To summarize, the FFTs also have a recursive nature, which becomes more fine-granular with each recursion depth in the original. For example, this trippy FFT https://raw.githubusercontent.com/mxmlnkn/fft-image-experime... shows that the hierarchical square pattern probably repeats ad infinitum. Note that those details that get added with more recursion get lost when downscaling the images and the nearest-neighbor-upscaled images almost look like they are dithered: https://github.com/mxmlnkn/fft-image-experiments/raw/master/...
It is interesting though that the Fourier transform has a recursive nature that is still visible when downscaling a larger image. When doing that for any of the space-filling curves you basically just get a gray image because they evenly fill the space. Well, the Dragon curve and the Gosper Diagram do have boundaries that are still visible even when downsampling large-resolutions versions:
Hilbert Curve that simply looks gray when downsampled:
Is there something interesting at the limit when the recursion level is close to infinity?