Random Fractals and the Stock Market

Symmetric Cartoons

Even with only two turning points (a, b) and (c, d), the totality of possible graph behaviors is difficult to visualize.

This is because a map of which turning points give rise to which behaviors is 4-dimensional: a, b, c, and d can take on any values between 0 and 1, except that we must have a <= c.

One way to get a manageable picture of this is to consider only symmetric cartoons.

That is, if the first turning point is (a,b), the second is (c,d) = (1-a,1-b).

In other words, the second turning point is determined by the first, so specifying (a,b) determines the whole generator.

The turning point (a,b) can lie anywhere in the rectangle 0 <= a <= 1/2 and 0 <= b <= 1.

In the animation below we sample some of the symmetric cartoons.

To make the symmetry more obvious, and the make comparisons clearer, here we do not randomize the choice of UDU.

Click the picture to animate.

Return to Random Fractal Cartoons.