polygoneja

Polygoneja is a theoretical concept in geometry used to describe a class of shapes formed by the union of two or more polygons that meet along their edges. In a polygoneja, the constituent polygons are connected, and their interiors may touch only along shared edges or at common vertices, ensuring the overall figure is a single connected region. The term combines poly- meaning many with the idea of a polygon, and the suffix -ja is used here to denote a class or collection.

Definition and structure: A polygoneja is specified by a decomposition into simple polygons P1, P2, ..., Pk such that the intersection of Pi and Pj is either empty, an entire shared edge, or a shared vertex, and the union of all Pi equals the polygoneja. The boundary of a polygoneja is a piecewise-linear closed curve, and the topology can be represented by the planar graph formed by the polygon edges.

Classification: Polygonejas can be convex, rectilinear, or non-convex, and they may be simple (with a single boundary) or have holes (leading to higher genus in a topological sense). Area and perimeter relate to their decomposition: the area is the sum of the areas of the constituent polygons minus any overlaps along shared edges, while the perimeter reflects the outer boundary after internal edges are merged.

Construction and applications: Polygonejas can be generated by tiling a region with polygonal pieces and merging adjacent pieces along common edges, or by taking unions of polygons that satisfy the adjacency rules. Applications appear in computational geometry, computer graphics, geographic information systems, and tiling theory, where understanding decompositions, boundaries, and dual graphs is useful.

Examples: A rectangle can be viewed as a polygoneja with a single constituent polygon; an L-shaped figure can be formed by three rectangles sharing edges. See also polygon, polygon union, tiling, planar graph, polyomino.