The interplay of science, art, and technology has long fascinated observers, sparking the creative imagination, fueling innovative breakthroughs, and shaping the trajectory of human progress. It is at the intersection of these realms that we uncover intriguing connections, fostering an appreciation of the beauty that emerges from the complex interplay of order and chaos. The present research aims to delve into this fascinating triad through the exploration of a captivating theme: the “forbidden symmetry.”

“Forbidden symmetry” refers to a category of patterns that, while maintaining a degree of symmetry, evade the expectation of periodic repetition characteristic of traditional symmetry. Quasicrystals, a form of matter that defies the norms of conventional crystallography, exemplify this intriguing type of symmetry in the realm of natural sciences. Simultaneously, in the domain of cultural arts, a striking parallel is found in the elaborate, non-repeating geometric patterns adorning Islamic architectural and design masterpieces.

Taking a step further, we delve into the realm of generative art — an art form that leverages algorithms and computational processes to create work that encapsulates the principles of both quasicrystals and Islamic patterns. Generative art represents an exciting fusion of creativity and automation, where the innate aesthetics of “forbidden symmetry” are translated into an evolving, algorithmic canvas.

In this research, we seek to elucidate the profound interconnections among quasicrystals, Islamic patterns, and generative art, underscoring how these distinct yet intertwined domains offer unique insights into the transcendent beauty of “forbidden symmetry.” Our exploration will lead us down the rabbit hole of complex patterns, into a world where science, culture, and technology meet, offering an enriching perspective on the enduring allure of complexity, order, and symmetry.

Quasicrystals: The Unveiling of the Forbidden Symmetry

Quasicrystals, a unique class of materials, represent a striking departure from our conventional understanding of crystalline structures. In standard crystals such as diamonds, quartz, or salt, the atoms are arranged in a periodic manner. This predictable, repeating arrangement gives rise to their inherent symmetry, leading to the recognizable and orderly shapes that are characteristic of these materials.

However, quasicrystals stand in stark contrast to this familiar paradigm. Despite being made of atoms, just like standard crystals, the patterns in which these atoms arrange themselves in quasicrystals are different. The patterns are deterministic and follow mathematical rules, creating an orderly structure. Yet, unlike standard crystals, these patterns never exactly repeat themselves. This results in a state that is known as quasi-periodicity, giving quasicrystals their unique and intriguing forms of symmetry, an example of the phenomenon referred to as “forbidden symmetry.”

This notion of “forbidden symmetry” in quasicrystals was brought to light in the 1980s by the scientist Dan Shechtman. In a daring challenge to the established views of crystallography, Shechtman discovered that quasicrystals could exhibit five-fold rotational symmetry, something previously considered impossible. In other words, unlike a square crystal that looks the same if you turn it by 90 degrees, or a hexagonal crystal that remains the same if turned by 60 degrees, a quasicrystal could be turned by only 72 degrees (a fifth of a full circle) and still appear the same. This kind of symmetry was once considered forbidden because it violated the rules of periodic crystal structure.

Shechtman’s groundbreaking discovery, for which he was awarded the Nobel Prize, not only revealed an entirely new form of matter but also expanded our understanding of what is possible in nature. It revealed the existence of a beautifully paradoxical combination of order and non-periodicity at the atomic level, embodied by quasicrystals. This discovery has since inspired new ways of thinking in various fields, including materials science, mathematics, and even art, providing a rich source of inspiration for those interested in the fascinating interplay between order and complexity.

Islamic Patterns: The Aesthetic of Non-Repeating Symmetry

Islamic patterns, as an essential aspect of Islamic artistic tradition, present an intriguing parallel to the world of quasicrystals. These ornamental designs, which grace the interiors and exteriors of Islamic architecture, from the majestic domes of mosques to the intricate surfaces of traditional crafts, carry a distinct aesthetic language marked by intricate geometric arrangements and a sense of infinite continuity.

A quintessential characteristic of Islamic patterns is their exquisite intricacy. The fundamental geometric forms — circles, squares, triangles, and polygons — are subdivided and multiplied to create an intricate web of interlacing lines, tessellations, and polygons. This multifaceted complexity makes the patterns visually arresting and imbues them with a sense of depth and dimensionality.

