Fun fact, the Mandelbrot set is a 2-dimensional set (because it’s defined in the complex plane). However, its boundary line is a fractal, which can be understood as having a non-integer dimension (i.e., between 1, the topological dimension of a line, and 2, the dimension of a plane). There are multiple ways to define fractal dimensions such as the Hausdorff dimension. For example, the Sierpinski triangle has a Hausdorff dimension of 1.58. But the Mandelbrot set is special here, too, as it seems to have a Hausdorff dimension of 2, meaning that its boundary is so curly that it fills “a plane’s worth of space” despite its line-like topology.
Fun fact, the Mandelbrot set is a 2-dimensional set (because it’s defined in the complex plane). However, its boundary line is a fractal, which can be understood as having a non-integer dimension (i.e., between 1, the topological dimension of a line, and 2, the dimension of a plane). There are multiple ways to define fractal dimensions such as the Hausdorff dimension. For example, the Sierpinski triangle has a Hausdorff dimension of 1.58. But the Mandelbrot set is special here, too, as it seems to have a Hausdorff dimension of 2, meaning that its boundary is so curly that it fills “a plane’s worth of space” despite its line-like topology.