Trajectories and Traces on Non-traditional Regular Tessellations of the Plane

Abstract

In this paper, shortest paths on two regular tessellations, on the hexagonal and on the triangular grids, are investigated. The shortest paths (built by steps to neighbor pixels) between any two points (cells, pixels) are described as traces and generalized traces on these grids, respectively. In the hexagonal grid, there is only one type of usual neighborhood and at most two directions of the steps are used in any shortest paths, and thus, the number of linearizations of these traces is easily computed by a binomial coefficient based on the coordinate differences of the pixels. Opposite to this, in the triangular grid the neighborhood is inhomogeneous (there are three types of neighborhood), moreover this grid is not a lattice, therefore, the possible shortest paths form more complicated sets, a kind of generalized traces. The linearizations of these sets are described by associative rewriting systems, and, as a main combinatorial result, the number of the shortest paths are computed between two triangles, where two cells are considered adjacent if they share at least one vertex.