Polygonina

Polygonina is a generalized geometric concept used to describe polygon-based polyforms. It unifies families such as polyominoes, polyiamonds, and polyhexes by using a fixed base polygon as a building block.

Definition and scope. For a fixed convex base polygon P with equal edge lengths, a polygonina is a finite, edge-to-edge connected union of translated copies of P in the plane. Copies may be rotated, and shapes are considered up to congruence. The number of constituent copies is called the polygonina’s order, denoted by n. The boundary of a polygonina is a polygonal curve whose edges are congruent to the sides of P. The concept is parameterized by the choice of base polygon P and by the arrangement of adjacency between copies.

Relation to other polyforms. When P is a square, polygonina coincides with polyominoes; when P is an equilateral triangle, it corresponds to polyiamonds; with P a regular hexagon, it relates to polyhexes. The polygonina framework therefore provides a common language for describing and comparing a wide range of edge-to-edge polygonal shapes.

Properties and operations. Key properties include connectedness and edge-to-edge contact. Shapes are built by attaching copies of P along shared edges, allowing growth in a modular fashion. Enumeration problems ask how many distinct polygoninas of order n exist for a given base polygon P, a topic of combinatorial interest. Transformations such as rotation or reflection preserve the classification up to congruence.

Applications. Polygoninas appear in puzzle design, tiling theory, and modular construction models, where complex shapes arise from simple, repeatable units. Related topics include tilings, polyforms, and symmetry groups.

See also. Polyomino, Polyiamond, Polyhex, Tiling theory.