Geometry in Geometry, Examination of Fractal Dimension in the Entrance of Sheikh Lotfollah Mosque and Its Effect on Redefining the Meaning of Place

The place has lost its meaning in both aspects of spatial communication and the evocativeness of space and, accordingly, in contemporary architecture. The current research tries to provide the possibility of evaluating our cultural heritage for today's architecture in completely scientific and defensible ways. The proposed method is to read a qualified place architecture from the past with new geometry. Fractal geometry, one of the branches of new mathematics, has been introduced as a measurement tool in this research. This geometry, introduced by Benoit Mandelbrot in 1975, is extension of the principle of parallelism in Euclidean geometry. In fact, by introducing a random seed, a fractal dimension, and a Longitudinal parameter, fractal geometry makes it possible to identify natural phenomena that could not be recognized and analyzed by Euclidean geometry before that. The relationship between fractal geometry and architecture can be examined from three perspectives. The first look is conceptual. In this view, fractal geometry has been a tool for understanding flat and spatial decorations in architecture. If we can prove the fractal pattern in this view, the result is not generalizable to the entire architectural space. The second look is formal. In this view, fractal geometry has led to the creation of parametric architecture by influencing construction technology. Of course, most of these buildings are reduced to minimalist sculptural buildings, and the space and architecture lack an architectural language. The third view is linguistic and methodological. In this view, it is possible to read the fractal pattern in the architectural facade using the fractal dimension with the square grid method. As the meshing accuracy increases, the fractal dimension also increases. In this view, we can read the existence of a fractal pattern on the facade's surface, but we cannot generalize it to the entire architectural space. None of these three views have considered the fractal pattern in the architectural space and its extension in the user's mind. The proposed strategy of this research, which can be considered the fourth look, is based on examining the possibility of people's life inside a fractal. Reinterpreting old architecture with a new method may restore the lost meaning of place to contemporary architecture. The research strategy is to use logical reasoning. This strategy used the architectural space's volume simulation with a spatial fractal called the Menger sponge. The full and empty volumes in the architectural space have been calculated in two iterations. The fractal dimension of the Menger sponge can be generalized to architecture due to the similarity of the subtracted and remaining architectural modules. The entrance space of the Mosque of Sheikh Lotfollah Isfahan was selected as a case study, and the fractal dimension was calculated for it. The space of the entrance arch was considered the first iteration, and the space after the wooden entrance door, plus the spaces on both sides of the facade on both floors, was considered the second one. The fractal dimension is approximately 2.8 in both iterations. The presence of a single decimal number in both iterations indicates the existence of a dimension of the fractal pattern in space. Therefore, there is no problem in proving the existence of a fractal pattern in this space. This pattern is maintained in the depth range of 6.13-6.21cm from the facade. The results show that the meaning of the place changes, accepting the presence of a fractal pattern in this space. The meaning of place no longer ends with the observer's presence in the center of space but begins with his movement from the center of space. The observer moves from in front of the big entrance arch (first iteration), and with his presence on the second floor (second iteration) and turning his gaze to the field, the renewal of memory does not end but begins again. His perception of the meaning of place continues through repeating spaces to infinity. Repetitive and self-similar spaces in the fractal pattern cause the perception of diverse information from the space. The observer's mind crosses the realm of visible spaces and enters the visualization of the invisible space and the immeasurable and unpredictable realm. Reading the spaces with this method shows that there is no need to break the structure and remove the architectural language from the space in order to understand the fractal pattern in the space. Instead, fractal patterns can be identified everywhere by changing the viewing point, provided that the matter is looked at from the correct insight. The present research results can be used to create a new design language called "Zaya" for creating design algorithms.