The global long-range order of quasi-periodic patterns in Islamic architecture.

Three decades after their discovery, the unique long-range structure of quasicrystals still poses a perplexing puzzle. The fact that some ancient Islamic patterns share similar quasi-periodic symmetries has prompted several scientists to investigate their underlying geometry and construction methods. However, available structural models depend heavily on local rules and hence they were unable to explain the global long-range order of Islamic quasi-periodic patterns. This paper shows that ancient designers, using simple consecutive geometry, have resolved the complicated long-range principles of quasi-periodic formations. Derived from these principles, a global multi-level structural model is presented that is able to describe the global long-range translational and orientational order of quasi-periodic formations. The proposed model suggests that the position of building units, locally and globally, is defined by one framework, and not tiled based on local rules (matching, overlapping or subdividing). In this way, quasi-periodic formations can grow rapidly ad infinitum without the need for any defects or mismatches. The proposed model, which presents a novel approach to the study of quasi-periodic symmetries, will hopefully provide a deeper understanding of the structure of quasicrystals at an atomic scale, allowing scientists to achieve improved control over their composition and structure.