Construction of Tilings with Transitivity Properties on the Square Grid

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.