinate systems for the Fractal space and the Design space indicate the scaling used in the Fractal and Design windows respectively.

The moves of a chaos game give rise to equivalent affine transformations. These transformations can be viewed by choosing either Matrix Compact Form, Matrix Function Form, or Scale/Rotation Form from the Code menu. This will lock those forms, however, so no changes can be made in those windows except for the assigned probabilities. Switch back to Fixed Point Form to make additional changes.

You can save the IFS created from a chaos game to the fractal list. This IFS will still be available if you switch back to the IFS Mode. If you later switch back to the Chaos Game Mode and the IFS is still part of the fractal list, the chaos game moves will still be available in the IFS window and design window. While in the Chaos Game Mode you can save an individual IFS or the fractal list to a file with extension ".chg". This file can then be opened at a later time to restore the chaos game moves. However, if you switch to the IFS mode and save the current IFS or the fractal list to a file, only the IFS code is saved and not the individual moves of the chaos game.