I believe there is a very minor typo: the flag should read: PlotStyle -> Texture[texture] ?

Thanks, Greg, for doing the Riemann sphere... a nice addition to this already colorful thread.

It's possible (probable?) that the visual similarity arises from expansions around a zero being dominated by low-order linear terms.

But what I was really hoping for may be too unrealistic--Is there a description of something like the Eisenstein series in terms of rotating propellers and sliding shutters?

Looks like a good entry for the "Wolfram One -liner competition"! It also solves my problem to convert any image into a "roller shutter captured" image. here is the transformation on he Wolfram logo using your code:

You can use the following code to transform an image in one step:

ImageAdd[ImageTransformation[img, Block[{x, y}, {x, y} = #; 
    RotationTransform[(550 - y*550)*Degree, {.5, .5}][{x, y}]] &], 
 Graphics@Disk[]]

manipulate 550 for different options.

Example:

And here's a way to do this with a continuous color wheel. This is sort of a blend between your idea and @Henrik Schachner's posted here.

The idea is with a standard color wheel we can exploit its cyclic nature and turn rotation into scalar addition. Just add a constant to the h channel to rotate the wheel.

hsbImage = Compile[{{r, _Integer}},
  Table[{(ArcTan[r - j - .001, i - r] + π)/(2π), 1, 1}, {i, 0, 2r}, {j, 0, 2r}]
];

r = 1024;
rps = 2.25;

base = Image[hsbImage[r], ColorSpace -> "HSB"];
mask = Image[DiskMatrix[r]];

splits = ImagePartition[base, {2r + 1, 1}][[All, 1]];

transform = ImageAssemble @ Table[
  {ImageAdd[splits[[k]], Hue[rps * k/(2r + 1), 0, 0]]}, 
  {k, 2r + 1}
];

Here we do a little trickery to renormalize the hues and turn the image into a disk:

final = ColorCombine[
  {
    ImageMultiply[Mod[#1, 1], mask], 
    ImageMultiply[#2, mask], 
    #3
  }& @@ ColorSeparate[transform], 
  "HSB"
]

To add discrete sectors, like in the original post above, we can simply round:

Round[final, 1/6]

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