Sampling fractal boxes, N-dimensional

Can we get more interesting clusters by sampling from boxes in higher dimensions?

For N=3 dimensions, we can still plot. The result is not very remarkable.

For higher dimensions, we have to project onto a subplane. I use sklearn's PCA to find the subplane with the highest variation. The result is that we get more noise points, so it's not more useful than the standard 2-dimensional box sampling.