Fourier transform of the Sierpinski carpet

[sheaf]

Hello,

I was curious to see what the Fourier transform of a fractal would look like.

I took a Sierpinski carpet:



and plotted the norm of the Fourier transform, with a logarithm on top of it to reduce the contrast (see the attachment).

The Fourier transform is itself a fractal, which mimics the structure of the original carpet. But the squares are rounded, and each of them is tangent to several other squares at several levels.

Do you have any explanation for this ?