[GIF] This is Only a Test (Decagons from stereographic projections)

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stereo29.gif

This is Only a Test

This one is fairly straightforward. Form 60 concentric circles on the sphere centered at the point (0,1,0). On each circle, take 10 equally-spaced points, stereographically project to the plane, and form a decagon from the resulting points. Now rotate the sphere and all the points on it around the axis (0,1,0). The result (at least after adding some color) is this animation. This is a sort of discretized companion to my old still piece Dipole.

Here’s the code:

Stereo[p_] := p[[;; -2]]/(1 - p[[-1]]);

With[{r = 2, n = 10, m = 60, 
  cols = RGBColor /@ {"#2EC4B6", "#011627", "#E71D36"}},
 Manipulate[
  Graphics[
   {EdgeForm[Thickness[.0045]],
    Join[{Reverse[#[[1]]], #[[2]]}]
      &[Partition[
      Table[
       {Blend[cols, ?/?], 
        EdgeForm[Lighter[Blend[cols, ?/?], .15]],
        Polygon[
         Table[Stereo[(Cos[?] {0, 1, 0} + 
              Sin[?] {Cos[t], 0, Sin[t]}).RotationMatrix[?, {0, 1, 0}]],
          {t, ?/2, 5 ?/2, 2 ?/n}]]},
       {?, ?/(2 m), ? - ?/(2 m), ?/m}],
      m/2]]},
   PlotRange -> r, ImageSize -> 540, Background -> Blend[cols, 1/2]],
  {?, 0, 2 ?/n}]
 ]
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