Classic Iterated Function Systems

The Golden Self-Contracting Angles

The golden ratio is \[ \phi = \frac{1+\sqrt{5}}{2}. \] It is the unique positive solution to the equation \(x^2-x-1=0\), and its reciprocal \(1/\phi\) is the unique positive solution to \({x^2+x-1=0}\). Two useful properties of \(\phi\) are that \[ \begin{align} \frac{1}{\phi} &= \phi-1 \\ \frac{1}{\phi^2} &= 2-\phi \end{align} \] See the Wikipedia article for more information.

Below is the graph of the self-contacting scaling factor as a function of $\theta$, along with the horizontal line at \(1/\phi \approx 0.61803\). We see that there are exactly four intersections. We claim that these occur at $\theta$ = 60°, 108°, 120°, and 144°.

Case: $\theta = 60^\circ$

When $0 \lt \theta \lt 90^\circ$, the self-contacting scaling factor $r$ is the unique solution [Proof] to the equation \[\sum_{k=0}^{N-1} r^{k+2}\cos(k\theta) = \frac{1}{2}\] where $N$ is the smallest integer such that $N\theta \ge 90^\circ$. If $\theta$ = 60°, then $N$ = 2, and the equation for $r$ is \[ \frac{1}{2} = r^2 + r^3\cos(60^\circ) = r^2 + \frac{r^3}{2} \\ \\ \Rightarrow 0 = r^3 + 2r^2 - 1 = (r^2 + r - 1)(r+1) \] Therefore $r$ satisfies \(r^2+r-1=0\), and hence \(r = 1/\phi\).

Case: $\theta = 108^\circ$

For $90^\circ \lt \theta \lt 135^\circ$, the self-contacting factor $r$ satisfies the equation [Proof] \[ (*) \quad 2\cos(2\theta)r^2 + 2\cos(\theta)r + 1 = 0 \]