monohedral

Monohedral tiling is a tiling of the plane by congruent copies of a single polygonal shape, called the monohedral tile. The copies are arranged so that the plane is completely covered without gaps or overlaps. In standard definitions, the congruence between tiles is taken to be any rigid motion of the plane, including translations, rotations, and reflections; some authors restrict to orientation-preserving motions, in which case reflections are not allowed.

Common monohedral tilings arise from regular polygons: the equilateral triangle tiling, the square tiling, and the regular hexagon tiling. Other monohedral tilings include the Cairo pentagonal tiling, using a single pentagonal shape arranged in two mirror orientations. More generally, many nonconvex polygons also tile the plane monohedrally.

Not every polygon tiles the plane monohedrally; determining whether a given polygon admits a monohedral tiling is a fundamental problem in tiling theory. For convex pentagons, there is a known classification of the families that tile the plane with congruent copies. The study of monohedral tilings intersects with related concepts such as isohedral tilings, where tiles are interchangeable by the tiling’s symmetry, and with the broader theory of tessellations and aperiodic tilings.

Monohedral tilings are of interest in mathematics, art, and design, illustrating how a single shape can generate complex and regular patterns through repetition and symmetry.