A quick note, as someone who occasionally fiddles with special functions:

The g2[] and g3[] functions in Clayton's post are effectively scaled versions of the built-in functions WeierstrassInvariantG2[] and WeierstrassInvariantG3[].

Additionally, the mapping under consideration involves (a scaled version of) the lemniscatic case of the $\wp$ function, such that a lot of things greatly simplify. In fact, one could avoid the Weierstrass functionality altogether, and just use Jacobi elliptic functions with parameter $m=\frac12$ instead, through well-known relations between them and the lemniscatic case of the $\wp$ function. Taking the Jacobi route has the advantage of being slightly more efficient in numerically evaluating the expressions involved.

More concretely, here is a simplified version of Clayton's original code:

DynamicModule[{ell = Function[z, 1 - 2 JacobiNS[EllipticK[1/2] z, 1/2]^2],
               width = .012, n = 8, cols = RGBColor /@ {"#21243d", "#88e1f2"}},
              Manipulate[Graphics[With[{c = 2 Cos[t] ((1 + I)/2)},
                         {FaceForm[cols[[-1]]], 
                          Table[Polygon[Join @@ Transpose[
                          Table[ReIm[-c + 2 Cos[t] # + Sin[t] Cayley[ell[#]]] & /@
                                {x +I (y - width), 1 - x + I (y + width)},
                                {x, -width, 1 + width, (1 + 2 width)/100}]]],
                                {y, 0., 1, 1/n}], 
                          Table[Polygon[Join @@ Transpose[
                          Table[ReIm[-c + 2 Cos[t] # + Sin[t] Cayley[ell[#]]] & /@
                                {x - width + I y, x + width + I (1 - y)},
                                {y, -width, 1 + width, (1 + 2 width)/100}]]],
                                {x, 0., 1, 1/n}]}], 
                          ImageSize -> 540, PlotRange -> Sqrt[3], 
                          Background -> cols[[1]]], {t, 0, π}]]

Thank you very much for the wonderful presentation Clayton!

No, it's based on ImageMagick, gifsicle, and ffmpeg, all of which I have installed already, so I prefer to just apply them directly myself.

It's basically replacing Manipulate with Table and then Export.

The only real trick is to make "DisplayDurations" smaller. "DisplayDurations" specifies how long each frame should be displayed; the default, I think, is not to specify anything, so then you just fall back to the default of the web browser you're viewing the GIF in, which is typically something like 1/10 second. This indeed makes for very choppy animations.

When possible I try to use "DisplayDurations" -> 1/50, so the animation is at 50 fps (which, due to the vagaries of the GIF format, is the maximum possible), but this obviously requires more frames and hence a larger filesize. To reduce the size I'll sometimes drop down to "DisplayDurations" -> 3/100 (again, due to the GIF format, the durations have to be multiples of 1/100), or about 33 fps, but this is about as low as I'm willing to go.

I didn't in this case, but sometimes I also need to postprocess the GIF in gifsicle or Gifski to try to reduce filesize in more sophisticated ways.

Anyway, here's the code I used to export the GIF:

squircle = 
  Block[{invts = {g2[2., 2. I], g3[2., 2. I]}, s, width = .012, n = 8, 
   c, cols = RGBColor /@ {"#21243d", "#88e1f2"}},
  s = WeierstrassP[1, invts];
  ParallelTable[
     c = 2 Cos[t] # + Sin[t] Cayley[-WeierstrassP[#, invts]/s] &[(1 + I)/2];
     Graphics[
      {FaceForm[cols[[-1]]],
       Table[
        Polygon[
         Join @@ Transpose[
           Table[
            ReIm[-c + 2 Cos[t] # + 
                Sin[t] Cayley[-WeierstrassP[#, invts]/s]] & /@
             {x + I (y - width), 1 - x + I (y + width)}, {x, -width, 1 + width, (1 + 2 width)/100}]]],
        {y, 0., 1, 1/n}],
       Table[
        Polygon[
         Join @@ Transpose[
           Table[
            ReIm[-c + 2 Cos[t] # + 
                Sin[t]  Cayley[-WeierstrassP[#, invts]/s]] & /@
             {x - width + I y, x + width + I (1 - y)},{y, -width, 1 + width, (1 + 2 width)/100}]]],
        {x, 0., 1, 1/n}]},
      ImageSize -> 540, PlotRange -> Sqrt[3], Background -> cols[[1]]],
     {t, 0., ? - #, #}] &[?/150]
  ];

Export[NotebookDirectory[] <> "squircle.gif", squircle, 
 "AnimationRepetitions" -> Infinity, "DisplayDurations" -> 1/50]

Thanks!