A digital distance on the kisrhombille tiling

• Published in: Acta crystallographica. Section A, Foundations and advances Vol. 80; no. 3; pp. 226 - 236
• Main Authors: Kablan, Fatma, Vizvári, Béla, Nagy, Benedek
• Format: Journal Article
• Language: English
• Published: 5 Abbey Square, Chester, Cheshire CH1 2HU, England International Union of Crystallography 01.05.2024
• Subjects: digital distance; digital geometry; hexagonal grid; digital geometry; digital distance; hexagonal grid
• ISSN: 2053-2733; 2053-2733
• DOI: 10.1107/S2053273323010628

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. The kisrhombille tiling is the dual of one of the eight semi‐regular tilings and is built up by right‐angled triangles in 12 orientations. In this paper, an appropriate coordinate system is presented and the digital distance is defined and computed by the number of steps of neighboring triangles, where two triangles are considered to be neighbors if they share at least one point on their border.