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

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