Every  projective threefold admits a fan structure on its moving cone coming from Mori theory. In particular, there is such  a fan on the Dolgachev-Nikulin-Voisin  K3 mirror family of degree 2d. Its relevance comes from applications to toroidal compactifications of polarised K3 surfaces, as carried out by Gross, Hacking, Keel and Siebert, and the fact that it gives a combinatorial description of the boundary of the KSBA compactification of stable K3 pairs.