  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.
