A digital distance on the kisrhombille tiling

Abstract

The kisrhombille tiling is the dual tessellation of one of the semi-regular tessellations. It consists of right-angled triangle tiles with 12 different orientations. An adequate coordinate system for the tiles of the grid has been defined that allows a formal description of the grid. In this paper, two tiles are considered to be neighbors if they share at least one point in their boundary. Paths are sequences of tiles such that any two consecutive tiles are neighbors. The digital distance is defined as the minimum number of steps in a path between the tiles, and the distance formula is proven through constructing minimum paths. In fact, the distance between triangles is almost twice the hexagonal distance of their embedding hexagons.