Polyedreen

Polyedreen is a term used in geometry to denote a class of space-filling polyhedra, i.e., polyhedra that can tessellate three-dimensional Euclidean space by copies of themselves with no gaps or overlaps. The concept emphasizes the cell as the fundamental unit in a polyhedral honeycomb and is used to study how different polyhedra can tile space and what symmetries such tilings exhibit. The name combines “polyhedron” with the -een suffix common in naming geometric forms.

In a polyedreen tiling, the faces of each cell are polygons that meet at edges and vertices so that around every edge the dihedral angles of neighboring cells sum to 360 degrees. The study of polyedreens overlaps with isohedral tilings (where the tiling has transitive symmetry on the cells) and with the broader classification of space-filling polyhedra. Classic examples of polyedreen cells used in tilings include the cube, the truncated octahedron, and the rhombic dodecahedron, each supporting a well-known space-filling arrangement.

Applications of polyedreen theory appear in crystallography, materials science, architecture, and computer graphics, where tilings by polyhedra model crystal frameworks, porous media, and polygonal meshes. The concept provides a framework for analyzing how local face- and edge-geometry determines global tiling possibilities and symmetry properties.

See also: polyhedron, tessellation, honeycomb, uniform tiling, symmetry group. References to historical work by Kepler and Coxeter appear in advanced texts on three-dimensional tilings.