  For minimal $\mathbb{Z}^{2}$-topological dynamical systems, we introduce a
cube structure and a variation of the regionally proximal relation for
$\mathbb{Z}^2$ actions, which allow us to characterize product systems and
their factors. We also introduce the concept of topological magic systems,
which is the topological counterpart of measure theoretic magic systems
introduced by Host in his study of multiple averages for commuting
transformations. Roughly speaking, magic systems have a less intricate dynamic
and we show that every minimal $\mathbb{Z}^2$ dynamical system has a magic
extension. We give various applications of these structures, including the
construction of some special factors in topological dynamics of $\mathbb{Z}^2$
actions, and a computation of the automorphism group of the minimal Robinson
tiling.
