Twist-and-Flip

Yet another chaotic algorithm is the twist-and-flip algorithm. Typically, you do something similar to the following process:

• Choose an initial rotator point r and a set {z_k} of other points, usually in a line or some other plane figure.
• Let z_k(n) denote point z₀ after n iterations. For each point:
  • Rotate point z_k(n-1) around the rotator point r, but not a constant amount (this is the twisting part). The amount a point is rotated will be some function of the distance between z(n-1)_k and r.
  • Flip the point through the origin; that is, if z_i was (x_i, y_i), then it becomes (-x_i, -y_i).
• You have the "lather" and the "rinse" of the algorithm; now, repeat.

So that's one standard algorithm. Just for fun, I decided to expand on this and make something so random-seeming that it might just get me extra-credit, so I wrote a complex number class and used that as the basis (The x-axis is the real part and the y-axis is the imaginary part for each z C). My algorithm is as follows:

• The rotator point r is at (-0.2 + 0.6i).
• The line to be iterated upon is 100,000 points long and has endpoints as determined by the user's initial two mouse clicks.
• For each point z_k, let mod_k be the modulus of (z₀ - r) (that is, the distance between the rotator and the rotatee) and let arg_k be the counter-clockwise angle between the x-axis and z_k. Then:
  • Multiply z_k by 1 / arg_k.
  • Rotate this new point by (mod_k / pi) radians to obtain the new point, z_k+1.
• And of course, repeat for as many iterations as desired.

My algorithm uses some nice properties of complex numbers that make twisting and flipping fairly easy. Let's assume we have to copmlex numbers z and w, which we shall express in polar form as z = r_z cis A_z and w = r_w cis A_w (sorry, but there are no thetas on the keyboard!). Conveniently, we can prove (but will not here) that wz = r_wr_z cis (A_w+A_z). You can see how this has some nice properties for twisting and flipping!

Okay, Gary, I hear you thinking, this is a great algorithm. But can I see your code? How about some pictures? Of course you can! Here they are:

My code: twistflip.cpp My complex class which I used in my code: complex.h Pictures! Both an animated GIF with 10 iterations and the individual iterations themselves.