Multipolige

Multipolige is a term used in geometry to describe a class of plane figures formed by joining multiple polygonal cells edge to edge. It generalizes polyominoes and polyforms by allowing the constituent cells to be arbitrary polygons rather than only squares or hexagons. Multipoliges are studied in combinatorial geometry and tiling theory as a way to model complex composites and layouts.

Definition: A multipolige of a given polygon family P is a finite, connected union U of cells from P, embedded in the plane, such that any two cells either do not intersect or meet along a full edge or at a vertex. Each interior edge is shared by exactly two cells, and the boundary of U is the outer boundary formed by the non-shared edges. If U has no holes, it is simply connected; otherwise it has genus > 0 or boundary components.

Classification: Multipoliges are categorized by order (the number of cells), by the polygon types used, by topology (number of holes), and by symmetry. They can be considered up to rigid motions of the plane (translations, rotations, reflections).

Construction and examples: A simple multipolige with four squares is a polyomino; replacing squares with regular pentagons yields a pentagonal multipolige. A tri-hex multipolige uses three hexagons arranged edge-to-edge. Mixed families may combine triangles, squares, and hexagons.

Applications: They serve as a generalization framework for tiling puzzles, materials design, and computational geometry, where the arrangement of polygonal units affects packing, coverage, and rigidity.

Etymology and related concepts: The name derives from multi- (many) and polygon, reflecting the use of many polygonal cells. Related notions include polyomino, polyform, tessellation, and polygonal tiling.