Surfacesuch

Surfacesuch is a term used in theoretical geometry to denote a class of two-dimensional smooth manifolds assembled from congruent surface patches. By construction, a surfacesuch is compact and orientable and can be embedded in three-dimensional Euclidean space. The defining feature is a decomposition into a finite set of patches, each congruent to a fixed model patch, joined along edges so that tangent planes align, ensuring at least C1 continuity.

Construction typically follows a patch tiling approach: select a model patch P on a base surface, create copies by isometries, and glue along matching edge curves. The global symmetry group of the resulting surface is determined by the arrangement of patches.

Properties include local geometric uniformity, since each point lies in a patch congruent to P; Gaussian curvature varies with the patch geometry; Euler characteristic and genus depend on the patch connections.

Examples: a sphere tiled by eight congruent spherical caps; a torus subdivided into twelve congruent patches. Extensions of the concept allow non-congruent but similar patches or higher-order continuity along seams.

Applications: used as a teaching device in differential geometry curricula to illustrate patchwise construction; used in computer graphics and architectural modeling to study smooth transitions across seams.

History: The term surfacesuch appears in speculative geometry literature to discuss surface assembly with symmetry, without claiming real-world mathematical consensus.

See also: tessellation, patching, surface tiling, differential geometry, geometric modeling.