Fractal Dimension and Perception of Order in Islamic Art

https://www.intechopen.com/online-first/86254

Abstract

In this article, I aim to draw attention to how human beings’ innate emotional closeness, connectional integrity, and creation feature toward other living things and the causality of coexistence are interpreted in the triad of mind, philosophy, and architecture. Esthetic appreciation for nature is one of the essential human tendencies, and it is biologically encoded in the human structure. The primary function of the structures of nature to create an esthetic effect in man is the maintenance of life. In addition to the geometric properties of biological forms, such as fractal formation and scale invariance, symmetry, self-similarity, and complex hierarchy, indirectly relate humans to patterns and properties of natural elements. The relationship between people and places is manifested not in spatial distinctions but in the transition between perspectives. In this article, I argue that the architectural, philosophical, and artistic triad of the Alhambra Palace is a magnificent labor that can correspond to the concepts of connective totality, biophilia, and fractal.

Keywords

• Islamic art
• biophilia
• fractal
• geometric synthesis
• philosophy of mind
• cognitive science
• perception

1. Introduction

As a result of the interaction and transformation of cultural dynamics, each period shapes its own ecology. The human brain, which can develop thoughts, emotions, and behaviors, has a unique connectivity with its originality, creativity, imagination, the potential to transform itself with each new learning, and the ability to create a communication network and pattern. The authoritative communication language of society can be defined as life. Philosophy, art, architecture, and science understand this language and discover new truths from experiences to be articulated on old truths, give meaning to lives and new “more truths.” The interconnectedness network of life, which consists of information, is modeled in the human mind with philosophy, art, architecture, and science. That area of the network of existence is again “created” through specified experiences with the premise of philosophy, art, architecture, and science. When we examine the existence of nature, from the veining structure of a leaf to the folds where streams meet and separate, from the images created by the lightning to the elegant patterns produced by the flowers in an artistic manner; we encounter various occurrences or actions, from the intricate form of mountains to the behavior of nonlinear weather conditions that we cannot precisely predict. When we pay attention to this production language witnessed by human beings, we can analyze how a process consisting of a few simple steps turns into magnificent formations when repeated, no matter how complex it may seem. The presence and richness of fractality in the forms produced by nature and observed behaviors draw attention. Connectivity is also hidden in geometry. Geometry maps are translated as measurements of the earth or earthly measurements. Geometry reveals parts of all forms of the part’s relation to the whole and the structure. Geometry, which forms the basis of the original structure of everything that exists, is a constant reminder to us of the sacredness of the relationship with the whole, its interconnectedness in the universe, and its constant interaction with the networks it sees with everything. Natural analogs deal with the organic, inanimate, and indirect connotations of nature. Living things, materials, colors, shapes, sequences, and patterns found in nature; In the built environment, it manifests itself as a work of art, ornament, furniture, decor, and weaving. Imitating flower and leaf patterns, choosing colors in the natural environment, or organically designed furniture provides an indirect connection with nature.

Man is a part of nature and is materially dependent on nature in terms of resources such as food, clothing, and medicine to maintain his life. Apart from this, nature is essential in terms of its effects on human beings’ emotional, cognitive, esthetic, and spiritual development. In theory and practice, a person’s physical, mental, and spiritual health depend on the quality and continuity of the interaction with the natural environment. This interaction increases the potential for a safe and fulfilling life. Man’s dependence on natural resources, satisfaction from interaction with nature, the development of exploration and skills, the effect of the physical appearance of nature on peace and inspiration, and emotional bonds with plants and animals have been studied in different subjects. The actual discovery area of humanity is life on earth and the transfer of related information to science, art, and daily life. Human beings’ ties to the rest of life are poorly understood and need new scientific research and bold esthetic interpretations. It is necessary to develop an intellectually more substantial and persuasive environmental ethic.

The symbolic experience of nature refers to the use of nature to facilitate communication and thought. Man’s mental and physical evolution has taken place together with environmental features such as light, sound, smell, wind, water, plants, and animals. The use of symbols related to nature showed itself in the development of spoken language. Problem-solving, critical thinking, and manual skills have emerged from centuries of interaction with nature. His closeness to natural elements is biologically coded into the human structure. It is known that animal and plant representations observed in historical buildings and places have long been used in architecture for decorative and symbolic decorations. The consistency of these natural themes, applied differently in each period, shows that biophilic design is not new. Instead, history supports that connection with nature is vital for modern humans as an urban species to maintain a healthy and vibrant existence.

Biophilia and biophilic design emphasize that physical and mental well-being depends, beyond us, on the quality of our relationships with nature, of which humans are ecologically a part. The biophilic design approach is shaped on the basis that people feel better in environments with sunlight, animals, trees, flowers, running water, birds, and natural processes. In buildings that have survived from ancient times, the concept of design connected with nature in a continuous state shows that although the biophilia hypothesis is a modern concept, it is not a new point of view. Communities in the past, which built shelters from local materials, have historically revealed their structures through biological orientations and spatial arrangement processes of their immediate environment and minds. Using what they had for shelter, protection, and worship, they instinctively built structures that provided the knowledge, form, and meaning their sense of well-being required. Therefore, design decisions have been realized as a natural extension of the neurological processes that make humans alive and human. The design framework of modern architecture has yet to develop in a way that includes nature and requires a relationship with the environment. However, current evaluations of historical buildings show that people perceived healing qualities better in the past and included them in their immediate surroundings.

