Forbidden Symmetries

Abstract

The emergence of quasi-periodic tiling theories in mathematics and material science is revealing a new class of symmetry, which had never been accessible before. Because of their astounding visual and structural properties, quasi-periodic symmetries can be ideally suited for many applications in art and architecture; providing a rich source of ideas for articulating form, pattern, surface and structure. However, since their discovery, the unique long-range order of quasi-periodic symmetries, is still posing a perplexing puzzle. As rule-based systems, the ability to algorithmically generate these complicated symmetries can be instrumental in understanding and manipulating their geometry. Recently, the discovery of quasi-periodic patterns in ancient Islamic architecture is providing a unique example of how ancient mathematics can inform our understanding of some basic theories in modern science. The recent investigation into these complex and chaotic formations is providing evidence to show that ancient designers, by using the most primitive tools (a compass and a straightedge) were able to resolve the complicated long-range principles of ten-fold quasi-periodic formations. Derived from these ancient principles, this paper presents a computational model for describing the long-range order of octagon-based quasi-periodic formations. The objective of the study is to design an algorithm for constructing large patches of octagon-based quasi-crystalline formations. The proposed algorithm is proven to be successful in producing an infinite and defect-free covering of the two-dimensional plane.