torus_ring.m

Torus[r1_, r2_, c_] :=
 (* Produces a torus with major radius r1 and minor radius r2,
    with surface color c and parallel with the xy plane. *)
 ParametricPlot3D[
  {Cos[t] (r1 + r2 Cos[u]), Sin[t] (r1 + r2 Cos[u]), Sin[u]},
  {t, 0, 2 Pi}, {u, 0, 2 Pi},
  ColorFunction -> (c &),
  Mesh -> None,
  PlotPoints -> 10,
  PlotRange -> {{-3, 3}, {-3, 3}, {-3, 3}}]
  
TorusRing[c_, dθ_,  dt_] :=
 (* Produces a ring of 12 tori with varying major radii.
 c - the color of each torus 
 dθ - the initial angular offset of the entire ring
 dt - intended to control the spinning of the ring as well as the
         growing and shrinking of each torus. *)
 Graphics3D[
  Table[
   Translate[
    First@
     Torus[1.5 + 1.5 Sin[2 θ] + 2 dt], 1, c],
     {15 Cos[θ] + dθ + dt], 15 Sin[θ] + dθ + dt], 0}],
   {θ, 0, 2 Pi, Pi/6}]]
  
 ThreeRings[c1_, c2_, c3_, dt_] :=
 (* Produces three intersecting rings, one parallel to the xy plane,
    one in the plane that intersects the line y=x, 
    and the third in the plane that intersect the line y=-x.
    c1, c2, c3 - are all Mathematica Hue or name color values
    dt - controls the rotation and size of each torus for each of the
    rings. *)
   Graphics3D[
     {First@TorusRing[c1, 0, dt],
      Rotate[
        First@TorusRing[c2, Pi/2, dt], Pi/2, {1, -1, 0}] ,
      Rotate[
        First@TorusRing[c3, Pi/2, dt], Pi/2, {-1, -1, 0}]}]

Export["/torus_rings.gif",
 Table[
  Graphics3D[
   First@ThreeRings[
     Hue[0.4, 0.7, 0.5], 
     Hue[0.05, 0.7, 0.9], 
     Hue[0.6, 0.7, 0.5], 
     dt],
   Background -> Black,
   Boxed -> False,
   ImageSize -> {600, 600},
   ViewAngle -> 0.25,
   ViewPoint -> {5 Sin[dt], 5 Cos[dt], 1}],
  {dt, 0, 2 Pi, 0.01 Pi}],
 "DisplayDurations" -> 0.05]