Since the periods of urbanization activities, they have built their shelters with biological guidance, and applications, such as gardens, courtyards, atriums, and inner gardens, have been made in all societies, and contact with nature has been achieved. Humanity, whose relationship with nature is based on existence, attributed holiness to these gardens by developing mythological and religious events in connection with nature. In Islamic civilization, courtyard design, which connects with nature and privacy in civil and public buildings, has been the most critical element determining the development of Islamic architecture and cities. The garden of a surreal palace depicted in Tales of One Thousand and One Nights, which is described as one of the seven wonders of the world, is the most important example of which the healing, purifying, and spiritualizing effect of nature has been proven in ancient communities where water plays with various plants and trees were masterfully constructed. The Palace, an excellent source of pride for all Islamic art and Andalusian architecture, is the Palace of Alhambra (Kasrü’l-hamrâ). The building, built for military use by the Andalusian Umayyads on a hill overlooking the rivers in Granada, Spain, in the ninth century, was expanded with towers, interconnected rooms, courtyards, and courtyard gardens transformed into a magnificent design. The Alhambra palace, which means “red” in Arabic and takes its name from the unique red clay soil of the geography, is a magnificent yet modest architectural design of the thirteenth century Andalusian period.

2. Approach to philosophy of mind with architectural concepts of biophilia and fractal geometry

It has been understood that especially the ornamental elements in the spaces constructed in connection with nature are geometric compositions that directly affect the nervous system. Traditional motifs, colors, articulated surfaces, and spatial formation emphasize design in connection with nature [1].

The dialog between architectural spaces, shaped as the composition of natural forms and patterns, and human beings’ innate commitments has been manifested in traditional architecture. Specific strategies for establishing this dialog include direct access to daylight, fresh air, plants, and green spaces [2]. The use of atrium and peristyle in Greek architecture was seen as the most appropriate method for incorporating light and nature into the building to create a qualified living space and incorporate natural elements into the plan of the building. A similar point of view is also seen in traditional buildings in Islamic architecture, where courtyard systems and cloistered corridors are used. The courtyard system, used in different forms in public and civil architecture, serves as a garden from nature by providing a central gathering area for the surrounding rooms, natural ventilation, daylighting, and planting [3].

It is possible to define the charm in traditional architecture as organic, lively, whole, comfortable, accessible, egoless, confident, and mostly “eternal” qualities that can also be defined as biophilic values. Because of these qualities, it reflects the feelings of infinity and is called “timeless architecture.” Christopher Alexander states that historic buildings are a timeless way of building and defines timeless architecture as spaces that give a feeling of comfort rather than restlessness since there are no internal contradictions incompatible with human nature [4].

He describes it as a timeless experience where individuals who experience these structures feel comfortable and live in harmony with themselves and without stress. The proportion and geometry in these buildings are associated with the scale of the building components and their positioning concerning the surrounding structures because one of the basic principles of timeless spaces is their positioning and shaping according to the built environment in which they are located [5]. It is thought to have physiological and psychological effects on humans due to the flawless processing of sacred geometric principles and their construction as a harmonious built environment for living [6]. The composition of fractal forms with natural textures and colors in the editing of cognizant natural images in the built environment is defined as the biophilic approach. In the discussion of healing space, there are also opinions that the biophilic design presented by the architects to the users is not real nature and will have a different effect. However, experimental studies have shown that the human mind interacts with nature indirectly and has similar effects with being in nature when in contact with distinctive forms and surfaces. The natural environment and the built environment, including natural forms, details, hierarchical division, fractal patterns, and colors, help elicit innate responses and trigger a sense of well-being.

Physical and biological sciences have utilized fractal geometry seen in ornaments since antiquity to model historical and mythological accurate, symbolic events and geometric compositions of the period. Fractal geometry was later used to establish the link between art and nature, serving esthetic concerns, so researchers determined that fractal patterns were linked with local vegetation [7]. In this context, recent studies have proven that fractal patterns and multidimensionality in spaces affect the parasympathetic nervous system and provide an impulse to relax and calm down [8]. It has been understood that the ornamental elements in the spaces constructed in connection with nature are geometric compositions that directly affect the nervous system. Traditional motifs, colors, articulated surfaces, and spatial formation emphasize design in connection with nature [1].

In historical, sacred, and secular spaces that integrate the logic of natural life into the built environment, bring the language of “life” to the spaces and thus provide neurological nourishment to the users, the mental and physical arousal experiences that people experience at a high level stemming from this visual/mental discourse, nowadays called biophilic design, but the architecture of architecture. Shows that the “healing architecture” attitude, which has been effective since its existence, is related to the qualities it provides to the building.

The building form of the madrasahs with courtyards, where both theoretical and practical training were held, developed over time and characterized by the use of water, shadow, daylight, color, texture, and the effect of environmental forces, dominated by square and rectangular geometry with iwans on four directions, the courtyard surrounded by a portico. Despite the sculptural stance of Western architecture, the harmonious integration of the exterior and interior spaces of traditional Islamic madrasas with natural elements and the harmonious use of natural materials and geometric forms in the building is referred to as nature-friendly designs in the field of today’s architecture [9].

