A Three-Dimensional Box-Counting Method to Study the Fractal Characteristics of Urban Areas in Shenyang, Northeast China

Modern expansion and three-dimensional growth are rapidly altering the morphological features of traditional cities. This morphological phenomenon fully reflects the internal organization mode and composition rules of modern cities. This study draws on the research method of three-dimensional fractal, focusing on the situation where there is less research on the fractal form at the block scale, and conducts a fractal research on the three-dimensional form of a city at the meso and micro scales, in order to reveal the fractal characteristics of modern urban density. Based on fractal theory, the urban form of Shenyang, Northeast China, was quantitatively analyzed using the box-counting (Minkowski–Bouligand) method to calculate the two-dimensional (2D) and three-dimensional (3D) box dimensions of urban areas. Next, by analyzing the correlations between morphological indicators and 2D and 3D fractal dimensions, this study proposes cluster features of the correlation between the 3D fractal dimension and floor-area ratio. Then, this study summarizes the fractal characteristics of Shenyang’s urban form, based on the cluster analysis and spatial features of various urban areas within the city. The analysis results show the fractal dimension of Shenyang’s urban form to have characteristic expected values; fractal dimension clusters reflect spatial differences in the forms of different urban areas. The 3D D value of architectural morphology fractal in urban areas of Shenyang is between 2.41 and 2.70. From this, the representative characteristics of Shenyang’s urban form were obtained: first, it has the fractal characteristics of morphological hierarchy and system embeddedness; second, under unified and standardized management, its basic urban form structure displays the fractal characteristics of morphological similarity and system hierarchy; and third, its 3D urban form characteristics include the spatial accumulation of clusters and morphological patches, creating a patchwork of different building heights and densities, with the spatial clustering of density form highly correlated with the fractal dimension. The results of this research will provide reference samples for the morphological identification, design control, and design review of modern cities, and enrich the research results of the application of fractal theory to urban morphology at the meso and micro scales.