polygonien

Polygonien are a class of geometric objects used in tiling theory and geometric topology to study surfaces assembled entirely from polygonal faces. They generalize simple polygon tilings by allowing arbitrary identifications of edges and faces, producing finite or infinite 2-dimensional cell complexes that locally resemble a mosaic of polygons.

A polygonien consists of a collection of polygonal faces, their edges and vertices, together with incidence relations describing how faces meet along edges and how edges meet at vertices. It can be described concretely by a tiling pattern on a surface or, equivalently, as a quotient of a regular tiling by a symmetry group. The dual graph of a polygonien encodes the adjacency of faces and provides a convenient combinatorial description.

Polygonien are categorized by symmetry and vertex configuration: regular polygonien have identical faces and uniform vertex figures; semi-regular have a limited set of face types and vertex types; irregular polygonien lack such uniformity. Finite polygonien have an Euler characteristic V − E + F equal to the characteristic of the underlying surface (for example, a sphere or torus), while infinite polygonien may tile the Euclidean or hyperbolic plane. Curvature distributions arise from local face angles.

Examples include the standard square tiling of the plane as a simple polygonien, and the surface tilings of polyhedra like the cube or dodecahedron when viewed as polygonien on a sphere. Polygonien are used in computer graphics for mesh generation, in architectural tiling, in crystallography as models of planar networks, and in pedagogy to illustrate tiling and topology concepts. The term is used in some expository works to describe such polygonal complexes and is related to tilings and polyhedral geometry.