Abstract
Periodic tilings play a role in the decorative arts, in construction and in crystal structures. Combinatorial tiling theory allows the systematic generation, visualization and exploration of such tilings of the plane, sphere and hyperbolic plane, using advanced algorithms and software. Here we present a “galaxy” of tilings that consists of the set of all 2.4 billion different types of periodic tilings that have Dress complexity up to 24. We make these available in a database and provide a new program called Tegula that can be used to search and visualize such tilings.
Highlights
Two dimensional periodic tilings play a role in the decorative arts, prominent examples being the euclidean tilings that cover the Alhambra palace in Granada, Spain, and M.C
Two dimensional euclidean tilings are used in the construction of floors, walls and roofs
Hyperbolic tilings are used in the analysis of minimal surfaces of three-dimensional crystal structures (Ramsden et al, 2009; Kolbe and Evans, 2018)
Summary
Two dimensional periodic tilings play a role in the decorative arts, prominent examples being the euclidean tilings that cover the Alhambra palace in Granada, Spain, and M.C. There are 2, 395, 220, 319 such tilings We refer to this collection as a “galaxy of periodic tilings” in the title of this paper, because, first, the number of tilings is very big ( not as large as the number of stars in a typical galaxy), and second, when viewing these tilings, the impression is that many look very similar to each other, much like stars in the sky. Each such tiling is represented by its Delaney-Dress symbol and we provide these in a SQLITE database. Tegula and the database of periodic tilings are open source and freely available
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