Experiments with plane-filling curves and Fourier transform

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?

It's not plane-fillingness, but fractalness of the inputs that makes the DFTs look interesting. So I'd experiment with other L-space system curves. A fractal has some periodicity that appears as bright lines on DFT, and when you add an extra level of recursion, the original periodicity remains and it gets supplemented with its harmonics. I'd also try to interpret a diagonal scan of the DFT images as a sound waveform, probably scaled down to fit it into the audible range. There's also a related so-called Radon transform that would make the DFT images more roundish.

Can you add some examples with different levels of recursion? It would be nice to see how the Fourier Transform changes when the level changes.

Is there something interesting at the limit when the recursion level is close to infinity?

Really great!

Curious: what about the reverse - draw the plane filling curves in the Fourier domain and then run an inverse transformation?

Edit: thanks for the reminder everyone, makes sense that the inverse would be roughly speaking “the same”! Cheers

The results of the Hilbert and Hilbert-like curves look similar to the Robinson’s aperiodic tiling of the plane[0]. In that picture they use squares and rounded squares, but if you did all rounded squares, I think it would look more similar.

[0] https://en.wikipedia.org/wiki/Aperiodic_tiling

I wonder what shape those squircles are?

Hmm, I was thinking about building a 2d game solver with a LLM and I was wondering the other day if there is a better way to represent 2D data instead of just raster scanning it to a vector. probably a hadamard transform might work out better but the hilbert pattern looks interesting too.

It is kind of striking how the images look like stars and galaxies

Could these be used to make dithering patterns?