👉 The math underlying the concept of fractals, particularly the Mandelbrot set, showcases the intricate beauty of self-similarity and infinite complexity. A fractal is a geometric shape that displays the same patterns at different scales, meaning it looks similar whether you zoom in or out. The Mandelbrot set is defined by a simple iterative formula: \( z_{n+1} = z_n^2 + c \), where \( z \) and \( c \) are complex numbers. For each point \( c \) in the complex plane, the sequence generated by this formula is examined. If the sequence remains bounded (doesn't go to infinity), \( c \) is part of the Mandelbrot set; otherwise, it's not. This process reveals an infinitely detailed boundary with infinitely many intricate structures, demonstrating how simple rules can produce complex and beautiful patterns. The self-similar nature of fractals, including the Mandelbrot set, highlights the deep connections between algebra, geometry, and analysis in mathematics.