However, the truly remarkable aspect of these patterns lies in their structural arrangement. Unlike many Western patterns, which tend to repeat in a predictable and periodic manner, Islamic patterns, like quasicrystals, do not merely replicate a simple motif over and over. Instead, they evolve across the surface in a way that, while maintaining overall coherence and symmetry, never exactly repeats. Each segment of the pattern can be seen as a unique interpretation of the overall geometric language. This creates a sense of infinite variation within the overarching order, a feature that resonates strongly with the concept of “forbidden symmetry” found in quasicrystals.

It’s fascinating to note that this aesthetic approach aligns with many of the philosophical and spiritual values embedded in Islamic culture. The designs symbolize unity in multiplicity, embody the principles of balance and continuity, and reflect a notion of the transcendent beauty of the infinite. They represent the harmony of the universe and the belief in an underlying order that binds all elements of creation.

Islamic patterns, therefore, stand as an enduring testament to a rich cultural and artistic tradition that, much like quasicrystals, celebrates the complexity and non-periodic nature of symmetry. This visual language provides an artistic expression of a principle that was only recognized in the material structures of quasicrystals in the late 20th century. Thus, the study of these patterns not only enriches our aesthetic understanding but also bridges cultural art with modern scientific discoveries, emphasizing the universality of the principles of order and symmetry.

Interplay of Quasicrystals and Islamic Patterns: Patterns of the Divine

The captivating congruity between quasicrystals and Islamic patterns transcends the boundaries of coincidence. The parallels seen in these unique forms of non-repeating symmetry are expressions of a fundamental principle that resonates in both the material and aesthetic realms. They epitomize an underlying ‘divine’ order that stretches across the natural and cultural world, which, while beautifully ordered, never succumbs to the banality of simple repetition.

Both quasicrystals and Islamic patterns carry the hallmark of complexity in symmetry. In quasicrystals, the atoms arrange themselves in a complex, quasi-periodic manner, creating a form of symmetry that is orderly yet non-repetitive. This arrangement of atoms is determined by precise mathematical rules, resulting in a structure that is inherently complex and yet beautifully symmetrical.

In parallel, Islamic patterns exhibit similar complexity in their designs. These geometric motifs, which are often based on mathematical rules, are generated through an intricate process of subdivision and multiplication of fundamental geometric shapes. They achieve a high degree of symmetry and order, but without resorting to simple repetition. Each part of the pattern, while connected to and coherent with the whole, exhibits a unique aspect of the geometric principle, contributing to the pattern’s overall complexity.

These similarities suggest that quasicrystals and Islamic patterns, though originating from disparate fields, share a common undercurrent. They both embody the ‘divine’ order — a term used here to describe an aesthetic principle that symbolizes balance, harmony, and infinite complexity within an overarching structure. This divine order is mirrored in the universe’s structure, from the complex formations of galaxies and the intricate patterns of biological life to the cultural expressions of art and architecture.

The shared theme of non-repeating symmetry between quasicrystals and Islamic patterns opens a riveting dialogue between natural science and cultural art. It illustrates how the notion of the “forbidden symmetry,” once thought impossible, permeates different domains, each one providing its unique interpretation and manifestation of this principle.

As such, the study of quasicrystals and Islamic patterns not only encourages a multidisciplinary understanding of our world’s inherent beauty and complexity but also fosters a shared language between science and art. It presents an opportunity to transcend the traditional boundaries of these fields, promoting a holistic perspective that intertwines the physical and the aesthetic, the tangible and the symbolic, the scientific and the divine.

Generative Art: Automation Meets Creativity

Generative art provides a unique synthesis of automated precision and creative imagination. It is an artistic practice that involves the use of a system — such as algorithms, computer programs, or machinery — that follows a certain set of rules or instructions to generate or significantly contribute to an artwork. Unlike traditional art forms where the artist manually manipulates the medium, generative art entrusts part of the creative process to an autonomous system, thereby introducing elements of randomness, unpredictability, and vast variability.

