tessellationsnormal

Tessellationsnormal is a term used in geometry to denote a class of plane tessellations that satisfy a specific normality condition. The name combines tessellation with the mathematical notion of a normal vector, emphasizing a balance of directional properties along tile edges.

Definition: Let T be a tiling of the Euclidean plane by polygons. T is tessellationsnormal if, at every vertex, the sum of the outward unit normal vectors to the edges incident to that vertex is the zero vector. Equivalently, the edge normals form a locally balanced vector field around each vertex. This condition is invariant under translations, rotations, and uniform scaling.

Examples and scope: Regular tilings by equilateral triangles, squares, and regular hexagons satisfy the condition. Many uniform tilings built from symmetric combinations also do. Some irregular or aperiodic tilings may fail the condition. The concept is primarily used as a diagnostic or design criterion in tiling studies and in computer graphics for mesh regularization.

Properties and implications: The tessellationsnormal condition implies a form of local equilibrium at vertices and is compatible with dual tilings. It interacts with symmetries and can facilitate certain vector-field constructions on the tiling. It is not universal across all tilings and may be sensitive to edge subdivision or vertex smoothing.

Relation and usage: In design and visualization, tessellationsnormal guides tiling choices to ensure balanced directional properties, useful in shading, finite-element meshes, and architectural patterns. The term is not universally standardized and may appear in speculative or pedagogical contexts rather than formal literature.

See also: tiling, tessellation, normal vector, dual tiling, uniform tiling, edge normal.