Quantitative analysis of the geometric complexity of ishak pasha palace using 2D and 3D fractal methods

This study aims to quantify the geometric complexity of the Ishak Pasha Palace, a prominent example of 18th-century Ottoman palace architecture, by integrating two and three-dimensional fractal analyses. Within this framework, orthogonal UAV photographs of the four main façades were processed into black and white contour maps, and façade-based two-dimensional fractal dimension (FD2) values were computed using the box-counting method via the ImageJ – FracLac plugin. For the three-dimensional assessment, a 3D digital model was generated from archival plans and elevations, and the building’s volumetric fractal dimension (FD3) was calculated using a cube-counting algorithm developed in Unity. Prior to the main calculation, the algorithm’s reliability was validated through calibration on a third-iteration Menger sponge with a known theoretical fractal dimension. The test produced FDtest = 2.7217, differing from the theoretical value (2.7268) by only 0.0051 (0.19%), indicating high accuracy. Results show façade FD2 values ranging from approximately 1.78 to 1.84 (mean = 1.8050), while the overall building volume has FD3 = 2.5774. These results extend FD-based quantification into an architectural interpretation of façade articulation and volumetric hierarchy, supporting systematic comparison across monumental historic complexes.