Derived from the traditional principles of Islamic geometry, a methodology for generating non-periodic long-range sequences in one-dimension for 8-fold, 10-fold, and 12-fold rotational symmetries

Abstract

This paper explores the use of traditional principles of Islamic geometry to generate the long-range order of different non-period sequences in one-dimension. The discovery of quasicrystals with forbidden long-range order has uncovered a range of puzzling non-periodic symmetries, triggering a variety of investigations into understanding their complicated structural logic. The discovery of historical patterns in Islamic architecture with similar quasi-periodic formations is providing new insights into understanding the mathematical properties of these complicated formations. This paper argues that the hierarchical self-similar logic underlying the construction process of the historical patterns can serve as the key ingredient for resolving the long-range order of the one-dimensional non-periodic sequences. A case study approach is employed to test the hypotheses. Three historical patterns with non-period symmetries were tested: 10-fold rotational symmetry, 8-fold rotational symmetry and 12-fold rotational symmetry. The results show that the methodology is successful in generating three different non-periodic sequences in one-dimension.