A journey from the octonionic ℙ² to a fake ℙ²

We discover a family of surfaces of general type with K 2 = 3 K^2=3 and p g = q = 0 p_g=q=0 as free C 13 C_{13} quotients of special linear cuts of the octonionic projective plane O P 2 \mathbb O \mathbb P^2 . A special member of the family has 3 3 singularities of type A 2 A_2 , and is a quotient of a fake projective plane. We use the techniques of earlier work of Borisov and Fatighenti to define this fake projective plane by explicit equations in its bicanonical embedding.