3Dtilasta

3Dtilasta is a theoretical framework for constructing and analyzing tilings of three-dimensional space using a finite set of polyhedral tiles. It generalizes two-dimensional tiling concepts to the volumetric domain by enforcing local compatibility conditions on tile facets to achieve gap-free assemblies. The framework emphasizes regularity and scalability, supporting both periodic arrangements and certain classes of aperiodic tilings suitable for complex volumetric structures.

Tiles in 3Dtilasta are polyhedra with labeled faces and oriented edges. Adjacent tiles must match face labels and orientation, preventing mismatches. Subdivision rules allow simple tiles to generate larger composites, enabling hierarchical tilings for efficient storage and rendering. The mathematical core combines graph theory with group actions to classify admissible tilings under symmetry constraints.

Algorithmically, 3Dtilasta uses constructive rules, local replacement schemes, and search-based generation to produce tilings. It also studies decidability and complexity of tiling problems in 3D, noting that verification of a given tiling can be computationally intensive. Tools from computational geometry and voxel-based representations support simulation and visualization.

Applications include architecture and digital fabrication, where repeatable 3D tilings can guide modular construction, as well as computer graphics and volumetric data compression, where compact tiling representations reduce storage while preserving topology. The concept remains primarily theoretical, with ongoing research exploring existence results, classification of tile sets, and practical algorithms for real-time generation.

In scholarly contexts, 3Dtilasta is discussed alongside classical tiling theory, convex polyhedra, and lattice-based methods. While not tied to a specific software project, it provides a framework for comparing 3D tiling techniques and exploring the limits of aperiodicity in three dimensions.