The Islamic era is the period of new s‌tyles and beyond its own time. Shamsa, Muqarnas, S‌tar, geometric motifs, and decorative elements have fractal nature, which was used as a tool to express the architect’s idea by repeating the same components in a spatial dimension. We will reach the effect of Islamic impacts on fractal architecture and semantic cognition, and by finding the roots of theories about fractal with an analytical-descriptive approach, we will unders‌tand the relationship between fractals and how they are used by architects. Our focus is on comprehensive s‌tudy and calculation, not only in ornaments but also in s‌tructure, to find patterns of fractal form, to represent coordination between components. Firs‌t, we proved and extracted the fractal properties and patterns between the cons‌tituent elements, including self-similarity, repeatability, small-scale, and non-integer dimension, using visual analysis, and then we set up the Box-counting analysis technique with two purposes of calculating fractal dimension and finding their relationship. Aware of the mathematical proportions and relationships between the components of nature, the architect of the Sheikh Lotfollah Mosque has displayed the geometric sequence limit with an ascending equation in the interior ornaments of the dome. His goal is both to create the world of spirituality on a micro-scale and make it unders‌tandable for an observer. Thus, in this way, he has sometimes expressed his concept through mathematics, proportions, and sometimes by showing beliefs in concepts such as unity in diversity, all of which in their essence have concepts consis‌tent with fractal forms. تفاصيل المقالة

The interaction and balance between fractal components, which maintain to function in nature in perfect harmony with the highest efficiency, has been the source of inspiration for architects in different eras. Today, contemporary Iranian architecture needs a comprehensive study and familiarity with world issues, hence the Khayyam’s mausoleum was chosen. the main purpose of this article is to analyze the geometry and numerical calculation of the fractal size of the building and its semantic knowledge in the contemporary building, which leads to finding the architect’s goals of how to design these forms and common concepts with Islamic architecture. After achieving how to use Islamic elements in a contemporary monument, we come to the interrelationship between fractal geometry and architectural design.With the box-counting technique, the fractal dimension was calculated by checking the two-dimensional plates in 4 different scales from the largest to the smallest scale to prove its decimal dimension. Then, information was collected by descriptive-analytic method and after determining the common features between fractal forms with analytical and comparative process, the obtained quantitative data resulted in extracting similar cases from the building. Finally, the relationship between the characteristics of fractal geometry and the architecture of the tomb was proved and the obtained patterns were matched with the concepts of self-similarity, iteration (proportion and symmetry), micro scale and non-integer dimension. The elements used such as tiled ornaments, the Islamic star and Shamsa, domed-shaped roof, regular numerical pattern of geometric shapes with symbolism, unity in diversity, infinity and dynamism are reminiscent of Islamic architectural style. Furthermore, fractal has been used in three methods: conceptual, mathematical and intuitive, and has been manifested in both structural and aesthetic aspects. تفاصيل المقالة