Evaluating the Fractal Pattern of the Von Koch Island Using Richardson’s Method

The principles of fractal geometry have revolutionized the characterization of complex geometric objects since Benoit Mandelbrot’s groundbreaking work. Richardson’s method for determining the fractal dimension of boundaries laid the groundwork for Mandelbrot’s later developments in fractal theory. Despite extensive research, challenges remain in accurately calculating fractal dimensions, particularly when dealing with digital images and their inherent limitations. This study examines the numerical artifacts introduced by Richardson’s method when applied to the Von Koch Island, a classic fractal curve, and proposes a novel approach for computing fractal dimensions in image analysis. The Koch snowflake serves as a key example in this analysis; it serves to assess the algorithm of fractal dimension calculation as his theoretical one is known. However, there is a fundamental difference between the theoretical calculation of fractal dimension and the actual calculation of the fractal dimension from digital images with a given resolution undergoing discretization. We propose eight different calculation methods based on Richardson’s area–perimeter relationship: the Self-Convolution Patterns Research (SCPR) method accurately estimates the fractal dimension, as the 95% confidence interval includes the theoretical dimension.