Human-centered ethics (anthropocentric), which is one of the three basic environmental and ethical approaches in explaining the relationship between humans and nature, is an approach that respects nature as long as it has been beneficial to humans since the ancient Greek period and is based on the domination of society over nature and argues that economic developments are the primary condition for human well-being. The second basic approach, biocentric ethics, accepts that other living beings other than humans have intrinsic value and argues that humans are in an equal position to all other living things. It is necessary to create an ecological citizenship model to minimize the damage to the natural environment through sustainable living and conscious consumption of natural resources. Accepting nature as a living organism, instead of the mechanical universe view that formed the foundations of science in the seventeenth century, will enable us to understand nature and accept it as a living entity. It is necessary to accept the existence of technology and industry and develop new approaches according to this reality to create and develop environmental awareness in humans, who are the biggest consumers of nature. Rapid economic and technological developments have caused irreversible damage to the ecosystem, and at this point, people have found the solution to seek ideal living spaces outside the world. However, the perfect balance and limitless possibilities the world offers for life have yet to be found anywhere else in the universe. In this context, the solution to ecological problems is to question our wrong relations with nature for centuries and to adopt and implement the understanding of development in harmony with nature.

The word biophilia, which is a combination of the words “bio” meaning life, and “philia” meaning sincere love-liking, means the instinctive love of all people for life and things related to life. The term “biophilia” literally means “love of life or living systems.” It was first used by Erich Fromm to describe a psychological orientation of being attracted to all that is alive and vital. Erich Fromm, who said there is an innate connection between the human self and other living things, used the concept of “biophilia” for the first time in 1964 as the opposite of necrophilia. He argued that being affected by vital things related to life and death is a psychological obsession and defined biophilia as “excessive curiosity and attraction to everything alive.” [10] Wilson defined the concept of biophilia as an innate tendency to focus on life and vital processes. He suggested that we value nature and ourselves more as we understand other living organisms. In the same book, Wilson stated that man’s tendency to relate to life and natural processes could express a biological need, including the desire to connect with life. Therefore, he argued that this need for nature is necessary for physical and mental growth in the developmental process. Further, Wilson argues that the biophilia hypothesis reveals human dependence on nature, whose ties extend far beyond simple matters of material and physical lateness and the craving for esthetic, intellectual, cognitive, and even spiritual meaning and fulfillment. Thus, it is inevitable for them to prefer parks, zoos, and environments where they can come into contact with water, with the idea of an uncertain future, in artificial environments offered by technology [11]. People connect with nature through emotions, such as curiosity, dominance, discovery, and fear. In their book The Biophilia Hypothesis (1993), Wilson and Kellert presented evidence confirming and opposing biophilia, explaining how these tendencies developed during the evolutionary process and why they are still evident in the modern age.

The universe is 2-dimensional and is perceived as 3-dimensional by the software of the brain. The language of the Hologram Universe is mathematics/geometry. Endless patterns transform each other from shape to shape; the pattern in the smallest ring is also evident in the largest; Everything is intertwined and connected. The knowledge that we live in a simulation is widely spoken about today and is scientifically proven day by day. In other words, life consists of the projection of our brain, and our brain consists of software that is the mathematical product of the universal program [12]. Different forms of biophilia are best seen in crops in the process of biocultural evolution. Intrinsic tendencies are shaped by the guiding influence of learning, culture, and experience. Biologically based, the tendency to associate with nature, behavior includes thoughts and feelings. It is a pattern of beliefs and tendencies toward nature, each reflecting human values and expressions. These tendencies manifest themselves directly or indirectly. They manifest themselves in human products, in manifest or hidden states of experience and understanding. In terms of content, intensity, and orientation, the role of society is enormous in the tendency to relate to nature. Various forms of biophilia are found deep within our biology and can be shaped by individual experience and cultural influences, not simply by reducing them to instinctive tendencies.

Man’s mental and physical evolution has taken place together with environmental features such as light, sound, smell, wind, water, plants, and animals. Problem-solving, critical thinking, and manual skills have emerged from centuries of interaction with nature. The closeness of man to natural elements is biologically coded into the structure of man. According to Kellert, people’s emotional, physical, and mental health and productivity today depend on their close relationships with nature. For this reason, it has been considered essential to examine how to maintain this relationship in the modern built environment and put it into practice. In the book, Biophilic Design, academic and scientific studies were compiled under the direction of Stephen Kellert.

3. Scientific research reveals how our environment affects the mind

Due to the basic need for shelter, a human cannot be thought of independently of the place and is constantly interacting with the environment in which he is physically and spiritually as an inseparable whole [13]. Undoubtedly, the alienation of people from nature began with the fact that they built cities to live in communities by adopting a settled life. The city-human relationship has reached the breaking point, especially with the industrial revolution. Since the adverse effects of urbanization on human health are a common problem, research from many branches of science has focused on this issue. “Ecopsychology,” which means ecological psychology, was popularized by historian Theodore Roszak in 1992 and explored the relationship between human behavior and the environment [14]. Ecopsychology examines the relationship between psychology and behavior in daily life practices and their environment. As a continuation of the Age of Enlightenment understanding, the field of architecture, which developed with the perspective of man’s sovereign authority over nature, has built cities, which are today’s living spaces, by ignoring the human-environment bond in the design processes. These studies, which seem to have achieved successful results, in theory, have created ecologically insensitive and unhealthy spaces where sociability is minimized in practical life. Ecopsychology, which examines the way people perceive their environment in daily life, and their interaction with the built and nature, says that the inclusion of social, cultural, and natural dimensions in the built environment is an important element [15].

