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

Posted 2 months ago
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Decagons formed from stereographically projected points

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, θ/π], 
        EdgeForm[Lighter[Blend[cols, θ/π], .15]],
        Polygon[
         Table[Stereo[(Cos[θ] {0, 1, 0} + 
              Sin[θ] {Cos[t], 0, Sin[t]}).RotationMatrix[ϕ, {0, 1, 0}]],
          {t, π/2, 5 π/2, 2 π/n}]]},
       {θ, π/(2 m), π - π/(2 m), π/m}],
      m/2]]},
   PlotRange -> r, ImageSize -> 540, Background -> Blend[cols, 1/2]],
  {ϕ, 0, 2 π/n}]
 ]

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