This magical trick is done with the helical coordinate system. The correspondences above are not limited to two dimensions. They also link multidimensional space to one-dimensional space. Although we can perform multidimensional filtering and spectral operations in one dimension, the gain is only conceptual. The cost is the same.
Recursion is a powerful operation but it can be dangerous.
With recursion, stability always needs to be considered. Multidimensional recursion would be devoid of power without stability theory and the useful techniques that go with it.