Quantitative Assessment of the Façades of Yıldırım Bayezid Bath Using Fractal Dimension and Shannon Entropy

Different façades of the same historic building may exhibit distinct geometric and informational characteristics as a result of variations in architectural configuration. This study quantitatively evaluates the north, south, east, and west façades of Yıldırım Bayezid Bath in Mudurnu, Türkiye, using fractal dimension (FD) and Shannon entropy (H) to examine geometric complexity and grayscale information diversity, respectively. Architectural façade drawings obtained from the General Directorate of Foundations were standardized using a common graphical protocol to ensure comparability. Geometric complexity was quantified through two-dimensional box-counting fractal analysis, while histogram-based Shannon entropy was calculated from the probability distribution of pixel intensities in standardized 8-bit grayscale images. The results revealed measurable differences among the four façades. The west façade exhibited the highest values for both FD (1.7959) and H (3.3059 bits), whereas the east façade showed the lowest values (FD = 1.7048; H = 2.3295 bits). Both measures produced an identical ranking of West > North > South > East. However, the magnitude of differentiation among the façades varied between the two measures, particularly for the north and south façades, whose FD values were relatively close while their entropy values showed greater separation. These findings indicate that agreement in façade ranking does not imply equivalence between the two measures. Rather, fractal dimension and Shannon entropy provide complementary quantitative information by capturing different properties of architectural façades. Their combined application offers a consistent framework for quantitatively examining façade characteristics within the same historic building and provides a methodological basis for further comparative investigations across broader samples of historic buildings.