Most of the research on the relationship between man and his environment has focused on cities that have turned into unhealthy spaces as an inevitable fact of our lives. In 1994, Rodhe and Kendle examined the positive effects of urban and green spaces on human psychology under five headings: emotional, cognitive, developmental, behavioral, and social; It reduces people’s stress and mental fatigue, improves their mental abilities, especially in children, strengthens their sense of self-confidence by balancing different emotional states, and positively affects human health by increasing socialization and communication [16]. Ulrich (1993) stated that the stress responses of individuals who are accustomed to living only in an urban environment without being exposed to the natural environment cause some cognitive losses. Unlike the life of ancient people, in urban environments where there are no physiological stimulants and no struggle for survival, mental stressors cause anxiety, and this becomes a cycle in the absence of positive and negative natural stimuli in urban life [17].

In addition to urban open spaces, the use of natural elements in closed spaces where people are in constant contact is also an important factor affecting human health, well-being, and productivity. Biophilia, sustainability, ecopsychology, and other research on the human-nature connection suggest that the human species respond positively to the forms, processes, and patterns that include nature. In this context, Heerwagen (2009) argues that in increasing the health and well-being of the spaces designed by using the knowledge of our inclination toward nature, working environments connected with nature will become both more productive and comfortable, homes will become more harmonious and livable with human nature, and public spaces will become more inclusive spaces [18].

4. Biophilic design parameters

After the biophilic design studies, the researchers focused on some design parameters for the applicability of the specified biophilic elements. In light of Appleton’s (1975) studies [19], listed physical elements and spatial features in preferred natural environments as mystery, danger, shelter and surveillance, confusion, and order [20] (Table 1).

Key dimension	Attributes and qualities
Prospect (ability to see into the distance)	Brightness in the field of view (windows, bright walls) Visual distance Ability to get to a distant point for a better view Horizon/sky imagery (sun, distant mountains, clouds) Strategic viewing locations View corridors
Refuge (sense of enclosure or shelter)	Canopy effect (lowered ceilings, screening, branch-like forms overhead) Variation in light levels (darkness suggests refuge) Enclosing surfaces (walls, partitions, screens) Penetrable barriers and surfaces for views out
Water (indoors or in view)	Glimmering or reflective surface (suggests clean water) Moving water (also suggests clean, aerated water) Symbolic forms of water
Biodiversity	Varied vegetation indoors and out (large trees, plants, flowers) Windows designed and placed to incorporate nature views Outdoor natural areas with rich vegetation and animals
Sensory variability	Changes and variability in environmental color, temperature, air movement, textures, and light over time and space
Biomimicry	Design derived from nature Use of natural patterns, forms and textures Fractal characteristics (self-similarity at different levels of scale with random variation in key features, rather than exact repetition)
A sense of playfulness	Incorporation of decor, artifacts, objects, and spaces whose primary purpose is to delight, surprise, and amuse
Enticement	Discovered complexity Information richness that encourages exploration curvilinear surfaces that gradually open information to view

Table 1.

Classification of properties of biophilic structures prepared by Heerwagen and Hase [21].

Heerwagen and Hase (2001) articulated that with the increase in knowledge and experience about the usability of nature’s principles in architecture and urban design, the characteristics that determine the quality and quantity of biophilic structures will expand.

With the development of computer technologies, the algorithms of the forms in nature are reproduced with numerous iterations and the fractal pattern of the form is modeled. Various fractal production methods are used in different disciplines such as architecture, medicine, design, digital installation, cinema, and animation. However, when the works of art of the past, intricate reliefs used by abstract geometric patterns, tile ceramics, woodwork, plaster ornaments, and marble columns are examined through computer programs, it is observed that they contain high mathematics, algorithm, symmetry, and geometry knowledge far beyond their period. The fractality of the esthetic language of nature draws attention to the patterns that are still not fully understood, deeply admired, and in forms with esthetic value. Behind the fractality produced by the knowledge of higher mathematics in the work of art, from form to essence, is a system of thought, an expression that leads from matter to meaning. Each work is a part of the cultural climate it is in and is a reflection of the artist’s world of thought. When higher mathematics knowledge turns into an artistic style, it interprets the universe’s existence from part to whole in the world of thought. Behind the ornaments, designs, architectural forms, and concrete elements that draw attention with their fractal patterns, there is a universe consciousness pointed out by the philosophy of mysticism.

In the case of more than one mode that oscillates independently from each other, the movement is not chaotic, but if the interaction of the combined connections of these modes, which oscillate in connection with each other, at least in groups of three, they are all affected by the changes in each of them at a specific moment and chaos occurs. Since repetition is possible in linear events, an estimation can be made from previous experience. In nonlinear events, the inability to determine the starting point, small changes in initial conditions turning into significant differences, and the absence of an analytical solution are the causes of chaos. It exists in all scientific fields where uncertainty is valid. Chaos theory deals with the order of chaos. In chaos theory, there is an orderly, beautiful, and solid structure within the irregular, complex, and unorganized structures. Fractal structures explain the order of chaos. Where chaos arises, dimensions become fractal, and fractal structures emerge as systems move from order to chaos. The most common principles of chaos theory; are sensitive dependence on initial state, iteration, chaos patterns, fractal and self-similarity, nonlinearity, strange attractors, turbulence, self-organization, and self-renewal. Chaos and uncertainty apply to all branches of science. The reason is that the number of variables required to make predictions about any event is too large, and it is impossible to create a system that includes all its variables.

