4.5 Spiral

Using rotations, you can get very interesting spiral shapes. An interesting effect can be obtained with just two transformations:

• Transformation 1: rescale the figure by 90% and rotate by 20 degrees.
• Transformation 2: rescale the figure by 10%, rotate by 20 degrees and shift by 1 in the horizontal axis.

Transformations for the spiral. Color indicates the images resulting from applying successive transformations.

Graph obtained with the chaos game algorithm with probabilities \(\pi = [0.9; 0.1]\).

\[ f_1(x, y) = \begin{bmatrix} 0.787 & -0.424 & 1.758647 \\ 0.242 & 0.859 & 1.408 \end{bmatrix} [x \ y \ 1]^T \] \[ f_2(x, y) = \begin{bmatrix} 0.181 & -0.136 & 6.086 \\ 0.090 & 0.181 & 1.568 \end{bmatrix} [x \ y \ 1]^T \]

By adding additional transformations, we can enrich this spiral with a second braid.

The reader is free to experiment with additional braids, just add another diminishing transformation.

Transformations for the double helix. Color indicates the images resulting from applying successive transformations.

Graph obtained with the chaos game algorithm with probabilities \(\pi = [0.7; 0.15; 0.15]\).

\[ f_1(x, y) = \begin{bmatrix} 0.787 & -0.424 & 1.758647 \\ 0.242 & 0.859 & 1.408 \end{bmatrix} [x \ y \ 1]^T \] \[ f_2(x, y) = \begin{bmatrix} -0.121 & 0.257 & -6.721 \\ 0.151 & 0.053 & 1.377 \end{bmatrix} [x \ y \ 1]^T \] \[ f_3(x, y) = \begin{bmatrix} 0.181 & -0.136 & 6.086 \\ 0.090 & 0.181 & 1.568 \end{bmatrix} [x \ y \ 1]^T \]