What makes generative art particularly fascinating is its capacity to mirror the complex order, the ‘forbidden symmetry,’ found in quasicrystals and Islamic patterns. By establishing and applying specific rules or algorithms, generative artists can design and produce intricate patterns. These patterns, reminiscent of their natural and cultural counterparts, exude symmetry and coherence but refrain from periodic repetition. Consequently, each representation of the pattern is singular and unique, yet it harmonizes with and contributes to the overall aesthetic unity, akin to each segment in Islamic design or each atomic arrangement in quasicrystals.

A potent tool for generative artists is p5.js, a JavaScript library specifically designed for creating graphic and interactive experiences on the web. Employing this tool, artists can write code that fabricates digital patterns inspired by quasicrystals’ intricate structures. For instance, the program could initiate with a simple geometric shape, such as a pentagon, then apply transformations to it by scaling, rotating, and placing it at different locations on the digital canvas while still preserving the quasicrystals’ characteristic five-fold symmetry.

With each iterative run of the program, new shapes are generated, interlocking seamlessly with the pre-existing forms, giving birth to an evolving pattern. Despite being highly orderly and coherent, the pattern never precisely replicates itself, much like quasicrystals. This process effectively emulates the natural formation of quasicrystals, where atoms position themselves according to strict bonding rules and the meticulous crafting of Islamic patterns following established design principles.

Furthermore, the artist can introduce elements of randomness into the code, which can lead to unexpected variations in the patterns, adding another layer of complexity and uniqueness to the output. These unpredicted elements within the structured framework echo the infinite variability seen in nature and Islamic patterns, underlining the interconnectedness of generative art with these themes.

Thus, through the practice of generative art and utilizing tools like p5.js, artists have the power to model the mesmerizing aesthetics of ‘forbidden symmetry’ digitally. They can explore and play with the intricate balance between order and complexity, rigidity and flexibility, predictability and surprise, thereby manifesting a digital equivalent of this fascinating concept. In doing so, they not only bridge the gap between the physical world of natural science and the abstract realm of artistic expression but also push the boundaries of creative possibilities.

Conclusion: The Enduring Allure of the Forbidden Symmetry

In weaving together the threads of this exploration, we arrive at a holistic understanding of how quasicrystals, Islamic patterns, and generative art are intricately linked, notwithstanding their apparent distinctiveness. These three realms, each in its unique way, embody the ‘forbidden symmetry’ — the complex, non-repeating symmetry once thought to be impossible.

Quasicrystals discovered only a few decades ago, stunned the scientific world with their paradoxical orderliness and non-periodic structure, exhibiting symmetries thought to be forbidden in the realm of crystallography. They stand as a testament to the infinite complexity and diversity in nature, challenging our conventional understanding of order and symmetry.

Simultaneously, Islamic patterns have for centuries mirrored this ‘forbidden symmetry’ in their geometric designs. Through meticulous crafting and arrangement of simple shapes, these patterns manifest a unique combination of symmetry, order, and non-repetition, echoing the aesthetic principles seen in quasicrystals.

Generative art brings these principles into a modern, digital context. Using algorithms and computational processes, artists create intricate, non-repeating designs that encapsulate the aesthetic essence of quasicrystals and Islamic patterns. This digital representation of ‘forbidden symmetry’ bridges the gap between the tangible world of physical materials and the abstract realm of art and design.

In the process of unifying these disparate domains, we have not only uncovered a profound commonality but also charted a new path for artistic creation and appreciation. This amalgamation broadens our perception, encouraging us to see beyond the confines of individual disciplines. It reveals how the principles that govern the atomic structures in quasicrystals and the designs in Islamic art can find a new expression in the digital world of generative art.

The ‘forbidden symmetry’ — the complex yet orderly, non-repeating symmetry — thus serves as a captivating motif, a shared language that resonates across the natural, cultural, and digital worlds. It invites us to embrace complexity, celebrate diversity, and find beauty in the balance between order and unpredictability.

In conclusion, the enduring allure of the ‘Forbidden Symmetry’ lies in its universal appeal and versatility. Whether it manifests in the atomic structure of quasicrystals, the geometric designs of Islamic patterns, or the algorithmic artistry of generative creations, it serves as a reminder of the inherent complexity and beauty of our world. Through its many manifestations, we are encouraged to continually expand our perspectives, explore novel connections, and appreciate the intricate interplay between science, culture, and art.

Research & Execution:
OMA Lab - Orkhan Media Art Lab. 2023-2024