Zometerns

Zometerns are a theoretical class of discrete geometric-topological constructs used to model spatial partitioning and information flow in networks. A zometern consists of a finite, connected collection of cells, each cell a polyhedral region, arranged so that neighboring cells share a boundary interface called a seam. Seams meet at junctions, and the incidence relations among seams and junctions define the connectivity of the structure. The term is used in discussions of mesh-like decompositions and has no single canonical form, allowing a variety of layouts to be studied under a common framework.

Construction and representation: A zometern is commonly described by a cell complex together with an adjacency relation that records which cells touch and along which seam. Each seam is tied to exactly two cells, or to a boundary in the case of open structures. Junctions represent the points where multiple seams converge. The geometry can be Euclidean or abstract, with a metric or purely combinatorial interpretation.

Properties: Zometerns admit dual descriptions, where cells and seams exchange roles; invariants such as the Euler characteristic and genus can be defined for closed structures. Operations include refinement (subdividing cells), contraction (merging adjacent cells along a seam), and seam rewiring (changing which cells meet along a seam) while preserving overall connectivity. Many zometerns can be embedded in surfaces of various genus, and their dual graphs provide alternate viewpoints on adjacency.

Applications and examples: As a modeling tool, zometerns are used to study mesh generation, geometric partitioning, and topological data analysis, where the goal is to balance cell quality with connectivity constraints. A simple example is a four-cell arrangement around a central seam, forming a cross-shaped pattern on a plane. More complex instances appear in simulations of spatial networks and architectural tiling.