Simple Fractal Tilings

Method 3

Take M to be this matrix. Note det(M) = 2. Here is the parallelogram, with a residue system, (0,0) and (-1,0), indicated by the circles.

Here is M-1

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

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

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

Here is the IFS table and corresponding tile

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

Here is another residue system, IFS, and tile.

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

Return to Method 3.