ettiler

Et tiler is a term used in tiling theory to describe a family of plane tilings generated by an abstract substitution process on a finite set of prototiles. In this context, a tiling is built by repeatedly replacing each prototile with a prescribed arrangement of tiles that belong to the same set, typically scaled by a common factor. The resulting pattern may exhibit various degrees of symmetry, and the infinite tiling can be either periodic or aperiodic depending on the chosen substitution rules.

Construction and properties are defined in terms of local rules and a global growth scheme. A single substitution rule assigns, to every prototile, a finite local pattern. Applying the rule iteratively yields larger and more complex regions that cover the plane without gaps or overlaps. If the rule enforces translational symmetry, the tiling is periodic; if it allows only non-repeating arrangements, the tiling is aperiodic. Et tiler thus serves as a framework for exploring how simple local constraints influence global structure, including questions of determinism, decidability, and tileability.

Usage and significance are mainly theoretical, with applications in mathematical modeling, computer graphics, and the study of symmetry. The term is often encountered in expository discussions or as a placeholder for illustrating substitution tilings, and it is not tied to a single canonical construction. Related topics include periodic tilings, aperiodic tilings, substitution tilings, and Wang tiles. Further reading generally covers tiling theory and substitution systems.