Scaling Ratios and Triangles in Siegel Disks

Let f(z)=e^{2i\pi \theta} + z^2, where \theta is a quadratic irrational. McMullen proved that the Siegel disk for f is self-similar about the critical point, and we show that if \theta = (\sqrt{5}-1)/2 is the golden mean, then there exists a triangle contained in the Siegel disk, and with one vertex at the critical point. This answers a 15 years old conjecture