The fractal dimension of Islamic and Persian four-folding gardens

Source: Nature


Since Benoit Mandelbrot (1924–2010) coined the term “fractal” in 1975, mathematical theories of fractal geometry have deeply influenced the fields of landscape perception, architecture, and technology. Indeed, their ability to describe complex forms nested within each other, and repeated towards infinity, has allowed the modeling of chaotic phenomena such as weather patterns or plant growth. Some human-designed patterns such as the ones developed by Islamic cultures have been found to follow similar principles of hierarchy, symmetry, and repetition. However, the application of these principles in the design of gardens is an underexplored field. This paper presents a comparative exploration of the four-fold garden design model—the chahár-bágh—typical of Persian and Islamic garden design by analyzing two case studies: Taj Mahal and Isfahan’s city plan. This four-fold pattern is known to not only have a religious reading but to be also linked with ideals of fair distribution. Using an innovative compositional fractal analysis inspired by architecture, our results demonstrate that these gardens contain a high level of self-replication and scale invariance and that they exhibit a high fractal dimension. The novel application of this method of analysis to historical landscape plans allows us to assess to what extent fractal concepts were already in use before the European Renaissance and Mandelbrot’s explorations, and to speculate on their symbolism in the context of Islamic and Persian garden design. Specifically, we conclude that the fractal characteristics of these gardens might be intended as a representation of the infinite divine but also of principles of fairness and equality. Moving forward, this approach could be applied to design spaces, namely in the infrastructural design of the urban fabric, which are both meaningful and environmentally just.


Fractal geometry is a theoretical framework formulated to unite forms and patterns previously considered too complex to be described (Mandelbrot, 1975). Specifically, the framework can be used to model natural forms such as coastlines and mountain ranges but also blood vessels. It is sometimes considered a universal pattern language as the forms it describes can be found in every living thing (Di Leva, 2016). Based on principles of self-similar symmetry and scale invariance, it is currently used in biological sciences such as medicine and ecology (Kenkel and Walker, 1996) as a way to understand complexity and structure chaos. Modern artists and designers have also been seduced by the approach’s innovative ability to create realistic organic patterns both in analog and digital platforms. However, there is no doubt that much like the shapes it describes, “fractal geometry dates back to centuries before the emergence of the fractal theory by Mandelbrot” (Abdelsalam and Ibrahim, 2019, p. 27).

Similarly, Islamic artistic expression follows three main modes: floral designs, calligraphy, and geometric patterns (Bonner, 2003; Khamjane and Benslimane, 2017). Out of these, at least the first and last of these modes echo elements of fractal geometry. Islamic patterns have also been found to exhibit high levels of self-similarity (Redies and Brachmann, 2017; Djibril and Thami, 2008; Aljamal and Banissi, 2003). In fact, “Iranian geometry using plant and geometric forms in Islamic buildings seeks to prove a special continuity in the life to plant and the human world” (Pudine, 2016, p. 257), similar to the unification of fractal patterns under one comprehensive framework. In addition, it has often been suggested that Islamic patterns are not merely intended as esthetic ornamentations but that behind each of them there is “a deep and subtle underlying spiritual philosophical message that has to be properly understood against the backdrop of the holy revelations of the Quran” (Aljamal and Banissi, 2003, p. 1). This principle extends to the design of gardens, such as the traditional chahár-bágh gardens which are rich in religious symbolism. Can the same be said about the use of fractal patterns within this cultural and artistic context?



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