2025/09/25 by Leigh Foster, Foster, Leigh
Computer Science · Materials Science · Mathematics · #05B45 #Advanced Combinatorial Mathematics #Cellular Automata and Applications #Combinatorics (math.CO) #FOS: Mathematics #Quasicrystal Structures and Properties
paper · pdf · doi:10.48550/arxiv.2509.21700
openalex publication_date 2025/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The question of whether a given region can be successfully filled by a finite set of tiles has been commonly studied, and there are many available arguments for whether a given finite region can be tiled. We can show that there is no domino tiling of the mutilated chessboard via a coloring argument, and a slightly more subtle argument for other two-colored square-grid regions using a height function of Thurston. In this paper, we examine finite regions of the hexagonal grid and a set of tiles known as the stone, bone, and snake. Using matrices in SL2(ℂ), we exhibit a new necessary criterion for a region to have a signed tiling by these tiles. This originally arose in a study of the double dimer model.