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.

Visually complicated patterns can be found in many cultural heritages of the world. Islamic geometric patterns present us with one example of such visually complicated archaeological ornaments. As long-lived artifacts, these patterns have gone through many phases of construction, damage, and repair and are constantly subject to erosion and vandalism. The task of reconstructing these visually complicated ornaments faces many practical challenges. The main challenge is posed by the fact that archaeological reality often deals with ornaments that are broken, incomplete or hidden. Recognizing faint traces of eroded or missing parts proved to be an extremely difficult task. This is also combined with the need for specialized knowledge about the mathematical rules of patterns’ structure, in order to regenerate the missing data. This paper presents a methodology for reconstructing deteriorated Islamic geometric patterns; to predict the features that are not observed and output a complete reconstructed two-dimension accurate measurable model. The simulation process depends primarily on finding the parameters necessary to predict information, at other locations, based on the relationships embedded in the existing data and in the prior -knowledge of these relations. The aim is to build up from the fragmented data and from the historic and general knowledge, a model of the reconstructed object. The proposed methodology was proven to be successful in capturing the accurate structural geometry of many of the deteriorated ornaments on the Minaret Choli, Iraq. However, in the case of extremely deteriorated samples, the proposed methodology failed to recognize the correct geometry. The conceptual framework proposed by this paper can serve as a platform for developing professional tools for fast and efficient results.

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