Fractal Geometry-Based Center Identification in Geographic Domains

When we talk about finding the geometric center of a region, we usually rely on traditional Euclidean measures like area or length. But natural shapes, like coastlines, often have a fractal nature that doesn’t quite fit into those classic definitions. So, we need to rethink what we mean by a ''center''. In this study, we deal with the concept of a \emph{fractal center}. Our approach, which uses Hausdorff measures, helps us pinpoint the geographic center of an irregular area by accounting for the complexity of its boundaries. We focus on the Iberian Peninsula and its various sub-regions, which, to our knowledge, have yet to be explored. To tackle this, we blend fractal dimension analysis with a multi-scale center-of-mass computation. First, we estimate the coastline's fractal dimension using the box-counting method. Then, we define a center of mass based on the coastline's fractal measure. We also use a multi-scale approach to ensure our findings are robust across data resolutions. As a result of this analysis, we have found that the fractal center of the Iberian Peninsula differs markedly from the traditional area centroid. It actually shifts toward areas with more irregular coastlines, highlighting the unique characteristics of these natural shapes.