Digital Geometry on the Dual of Some Semi-regular Tessellations

Abstract

There are various tessellations of the plane. There are three regular ones, each of them using a sole regular tile. The square grid is self-dual, and the two others, the hexagonal and triangular grids are duals of each other. There are eight semi-regular tessellations, they are based on more than one type of tiles. In this paper, we are interested to their dual tessellations. We show a general method to obtain coordinate system to address the tiles of these tessellations. The properties of the coordinate systems used to address the tiles are playing crucial roles. For some of those grids, including the tetrille tiling D(6, 4, 3, 4) (also called deltoidal trihexagonal tiling and it is the dual of the rhombihexadeltille, T(6, 4, 3, 4) tiling), the rhombille tiling, D(6, 3, 6, 3) (that is the dual of the hexadeltille T(6, 3, 6, 3), also known as trihexagonal tiling) and the kisquadrille tiling D(8, 8, 4) (it is also called tetrakis square tiling and it is the dual of the truncated quadrille tiling T(8, 8, 4) which is also known as Khalimsky grid) we give detailed descriptions. Moreover, we are also presenting formulae to compute the digital, i.e., path-based distance based on the length of a/the shortest path(s) through neighbor tiles for these specific grids.