Simple Fractal Tilings

Method 3

Take M to be this matrix. Note det(M) = 4. Here if the parallelogram, with a residue system, (0,0), (-1,0), (0,1), and (1,1), indicated by the circles.

Here is M-1

To put this in the standard IFS formalism, we must solve

0 = r*cos(theta) 1/2 = -s*sin(phi) -1/2 = r*sin(theta) 1/2 = s*cos(phi)

obtaining theta = 90, phi = -45, r = -1/2, and s = 1/sqrt(2).

Here is the IFS table and corresponding tile

R S Theta Phi E F -.5 .707 90 -45 0 0 -.5 .707 90 -45 -1 0 -.5 .707 90 -45 0 1 -.5 .707 90 -45 1 1

Here is another residue system, IFS, and tile.

R S Theta Phi E F -.5 .707 90 -45 0 0 -.5 .707 90 -45 -1 0 -.5 .707 90 -45 0 1 -.5 .707 90 -45 -1 -1

Return to Method 3.