4.3 McWorter’s pentigree

William McWorter in 1990 proposed an interesting family of fractals based on very simple geometric transformations. The fractal below uses six transformations, each of which scales the initial figure by \(\frac{3 - \sqrt 5}{2} \approx 0.38196\), and then rotates it by \(36^\circ\); \(108^\circ\); \(-36^\circ\); \(-108^\circ\); \(-36^\circ\) and \(36^\circ\) degrees and moves it accordingly.

Transformations for McWorter’s pentigree. Color indicates the images resulting from successive transformations.

Graph obtained with the chaos game algorithm with probabilities \(1/6\).

\[ f_1(x, y) = \begin{bmatrix} 0.309 & -0.255 & 0 \\ 0.255 & 0.309 & 0 \end{bmatrix} [x \ y \ 1]^T \] \[ f_2(x, y) = \begin{bmatrix} -0.118 & -0.363 & 0.309 \\ 0.363 & -0.118 & 0.225 \end{bmatrix} [x \ y \ 1]^T \] \[ f_3(x, y) = \begin{bmatrix} 0.309 & 0.225 & 0.191 \\ -0.225 & 0.309 & 0.588 \end{bmatrix} [x \ y \ 1]^T \] \[ f_4(x, y) = \begin{bmatrix} -0.118 & 0.363 & 0.500 \\ -0.363 & -0.118 & 0.363 \end{bmatrix} [x \ y \ 1]^T \] \[ f_5(x, y) = \begin{bmatrix} 0.309 & 0.225 & 0.382 \\ -0.225 & 0.309 & 0 \end{bmatrix} [x \ y \ 1]^T \] \[ f_6(x, y) = \begin{bmatrix} 0.309 & -0.225 & 0.691 \\ 0.225 & 0.309 & -0.225 \end{bmatrix} [x \ y \ 1]^T \]