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markdow's avatar
DFT space-filling tiles
3 3 289 (1 Today)
By markdow   |   Watch
Published: January 26, 2014
A 2D discrete Fourier transform of an aperiodic square tiling, colored by phase. The original tile motif was an interlocking boundary. In principle the original image could be reconstructed as an inverse Fourier transform of this image.


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2187x2187px 1.88 MB
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anonymous's avatar
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anseldow's avatar
What causes the continuous dark slashes across the 2nd and 4th quadrants?
markdow's avatar
Good question, and the answer is 'I don't know'. I started to track this down, but can't explain it well. I do know that it is not a mistake.

I have done a whole slew of these DFTs on similar aperiodic patterns, and this exact feature appears in many/most of them. See, for example,  DFT of the Thue-Morse pattern by markdow  DFT-of-the-Thue-Morse-pattern-116830280

The pattern for which this is the DFT is very simple, so it is a good case to consider. Roughly speaking, "there is zero amplitude when the phases in the x and y direction add up to +-pi, iff the signs of the phases are the same". This must have something to do with the symmetries of the patterns, e.g. whether they have mirror/rotation symmetries. But I haven't pinned that down.
anseldow's avatar
Thanks! I think I'm going to have to go back and brush up on my DFT mechanics before I can fully grasp all that.

Both are gorgeous images!
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