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

Abstract

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.

Affine coxeter group Wa(A4), quaternions, and decagonal quasicrystals

“…In this work we study the orthogonal projection of the root and weight lattices of 4 A with a different technique and show that it has a close correspondence with the pentagrid method of de Brujin [8] and the orthogonal projection of the Voronoi cell of the root lattice exhibits the nested cartwheel structure as suggested by Al Ajlouni [14]. We note that the projection of the hypercube in D5 can be explained in terms of the projection of the 5 B lattice generated by its short roots with a Coxeter-Weyl point group 5 () WB which admits the group 4 () WA as a maximal subgroup.…”

“…Let us remember that the facets of the Voronoi cells here are the rhombohedra as was shown in Figure 4. The work of R. A. Al Ajlouni [14] has pointed out that the ancient Muslim designers were able to construct a variety of quasi-periodic patterns. After examining a large number of Islamic patterns she concludes that the underlying basic structure, invariant under 10-fold rotations, is the quasi-periodic cartwheel pattern just like the one which we illustrated in Figure 11.…”

Tiling approach for the description of the sevenfold symmetry in quasicrystals

“…We expect that our mathematical models will help researchers to better understand the ordering phenomena in soft matter and relating materials, to explain the appearance of anomalous symmetries in colloidal layers, and to design new types of photonic crystals, artificial solids, metamaterials, and so forth [46][47][48][49][50][51][52][53]. We believe that the nature of the sevenfold symmetry will become a little less puzzling in both science and art [54][55][56][57].…”

Cognitive Bias and Claims of Quasiperiodicity in Traditional Islamic Patterns

“…1 Levine and Steinhardt [16] modelled quasicrystals as solids with three properties: long-range correlation in the orientation of atomic bonds, a mass density function that is quasiperiodic (a sum of periodic functions with incommensurate periods), and a local finiteness condition. On a global scale, such structures have no translational symmetry-each atom occupies a unique position different from any other.…”