The golden $L$L is a translation surface that has received much attention recently in the theory of billiard dynamics. It is built out of a $1 \times 1$ square with $1 \times \phi$ and $\phi\times1$ rectangles glued to it along their length-$1$ sides. See for example thisthe preprint preprintDavis and Lelievre - Periodic paths on the pentagon, double pentagon and golden L.