Is existence nothing more than the repetition of patterns and fundamental orders? This is the question that patterns pose, like a mirror reflecting reality. Patterns are not merely recurring motifs in the environment; they are ontological structures that allow us to search for order and meaning in the hidden layers of reality. From cosmic orders to human behavioral patterns, patterns display the invisible traces of this ontological question in human-made environments.

The problem of patterns and the questions they raise have been a subject of philosophical inquiry throughout history. Do patterns exist independently, or are they constructs of the mind? Is the world a collection of eternal patterns? Can all phenomena of existence be reduced to a set of fundamental patterns?

Repetition is one of the most enigmatic features of the world. On the one hand, recurring patterns make it possible to understand phenomena, bring order, and enable the continuity of events. On the other hand, these very patterns can turn into closed cycles that make transformation difficult. This raises a fundamental question: Is repetition merely a mechanism for preserving the status quo, or does it also contain within itself the possibility of change? Can we imagine a world in which, at no level, any kind of recurring pattern exists?

This paper presents a morphological study of Islamic geometric patterns (IGP) and their role in application of formal grammar in computational modeling of IGPs. Through a comprehensive literature review and data collection, we analyze the morphological properties of these patterns using techniques such as geometric transformations, pattern classification, and symmetry analysis. Based on our findings, we explore how these properties can be used in constructing a formal grammar through string rewriting system for a CAD application. Building on a study on the potential of a string rewriting system for modeling IGPs, the current research suggests an update to the previous system and introduces a new morphological structure for IGPs. The new method has an expanded sample set and is tested on a class of 5-fold star patterns with 12 members and demonstrates successful development. The results are implemented in a Grasshopper add-on, providing a flexible platform to generate the patterns through strings and to control their parameters. This tool opens up new possibilities to bridge traditional patterns with contemporary technologies and make them more accessible. Furthermore, it contributes to the preservation of IGPs as a significant cultural and architectural heritage, while also advancing the evolution of these patterns to a new and contemporary generation.

In the oriental practice of art and architecture, and among the regions under their influence, Islamic geometric patterns (IGPs) have been widely used, not only due to aesthetics and decoration but also to make it possible to cover wide flat surfaces, curved surface of domes, and perforated surfaces of window and partitions, with perfectly tessellated shapes. However, with advances in time and technology, these techniques could not connect to the new technologies and benefit from the capacities of digitalization. Recent progress in science and technology tends to open new doors to study geometrical patterns by digitalizing the old ones and developing new variations. This study looks at formal grammar and computer science to introduce a new approach to digital visualization of available IGPs, particularly, star patterns. We investigate the potentials of developing a re-writing system for simulation of IGPs to provide a flexible platform, which allows introducing IGP to CAD/CAM software without previous knowledge on their design or drawing techniques. This methodology allows designers to directly develop various scenarios of IGP applications and implement them on related CAD/CAM tools. Formal language and grammar theories, based on applied mathematics are contributing to the advancements of computer science and digital modeling. They can provide an opportunity to express relational definition and written equivalents of the geometries by using strings and symbols. It is supposed that by using the formal grammar frameworks, certain languages could be developed to visualize IGPs in a machine-friendly way, and consequently, this computational interpretation of IGPs facilitates their application and further developments, for example, regards to digital fabrication. The presented method of IGP visualization is developed as a C#-based add-on for Grasshopper in Rhino3D, one of the main modeling tools used by architects and product designers.