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Comment by wirrbel | Hacker News Reader

I wonder whether one could show this. maybe by pretending the image is actually a cell in a periodic grid, then transform this grid, and rotate this grid. The periodicity would make roation possible without getting empty spots.

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That's a good idea. The condition implicitly assumed to get the result mentioned ("A rotation in real-space is equivalent to one in reciprocal space") is that the original image is periodically tiled in all directions. Your suggestion mimics this condition.

Since your other comment mentioned limitations of optical apertures, maybe it's worth noting that the way the 45 degree rotation was done here (plunking the rotated image into a black background) mimics the action of an optical aperture -- that is, clipping the observed image along that rectangle.

This causes 45 degree spikes, similar to: https://en.wikipedia.org/wiki/Diffraction_spike#Diffraction_....

Yes. I was going to add that the lines in the original image are more likely due to the discontinuity between the top, bottom, left, right edges. The other comment wasn’t mine, but is right that an aperture in the back focal plane of a lens would cut out image information at higher resolution.

Something I find interesting is that, although on the surface this seems like a way to deal with rotations in general, it only works for certain angles. For instance, with a 45 degree rotation, you have to scale the image, too, and the resulting tile contains two of the original image.

Topologically, an underlying problem is that there is not a continuous family of rotations for a torus (the topological space corresponding to two independent periodic dimensions).

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