Construction of Tilings with Transitivity Properties on the Square Grid

• Main Authors: Tomenes, Mark D, De las Penas, Ma. Louise Antonette N
• Format: text
• Published: Archīum Ateneo 2024
• Subjects: Square grid; Square lattice; Tilings; Vertex-edge-tile transitivity; Discrete Mathematics and Combinatorics; Geometry and Topology; Mathematics; Physical Sciences and Mathematics
• Online Access: https://archium.ateneo.edu/mathematics-faculty-pubs/269; https://doi.org/10.1007/978-3-031-57793-2_10
• Institution: Ateneo De Manila University

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 a,b,c 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 a,b,c 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.