Oh! This is applying FFT to the images! I thought it was going to be applying to functions f_n : [0,2pi] -> [0,1]^2 where f_n is the n-th level version of the curve.

I guess in that view, higher order curves would have higher frequencies have higher amplitudes, on account of changing direction more often.

I guess that view of things wouldn’t have much visual appeal, as it would just be associating to each integer frequency, an individual vector. (I guess a 2d complex-valued vector?)

Hm, still, I think there would probably be uh, something to see in the directions of these vectors?

And to handle the complex-valued stuff I would think that if you moved from the e^{i t n} basis to the cosines and sines basis, that one could maybe thereby do-away with that? Or... Well, yeah, that should be true, expressing a real valued periodic function as a weighted sum of sine and cosine functions doesn’t require any complex coefficients, but would doing so actually make the vectors which are the pairs of the coefficients for the two axiis, have a particularly clear meaning?

I would imagine the direction of these vectors should correspond to something like the different directions the curves move in, or differences between these directions.

Oh! One thing that might be nice visually, If you got the Fourier coefficients for the n-th level of the curve, and then took only the first k coefficients and constructed the curve associated with that, then that might be cool looking. I wonder, if you held k constant, but increased n, would the first k coefficients converge? If so, what would the curve from those first k coefficients look like?