As in the example of the Sierpinski triangle (Figure 1), the chaotic structure formed by the iterations of shapes, forms, or systems is an example of chaos.

An inclusive research area examines the part-whole relationship of fractal geometry. It is the analysis area of fractal geometry from honeycomb to leaf veining, computability and unpredictable weather events to the intricate forms of mountains, from the connectivity of the brain to the network structure of the universe’s simulated energy fields, from statistics to the urban transportation network. When the fractal pattern of nature is examined, simplicity, esthetics, functionality, and “high design” are witnessed. Each part of the whole carries the self-knowledge of the whole. As in the DNA sequence of a single human cell or a plant seed, the information that exists in essence from the micro to the macro universe repeats itself in each piece. The fractal texture of nature Figures 2 and 3 develops from the inside out. In this development process, a production method is seen in which the information about the essence repeats itself at smaller ratio, and this production algorithm of vitality has a unique form and esthetic integrity. The most striking fractal pattern is the “Fi number,” also known as the “golden ratio,” which reveals the perception expressed as “beautiful” and the mathematics of esthetics, giving the value “1.61804.” This ratio, which was introduced by the Italian mathematician Leonardo Fibonacci, is also known as the “peak form of the fractal.” Although we state that fractal geometry was defined at the end of the twentieth century, it is seen that thinkers, architects, scientists, designers, and painters are not unfamiliar with this production language of nature.

“Pattern repeats of nature’s fractal objects”.

“The fractal geometry of nature”.

4.1 The fractal geometry of nature

The word that Anaxagoras pointed out in 500 BC: There is a part of everything in everything, all things are found in everything to a certain extent and leads us to the part-whole relationship that fractal patterns transplant. Neurology studies using the most comprehensive and systematic data today show us that the layered structure of the brain has a different mathematical system, that the communication network realized in line with this mathematical system creates a connectivity map specific to the individual, and that the connectivity map can change with each new learning. The discoveries in the field of neurology and the “connective holism” methodology brought to the part-whole relationship reveal everything from the brain’s communication network to the universe, from science to art, from individual life to cultural structure, is in integrity. [22]

While the esthetics and mathematics of the unique fractal patterns produced arouse a deep sense of admiration and amazement in us, the meaning indicated by this symbolism is met in a piece of stone, wood, tile ceramics, literature, music, and plastic arts. Apart from the symbols that invite us to unity from multiplicity in patterns, an architectural arrangement can also be considered as a reflection of fractality. The understanding encountered in the portal designs or courtyard arrangement of many Islamic architectures refers to the concepts of “inner and outer,” in other words, “essence and shell,” which are related to Islamic Sufi philosophy.

When we focus on the details of nature, we can see that exploring the fractal forms of nature is actually a reflection of the macro level, rather than what is visible on the outside. Because the ontological source of the outside is inside, its meaning is inside. With a detailed look at a simple fern, we witness that it repeats itself at smaller rates and reflects self-similarity in each new branch. Similarly, the fact that a seed contains a tree with its branches, leaves, flowers, and fruits in its seed expresses the depth of the essence, as well as the simplicity of the exterior. The hermetic structure, in which the shell, that is, the form, which is expressed as matter, body, or skin cage, is described as external, contains an essence, a spirit, a meaning that is described as internal. As the path from the shell progresses inward, discoveries, curiosity, and wonder point to the knowledge that development is from the inside out. This “adverb-adjective” relationship presents “a level of principles that support synthesis, integrity, and harmony,” not the contrast of inside and outside [23].

In systems that do not show linear behavior, sensitive changes in the initial parameters can create unpredictable results in the system behavior. Incalculable behavior patterns such as the formation of hurricanes, weather events, the path of mist in the air, and traffic flow in the city reveal the uncertainty and unpredictability emphasized by chaos theory. However, the emerging behavior pattern is a situation that occurs among the possibilities of the whole. Similarly, there is the potential to create unlimited variations when designing patterns. However, the fractal pattern that emerges from the sea of possibilities is again about the whole. This relationship, which is explained in Sufism, points to the existence of an information system that includes the visible behind it.

Sufism tends toward “the search for permanent principles that are inherent rather than temporary and variable.” In this sense, Sufism is a way of thinking that offers a methodology and an explanation model at the point of reaching knowledge. Art in Sufism is a journey of truth and truth does not mean the truth of a proposition. In a deeper sense, it is the revealing of the hidden, the transparency of being [24]. In this sense, there is a meaning to be expressed, and a truth pointed out with geometric patterns and motifs consisting of enormous fractal patterns. The fractal patterns of the Alhambra palace, the Taj Mahal, the Divrigi Dârü’ş-Şifa crown gate, the “dıraht-ı cân (tree of life) motif in the Konya Mevlana Tomb. and the star detail of the Sivas Gokmedrese also point to the understanding of unity emphasized by the Islamic mysticism behind the unique artistic and cultural accumulations and symbols of the geography. The meaning implied by fractal patterns draws attention to the metaphor of a mirror in the works of Andalusian Ibn Arabi.

