tilingsarrangements

Tilings arrangements refers to the systematic study of covering a surface with shapes, called tiles, without gaps or overlaps and often under specified matching rules. In mathematics, it encompasses tilings of the plane or of other surfaces, using one or more tile shapes, and focuses on the possible layouts or arrangements rather than the tiles alone. Tilings may be edge-to-edge or allow tiles to meet at vertices, and they can involve congruent tiles (monohedral tilings) or multiple shapes (polyomino tilings, isohedral and more general tilings).

Common questions include whether a region can be tiled (tileability), how many distinct tilings exist (enumeration), and how the tilings behave under symmetry. Periodic tilings exhibit repeating patterns, while aperiodic tilings admit no translational symmetry. Notable examples include Penrose tilings, which use two shapes with matching rules to force nonperiodicity, and sets of Wang tiles that enforce aperiodicity. Domino tilings, covering regions with 2×1 rectangles, form a central, tractable subproblem studied with algebraic methods.

Approaches to study tilings include formulating the problem as an exact cover instance, using algorithms such as Algorithm X with Dancing Links to find explicit tilings. Counting tilings can be done with transfer-matrix methods, generating functions, or determinant-based formulas in special cases. The general tiling problem is computationally challenging; tileability questions for general polyominoes are NP-complete.

Applications of tilings arrangements appear in architecture and art, as well as in physics and materials science, where tilings model patterns in quasicrystals and composite materials. The field, known as tiling theory, intersects geometry, combinatorics, algebra, and computer science, studying how local matching rules shape global structures.