polygonforms

Polygonforms is a term used in geometry and digital art to describe families of polygon-based shapes generated by formal rules applied to a base polygon. A polygonform begins with a simple polygon and undergoes transformations that introduce new polygonal components while preserving the underlying polygonal character. Depending on the method and dimensional context, polygonforms can exist as planar figures or as polyhedral surfaces assembled from polygonal faces. The concept emphasizes combinatorial structure and constructive rules rather than smooth curves.

Common construction methods include substitution on edges, where each edge is replaced by a motif consisting of smaller polygons, and vertex-driven transforms, where vertices are relocated along specified directions to form a larger network. Iterative application of these rules yields complex, self-similar patterns akin to fractals, but built from polygons. Parameters such as scaling factors, rotation angles, and replacement motifs control the shape family and determine convexity, symmetry, and tiling behavior.

Polygonforms are defined by their incidence structure—the way vertices, edges, and faces meet. Many exhibit discrete symmetries described by dihedral or cyclic groups; in 3D realizations, they form polyhedral meshes that retain polygonal faces. The mathematical study often uses graph theory and polyhedral combinatorics, with attention to planarity, tiling, and Euler characteristics in closed networks.

Simple examples include squareforms, where a base square is recursively subdivided into smaller squares, and hexiforms, built from hexagonal motifs. Variants may emphasize convex or concave shapes, regular or irregular tilings, and different iteration depths. Some polygonforms approximate smooth surfaces through dense polygonal meshes.

In practice, polygonforms appear in mathematical visualization, algorithmic art, and architectural tiling studies. They offer a framework for exploring how local polygonal rules produce global structure and provide educational models of recursion and symmetry. The term is more common in informal or digital-art contexts than in formal geometry literature.