Pack It In

Consider the eight vertices of the cube as points on the unit sphere, then make (spherical) circles centered on those points with radii chosen so that the circles are just tangent. Then this gives a packing of the sphere by 8 circles of (spherical) radius $\frac{1}{2} \arccos \left(\frac{1}{3}\right) \approx 0.61548$. (Of course, this is not the optimal packing by 8 circles; that is given by circles centered on the vertices of the square antiprism.)

Now, rotate the sphere and the eight circles around the $x$-axis and stereographically project down to the plane. After adding some color, the result is this animation.

Here's the code:

Stereo[{x_, y_, z_}] := 1/(1 - z) {x, y};

DynamicModule[{p, b, 
  verts = Normalize /@ PolyhedronData["Cube", "VertexCoordinates"], 
  t = 1/2 ArcCos[1/3], 
  cols = RGBColor /@ {"#EF6C35", "#2BB3C0", "#161C2E"}},
 Manipulate[
  p = RotationMatrix[?, {1, 0, 0}].# & /@ verts;
  b = Orthogonalize[NullSpace[{#}]] & /@ p;
  Graphics[{EdgeForm[None],
    Table[{cols[[Floor[(i - 1)/4] + 1]],
      Polygon[Table[Stereo[Cos[t] p[[i]] + Sin[t] (Cos[s] b[[i, 1]] + Sin[s] b[[i, 2]])], {s, 0, 2 ?, 2 ?/200}]]},
     {i, 1, Length[verts]}]}, PlotRange -> Sqrt[6], 
   Background -> cols[[-1]], ImageSize -> 540],
  {?, 0, ?/2}]
 ]