Patterns with enormous symmetry, which can be shown as the pinnacle of fractality, can be found in different geographies; Although they are created with materials such as stone, wood, marble, plaster wall, tile, and ceramic and with different cultural backgrounds, they are completely different forms of inspiration reflected from a single image. Today’s research indicate that the fractal pattern of rumi motifs and geometric patterns in the Alhambra palace inspired Escher’s drawings in which he describes infinity and cycle; It shows that it was reflected in the decorations of the Istanbul Beylerbeyi Palace as it affected many Islamic architectures and gave it to the architects of the Taj Mahal (Figure 3) with its garden arrangement [25]. Similarly, the magnificent architecture of the Taj Mahal points us to the schema of the heavens and the heavens on a page in the third volume of Ibn Arabi’s Fütûhât-ı Mekkiye (Figure 4) [26].

“Taj Mahal as an example of fractal structure”.

5. Fractality and symbolism in Alhambra palace wall decorations

When viewed from the outside, this palace reflects a reassuring, brave protective spirit, and its garden, like a mother’s heart, contains a sensitive, compassionate, peaceful home, a loving mystery. Between the two rivers, Darro and Genil, that carry the healing waters of the charismatic peaks of the Sierra Nevada and Sabika Hill rises. It makes you dream of a silent, wise, and generous soul who lived on its own on this steep plain, whose existence was mixed with legends due to its beauty.

To be recognized as a civilization, it must show a rise and progress in meaning and matter, abstract and concrete, thought and action. Andalusian Islamic civilization is an example where this progress is felt in the idea and in the works that are copies of the idea. Ibn Tufail Abentofail, Ibn Rushd Averroes, and Nureddin Batruci Alpetragius, and many other eminent scientists, philosophers, who grew up under the influence of Cordoba academies, the science center of Andalusia, reached a superior level in both positive and human sciences and trained Christian clergy members and paved the way for scholastic philosophy and therefore the Renaissance. Scholastic is the reinterpretation of the Christian creed with Aristotelianism. It is not a dark period as it is thought; on the contrary, it is a period when the ancient philosophy enriched with the translation-interpretations of Arab scholars was learned from the Arabs and adapted to Christianity, thus laying the foundations of European thought. For example, the Gothic style, which reflects scholastic architecture, realized itself only when the walls were shrunk to the last limit of technical possibilities. This goal of Gothic architecture is seen in the effort of the Arab architects who built the Alhambra, which we will read about in a moment from Kazantzakis. In short, the influence and importance of Andalusia can be understood when the history of this implicit civilization, which was left in the dark due to the excessive light of the Enlightenment and reduced to a legendary/exotic level, is examined [27].

Andalusians, highly developed in mathematics and geometry, have proven their success in their field in the Alhambra Palace they have designed. In the palace, with numerous rooms connected with expansive courtyards, everything is calculated with assured mathematics. Alhambra Palace Architecture in Mural Art with Fractalism and Symbolism In many points of the Alhambra Palace, the construction of which was completed in 250 years, “There is no victor/conqueror except Allah.” It has the inscription “La Galiba Illallah” which means. (Figure 5) The palace, which used to have the rule “No one who does not know the five pillars of Islam can enter,” is one of the best examples of “Mudéjar” architecture inspired by the belief in the “afterlife and heaven.” [28]

“Alhambra palace architecture in mural art with fractalism and symbolism”.

The technology developed for water transportation, the mathematical knowledge used in architecture and garden designs, the geometry encountered in ceiling and wall decorations, and the dazzling unity of the forms that attract attention in room designs and domes are far beyond its era. It is emphasized that the unique style of Andalusian art differs from “the magnificence of the Umayyad dynasty, its luxury-loving attitude and the dignified attitude of the Maghreb rulers.” Its architecture makes the spaces feel “the majestic air seen in heavy fabrics and carpets.” [29] According to Owen Jones, the Alhambra is the artistic language of the ancient Egyptians, the natural beauty and refinement of the Greeks, and the geometrical arrangements of the Romans, Byzantines, and Arabs [25].

In addition to the ancient wisdom underlying its esthetic depth, the magnificence of the palace, the richness of the decorations, and the harmony and elegance of the interior attract attention with its simplicity, tranquility, and modesty in the green of the hill outside. The owner of this magnificence is pointed out with calligraphy repeated thousands of times on the walls, where the esthetics and elegance of the Alhambra, the most magnificent architecture of its period in every respect, are delicately worked out.

The principles of the Alhambra style are included in the chapter “Moresque Ornament” in the book “The Grammar of Ornament,” in which architect, designer, and design theorist Owen Jones explores patterns and design principles from different cultural periods. The unique patterns created by the symmetry, reflection, and repetition of the relations between straight and curved lines are integrated with the architecture. The simplicity of the principles utilized in these complex patterns and the sense of balance arouse great admiration. Professor Antonio Fernandez Puertas phrases that everything is based on a single ratio and that the Alhambra exists from the floor to the wall decorations. The relationship between the side of a rectangle and the diagonal lengths is formed by a simple proportion of the heights calculated by the square roots of numbers such as 2, 3, 4, and 5. It is seen that various variations that give this ratio were used in every part of the palace, in interconnected rooms, in all courtyards, corridors, column lengths, placement, and decorations on it [30]. In the studies of Spanish academicians on Alhambra, it is stated that floor and wall decorations contain original solutions to mathematical problems. It is thought that the development from the inside out in fractal structures coincides with the evolutionary journey of human beings from the shell to the essence in Sufism.

