Comparative analysis of geometric forms and numerical calculation of fractals in Khayyam’s Mausoleum based on the Islamic architecture’s definitions and box-counting method

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 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 understand 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 the descriptive-analytic method after determining the common features between fractal forms with the 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), microscale and non-integer dimension. The elements used such as tiled ornaments, the Islamic star and Shamsa, domed-shaped roof, a 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.