Abstract
Tilings have been a subject of interest in discrete geometry for their algebraic and geometric properties. Their applications extend to other areas such as crystallography and mathematical morphology. In this paper, we study a class of tilings called \(\left(a,b,c\right)\) tilings where their respective symmetry groups form \(a\) orbits of vertices, \(b\) orbits of edges and \(c\) orbits of tiles. A method is presented where \(\left(a,b,c\right)\) tilings are derived and enumerated from the square grid using groups of symmetries involving 4-fold rotations. Conditions on \(a\), \(b\) and \(c\) are provided for \((a,b,c)\) tilings that arise. Using these conditions, examples of \((a,a-1,c)\) tilings are obtained.
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Tomenes, M.D., De Las Peñas, M.L.A.N. (2024). Construction of Tilings with Transitivity Properties on the Square Grid. In: Brunetti, S., Frosini, A., Rinaldi, S. (eds) Discrete Geometry and Mathematical Morphology. DGMM 2024. Lecture Notes in Computer Science, vol 14605. Springer, Cham. https://doi.org/10.1007/978-3-031-57793-2_10
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DOI: https://doi.org/10.1007/978-3-031-57793-2_10
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