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.

自从Benoit Mandelbrot（1924–2010）在1975年提出“分形”一词以来，分形几何学的数学理论就深刻地影响了景观感知，建筑和技术领域。确实，它们描述彼此嵌套的复杂形式并向无穷大重复的能力使人们能够对诸如天气模式或植物生长之类的混沌现象进行建模。人们发现某些人类设计的模式（例如伊斯兰文化所发展的模式）遵循类似的等级，对称和重复原则。但是，这些原理在花园设计中的应用尚待开发。本文对四倍花园设计模型chahár-bágh进行了比较探索。-通过分析两个案例研究来分析典型的波斯和伊斯兰花园设计：泰姬陵和伊斯法罕的城市规划。众所周知，这种四重模式不仅具有宗教意义，而且还与公平分配的理想联系在一起。使用受建筑启发的创新成分分形分析，我们的结果表明，这些花园包含高水平的自我复制和尺度不变性，并且具有较高的分形维数。这种分析方法在历史景观规划中的新颖应用，使我们能够评估欧洲文艺复兴和曼德布罗特探索之前分形概念已经使用到何种程度，并在伊斯兰和波斯花园设计的背景下推测它们的象征意义。具体来说，我们得出的结论是，这些花园的分形特征可能旨在代表无限的神性，也可以代表公平与平等的原则。展望未来，这种方法可以应用于设计空间，即在有意义且环保的城市结构基础设施设计中。