This understanding, which is classified as shell and core, exterior and interior, and matter and meaning, is reflected in detail, such as the portal, the arrangement of the courtyard that opens inward, and the decoration of the pulpit sections in the temples in Islamic architecture. It is seen that the decorations were not made outside the palace and the visitors were invited inside by drawing attention to the portal. The modest design surrounded by trees has a magnificent portal that opens inwards, as seen in many Islamic architectures. Behind the crown door, the courtyards of Jannat-ul Arif (Figure 6), which depict the verses of the Qur’an about paradise, and a garden arrangement in harmony with its architecture, meet. Due to the reflection of the flow, cleanliness, and clarity of the water, the Alhambra is also depicted as a palace in the water. In the pools surrounded by small plant groups, the sounds of water flowing from the fountains and the plays of light create a rhythm in harmony with the visual integrity; Dark green leafy plants that give coolness and peace, trees that give a sense of smell and freshness increase the depth of the courtyards. The inspiration of the opposite of water is also reflected in the garden and pool arrangement of the Taj Mahal 300 years later [31].

Figure 6, an exemplary garden of fractalism and symbolism in the Alhambra palace and “Jannat-ul Arif Generalife”.

The extensive garden named Jannat-ul Arif, located behind the palace and used as a place of rest in summer is considered one of the most successful landscape examples of Islamic architecture today. The construction of the garden, known as the Generalife, began in the thirteenth century and the work in the garden continued until the twentieth century. The construction of the garden known as the Generalife began in the thirteenth century, and the work continued until the twentieth century. Colorful flowers, plants, and trees were sent from various parts of the world for this unique garden. For this reason, Jannat-ul Arif is decorated with flowers and trees. The most attentive plant in the garden is the eight-century-old cypress tree. With its fountains, pools, pavilions and walking paths, colorful plants, and flowers, Jannat-ul Arif hosts the annual music and dance festival in Granada.

The muqarnas in the portal of the Alhambra palace, in the windows, in the arches, and in the ceiling decorations of the rooms, reflect the light from different angles with their intricate forms, creating a perception of perspective. In the structure, which is gradually shrinking from the outside in a layered way, each layer intertwines with the next, forming fractal patterns that seem to progress forever, and the light and shadow plays formed by the niches with different colors of muqarnas evoke a feeling of depth that makes the viewer think of the infinity and enormous order of the universe. Jones states that primary colors, such as blue, red, and yellow, gilding in the Alhambra were seen in the early art phase of civilization, and primary colors were used in the upper parts of the architecture [25].

This harmony, which gives the feeling of infinity, and the relationship established between the interior and the exterior includes the viewer, and it also assumes a place between the interior and the audience [29]. The layered structure of muqarnas and honeycomb wooden structures in different colors was designed with inspiration from the sevenfold heavens phrase of the Surah Mulk [32]. The high ceiling decorations consist of cedar wood pieces painted with colors representing the seven layers of the sky and pointing to the universe’s conception in Islamic mystic philosophy.

The Alhambra palace has geometric motifs and patterns with symmetrical groups, from the floor coverings to the domes. Unlimited variations of patterns can be produced with steps such as translation, reflection, rotation, and shift reflection of a single symmetry group. Intricate network structures formed by simple forms such as triangles, squares, and circles cover the surfaces and create an esthetic harmony that depicts infinity. There are two different classifications for the fractal network structure of patterns. The rooms of the palace, where visitors fall into the Lion’s Garden, are decorated with the fantastic nature of Arab architects—the stalactite arch, the nature of which is difficult to understand, and its accurate proportions are difficult to judge. In the daytime, besides the change in lighting, the appearance of the ceiling also changes, creating the illusion of movement. The most complex structure is based on accurate mathematical calculations and philosophical representations of Arab scientists. On the south side of the lion’s courtyard is the Hall of Abenserrach (Sala de Los abencerrajes), named after a tragic event—the slaughter of 37 members of the Abenserrach family. This structure has an unusual dome in the shape of an octagonal star decorated with stalactites. Soft light passes through the windows in the dome (Figures 7–9).

In drawing star patterns, vertical and curved lines are produced using net guides called construction lines, and then laying on the surface called tessellation is performed. While this algorithmic process, which was carried out with tools such as rope and pencil in the past, was designed with a compass and ruler before computer technologies, it can be produced through programs such as Geometer’s Sketchpad together with computer technologies. While a square mesh grid is used in four-layer star patterns, a pattern is created by surrounding other patterns because five-layer patterns do not fit on the square surface. Six-ply patterns are obtained with hexagonal mesh guides [33].

Each corner of the eight-pointed star patterns formed as a result of the rotation of the square, also known as the Seljuk star, at an angle of 45⁰, forms an angle of 90⁰. While the algorithmic steps of the eight-pointed star from the mesh grid to the star design are given in image A, variations of the star patterns can be produced from the hexagon mesh grid in image B. Infinity flow is seen in all patterns that transform from simple geometric shapes, such as triangles and squares, to unique fractal patterns. This sense of eternity and esthetic harmony, which arouses admiration in the audience, reflects the depiction of unity in the universe. These patterns indicate an excellent divine order dominating the material world. The transition from one star to another is continuous, and all the arms that make up the star jump from one star to the next and travel to infinity. There is no space in the surface coating of the patterns or randomness in the movements of the lines; on the contrary, the unity of many lines evokes the harmony of the universe. The fact that no part existing in the layered realm is idle reflects the moderation and unity mentioned in verse. There is a constant movement in the universe from multiplicity to unity and from unity to multiplicity. Although the variations created by fractal patterns are unlimited, they are derived from the same geometric forms [23]. As in Plato’s example of the broken mirror, we can look at the sun through shattered mirrors and see hundreds of suns, but when we want to see the actual sun, we raise our heads and witness that there is a sun.

