Julia Cubes

S[q_] := Module[{x, y, e = Sqrt[Plus @@ Map[#^2 &, q]]},
   x = Sqrt[(q[[1]] + e)/2];
   y = (e - q[[1]]) x/(q[[2]]^2 + q[[3]]^2 + q[[4]]^2);
   {x, y q[[2]], y q[[3]], y q[[4]]}
   ];
Q[qc_, qlist_] :=
  Flatten[
   Map[
    Module[{q = S[# - (qc~Join~{0, 0})]}, {q, -q}] &,
    qlist
    ], 1];
MakeFractal[qc_] := Module[{qlist = {{1.0, 1.0, 1.0, 1.0}}},
   Do[qlist = Q[qc, qlist], {6}];
   Graphics3D[
    {Specularity[White, 6],
     {Hue[#[[4]]], Cuboid[Delete[#, 4], Delete[#, 4] + 0.06]} & /@
      qlist
     },
    Lighting -> "Neutral",
    Boxed -> False,
    Background -> RGBColor[0, 0, 0],
    ImageSize -> {800, 800}
    ]
   ];
DynamicModule[{pt = {0, 0}},
 LocatorPane[Dynamic[pt], Dynamic@MakeFractal[pt], {{-1, -1}, {1, 1}}
  , AutoAction -> True, ImageSize -> {800, 800}]
 ]