The 3-dimensional Lyness map and a self-mirror log Calabi-Yau 3-fold

The 2-dimensional Lyness map is a 5-periodic birational map of the plane which may famously be resolved to give an automorphism of a log Calabi-Yau surface, given by the complement of an anticanonical pentagon of $(-1)$-curves in a del Pezzo surface of degree 5. This surface has many remarkable properties and, in particular, it is mirror to itself. We construct the 3-dimensional big brother of this surface by considering the 3-dimensional Lyness map, which is an 8-periodic birational map. The variety we obtain is a special (non-$\mathbb Q$-factorial) affine Fano 3-fold of type $V_{12}$, and we show that it is a self-mirror log Calabi-Yau 3-fold.