A Combinatorial Coordinate System for the Vertices in the Octagonal $C_{4}C_{8}(S)$ Grid

We define a new integer-valued combinatorial coordinate system for the vertices in the octagonal C 4 C 8 (S) grid. We review the existing coordinate systems proposed in the literature and provide formulas for the conversion between our and existing coordinate systems, as well as with Cartesian coordinates. The neighborhood relation between vertices can be obtained through arithmetic operations on vertex coordinates.