Spacetiling

Spacetiling, also referred to as space tiling, is the partition of Euclidean space into a set of non-overlapping cells that completely cover the space. A tiling is periodic if it repeats under translations; otherwise it may be aperiodic. In two dimensions, common examples include the square tiling, the equilateral triangular tiling, and the hexagonal tiling. In three dimensions, space tilings include the cubic honeycomb by cubes, the tetrahedral-octahedral honeycomb, and tilings by rhombic dodecahedra or truncated octahedra. Some tilings are face-to-face, meaning every shared boundary between adjacent tiles is a full face of both tiles.

Constructing tilings can involve lattice-based methods, such as Voronoi or Delone tilings derived from a point lattice. Finite polyhedra capable of tiling space are called space-filling or tessellating polyhedra; many examples exist, including the cube, truncated octahedron, rhombic dodecahedron, and certain pentagonal shapes in non-Euclidean spaces. In two dimensions, non-periodic tilings such as the Penrose tiling show that aperiodic tilings can cover the plane without translational symmetry. In higher dimensions, tiling theory studies space groups and symmetry constraints that govern tilings.

Applications of spacetiling appear in crystallography, where atomic arrangements approximate tilings of space, in computer graphics and architectural design for efficient packing and texture synthesis, and in mathematical research exploring tiling invariants and tiling theorems.