Digital distances on the 4-fold pentille tessellation

Abstract

4-fold pentille tessellation is in two of the known eight classes of periodic tessellations with convex pentagons with side-to-side property and it is also known as Cairo pattern. In these tessellations, the tiles occur in four different orientations and each pentagon tile has five side-neighbors (neighbor pentagons such that a full side is shared) and seven corner-neighbors (pentagons such that there is at least one point shared on the boundary, i.e., at least a corner). Digital, i.e., path-based distances were recently computed on this grid for side-neighbor criteria in Kov & aacute;cs et. al.: Distance on the Cairo pattern. Pattern Recognit. Lett. 145: 141-146 (2021),1, where the tiles are addressed by coordinate six-tuples. In this paper our aim is twofold, on the one hand, we give a much less complex description of the grid based on coordinate pairs, on the other hand, we compute also the distance based on corner-neighbors. Our result applies to two of the classes of the known 8 periodic side-to-side pentagonal tessellations based on a sole convex pentagon tile.