You tend to get these sorts of spirals in all kinds of situations where the curvature is continuously varying in some way. In the case of my Riemann turtles that condition seems to be induced by a periodic variation in the zero positions that itself slowly varies.
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I need to think more about it, though, because, as you recall, when I pasted bits of the critical strip together and just showed the zeros, I didn't see the arcs you saw from the phase variations--instead I saw straight diagonal streaks, which got me onto Fourier transforms.
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