Local algorithm for constructing the derived tilings of two-dimensional torus

The local structure of the derived tilings $\mathcal{T}$ of two-dimensional torus $\mathbb{T}^2$ is investigated. Polygonal types of the stars in these tilings are classified. It is proved that in the nondegenerate case the tilings $\mathcal{T}$ contain 7 different types of stars and all types are representable by the stars with inner vertices from the crown $\mathbf{Cr}$ of the tiling $\mathcal{T}$. There sets the maximum principle being the basis of the $LLG$ algorithm for layer-by-layer growth of the derived tilings $\mathcal{T}$.