In this paper, we give several pictorial fractal representations of the ample or Kähler cone for surfaces in a certain class of $K3$ surfaces. The class includes surfaces described by smooth $(2,2,2)$ forms in ${\mathbb P^1\times\mathbb P^1\times \mathbb P^1}$ defined over a sufficiently large number field $K$ that have a line parallel to one of the axes and have Picard number four. We relate the Hausdorff dimension of this fractal to the asymptotic growth of orbits of curves under the action of the surface's group of automorphisms. We experimentally estimate the Hausdorff dimension of the fractal to be $1.296 \pm .010$.

We investigate the number of integral solutions possessed by a pair of diagonal cubic equations in a large box. Provided that the number of variables in the system is at least fourteen, and in addition the number of variables in any non-trivial linear combination of the underlying forms is at least eight, we obtain an asymptotic formula for the number of integral solutions consistent with the product of local densities associated with the system.