Geodesic
Local algorithm for constructing the derived tilings of two-dimensional torus
Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 2, Tome 479 (2019), pp. 85-120

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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 $\mathcal{T}$. 
@article{ZNSL_2019_479_a3,
     author = {V. G. Zhuravlev},
     title = {Local algorithm for constructing the derived tilings of two-dimensional torus},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {85--120},
     publisher = {mathdoc},
     volume = {479},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2019_479_a3/}
}
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V. G. Zhuravlev. Local algorithm for constructing the derived tilings of two-dimensional torus. Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 2, Tome 479 (2019), pp. 85-120. http://geodesic.mathdoc.fr/item/ZNSL_2019_479_a3/