geometioiden

Geometioiden is a term used in geometric discussions to refer to a broad class of objects that generalize polygons and polyhedra by incorporating geodesic boundaries on curved spaces. In this article, geometioiden denotes the plural form, while a single object is called a geometioide. The concept sits at the intersection of polyhedral theory and non-Euclidean geometry, and is used mainly in exploratory or theoretical contexts rather than as a standard category in mainstream literature.

A geometioide is defined as a finite union of geodesic cells glued along shared faces, edges, and vertices, forming a finite cell complex embedded in a space with constant or variable curvature. The combinatorial type is described by incidence relations among cells, analogous to a polyhedron’s face–edge–vertex structure, while the geometric realization depends on the ambient curvature. When the ambient space is Euclidean, geometioiden reduce to ordinary polytopes; in spherical or hyperbolic settings, they become curved analogues with dihedral angles constrained by curvature.

Properties of geometioiden mirror those of classical polyhedra but with curvature effects. Local geometry near a vertex is governed by the sum of incident dihedral angles relative to the ambient curvature, which can differ from 360 degrees in non-Euclidean spaces. Symmetry groups extend traditional polyhedral symmetry to isometries of the ambient space. Operations such as duality, truncation, and tiling constructions have natural generalizations within this framework.

Examples include geometioid triangles and geometioid tetrahedra, as well as tilings of surfaces or three-dimensional manifolds by such objects. In practice, geometioiden serve as a language for studying polyhedral shapes in curved spaces, mesh generation, and computer graphics, while remaining primarily of theoretical or exploratory interest. See also polyhedra, tessellations, non-Euclidean geometry, and Coxeter groups.