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[GIF] Rotation Redux (Inverse Cayley transform of rotating grid of circles)

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Inverse Cayley transform of rotating grid of circles

Rotation Redux

A recent post by Dave Whyte (a.k.a. @beesandbombs) on Twitter reminded me of my old GIF Rotation, which is almost 5 years old (and dates back to before I was posting on the Wolfram Community). I figured it would be fun to do a new twist on the same basic animation.

The basic idea is fairly simple: start with a rotating grid of circles:

Rotating grid of circles

Apply the map $z \mapsto 1/z$ to the circles to invert them inside the unit circle:

Rotating grid of circles, inverted inside unit circle

(Notice that the inside circles are rotating the other way; not surprising since $1/z$ is proportional to $\bar{z}$.)

Now just apply the inverse of the Cayley transform which maps the upper half-plane to the interior of the unit circle (and the lower half-plane to the exterior) and you get the animation from the top of the post.

InverseCayley[z_] := I (1 + z)/(1 - z);

Here's the rest of the code:

With[{r = 12., cols = RGBColor /@ {"#381460", "#f4f4f4"}},
 Manipulate[
  Graphics[
   {cols[[1]], Thickness[.003], Line[{{-8, 0}, {8, 0}}],
    Table[
     Polygon /@
      Transpose[
       Table[
        {ReIm[InverseCayley[E^(I t) + #]], 
           ReIm[InverseCayley[1/(E^(I t) + #)]]}
          &[E^(I ?)*(a + I b)], {t, 0., 2 ?, 
         If[Norm[a + I b] < 8, 2 ?/400, 2 ?/20]}
        ]
       ],
     {a, -r, r, 2}, {b, DeleteCases[Range[-r, r, 2], If[a == 0., 0., 100]]}],
    If[Abs[? - ?/2] < .01,
     {Polygon[{{-8, 2.1}, {8, 2.1}, {8, 10}, {-8, 10}}],
      Polygon[{{-8, -2.1}, {8, -2.1}, {8, -10}, {-8, -10}}]}]
    },
   PlotRange -> 4.5, Axes -> False, ImageSize -> {540, 540}, 
   Background -> cols[[-1]]],
  {?, ?/4, 3 ?/4}]
 ]

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