Diagonios

Diagonios is a theoretical construct used in geometry and combinatorics to generalize the notion of a diagonal from polygons to more complex tilings and surfaces. The term is used mainly in speculative or didactic contexts to explore how diagonal-like connections can organize the structure of a tiling or network.

Definition: Given a finite tiling of a surface by polygons and a set of marked boundary points, a diagonios is a connected sequence of interior diagonals that joins two marked points. Each segment lies inside a tile and follows a diagonal of that tile; consecutive segments meet at tile edges but do not pass through tile corners except at endpoints. A diagonios is simple if it does not self-intersect, and closed if it starts and ends at the same point.

Variants and operations: Diagonios can be combined by concatenation, or transformed by local flips that replace a pair of adjacent diagonios with an alternate pair around a quadrilateral region. Two diagonios are considered equivalent if they are related by an isotopy that preserves the marked points and tiling structure.

Applications and examples: In educational settings, diagonios illustrate how diagonal pathways can trace connection patterns in grids. In theoretical work, they serve as a tool for counting path families, studying tiling symmetries, and modeling networks. The concept is not standard in formal geometry and is primarily found in teaching materials or speculative literature; it derives its name from the diagonal elements of tiles and an -ios suffix.