In addition to geometric patterns, vegetal motifs are also seen in the decorations of the Alhambra. The motif emerging from the main body is veined like a leaf and covers the surface, and the area it borderline is filled with various patterns, crests, or lines. It is known that stucco or pseudo-marbles made of material thought to be obtained by mixing marble dust, lime, gypsum, and egg white are used on these surfaces. The reliefs synthesize Greek, Visigoth, and local Iberian art styles. While encountering plain-looking vegetal and geometric patterns drawn in more general lines in the early period drawings, they were enriched with fine lines over time. Plaster decorations are created by using molds in larger spaces and surfaces. It is seen that while marble was used for the magnificent reliefs in the Umayyad period, plaster was preferred in the later periods. Plaster decorations made with the cladding technique on stone or brick walls allowed the production of new motifs and patterns due to the material’s ease of use. It is seen that motifs such as palmettes, volute motifs with snail folds, and pinecones are included in the reliefs [29]. In Alhambra architecture, the building design offers magnificent integrity with columns, garden arrangements, courtyard designs, ornaments in which geometric and floral patterns are integrated, and star systems. It is seen that every particle of this splendor points to infinity, in other words, to the whole. The richness created by fractal patterns opens a door from concrete to abstract with the inscriptions at its heart.

5.1 Geometric patterns in Islamic architecture

Different types of Islamic ornaments are based on geometric rules, the esthetics of this art is geometric in general. Islamic geometric design (Figure 10) is a system of abstraction from Islamic beliefs [34]. Islamic art has created a unique geometry while benefiting from the geometric forms of previous traditions. These forms gave identity to Islamic art, not with the reflection of the forms found in the real world, but perhaps with the forms determined by certain lines and borders. Traces of this can be found on geometric surfaces designed by early Islamic artists. The traces of the artistic and architectural heritage of ancient cultures, especially central asian and middle eastern societies, are evident in the art and architectural works of the Islamic period. The circle and line are the basis of the proportional system, also utilized in Islamic calligraphy.

Artists used these patterns not only to create visual beauty but also to convey the divine concepts of Islam. Often following religious and philosophical teachings, these artists used visual symbolism to place these divine concepts in patterns. The pentacle was used to symbolize the five teachings of Islam, furthermore, most Islamic geometric patterns artists used the symbolism of numbers and geometric figures, a continuation of the Pythagorean teachings, in creating their art. However, the symbolism is often set in the context of the piece [35].

6. Conclusion

The fractal pattern of nature, which repeats itself and carries the self-knowledge of the whole in every part, offers us a high design example with its simplicity, esthetics, and functionality. From the past to the present, it is seen that fractal geometry, which reflects the production language of nature, has been used in different disciplines, such as architecture, design, art, medicine, and statistics. The essential artistic productions of fractal geometric patterns were realized in thirteen-century Andalusia, and the peak form of fractality was reflected from the floors to the architecture of the Alhambra Palace. Alhambra is where magnificence meets simplicity, landscape with architecture, art, and mathematical knowledge. However, the calligraphy repeated thousands of times on the palace walls emphasized its persistence for centuries. Surfaces where gypsum is mixed with different materials on which geometric patterns are processed, tile ceramics, marble structures, and wood carvings. There is a thought system that is indicated behind each fractal pattern in every corner with its walls, portals, windows, and muqarnas domes, which are equipped with soft and esthetic calligraphy. In this sense, the philosophy of Sufism is seen as a methodology to reach the unity behind fractal patterns. The fractality, mathematical ratio, and esthetic integrity of each work are actually different parts that reflect the influencer, and at the same time, they are interpretations of the whole. It is seen that each pattern transforms from a grid of construction lines into unique fractal patterns that contain various symmetry groups such as straight lines, curves, intersections, integration, reflection, translation, and rotation. It is seen that symbols and symbols are preferred instead of direct expression in Sufism. Symbols indicate the order of the universe, grace in creation, perfection, eternity, and unity. The visual integrity created by the fractal patterns and the feeling of flow to infinity describes the journey and discovery of the viewer himself by drawing the viewer into himself. Hundreds of manifestations of a single ratio are reflected with honeycomb-shaped muqarnas ceilings, wall decoration, calligraphy, courtyard arrangements, landscape, and architecture. Each fractal pattern will provide a mental and spiritual opening in humans; it invites an effort to make sense of relations such as inner and outer, shell and essence, and matter and meaning. The fact that hundreds of fractal forms can be produced from a single mesh grid has the same meaning as the emergence of only one piece out of endless possibilities.

It is thought that this is the expression of being in a new existence every moment as a reflection of the continuity from multitude to unity, from unity to multiplicity. The visible, emerging pattern has the same essence as all patterns that have the potential to be visible. In the essence of concrete elements, unique forms produced with marble, plaster, wood, or tiles, that is, thousands of beings, there is an abstract system of thought, a sea of meaning, and the reflection of the characteristics of the being that integrates each piece, inviting to interiority and admiring the layered structure of the universe while confronting it with the layered structure of the self. The visible, emerging pattern has the same essence as all patterns that have the potential to become visible.

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