Pathways

This was inspired by some very cool pieces by Caleb Ogg.

The basic setup is that I choose 20 random points on the sphere...

spherepoints = Normalize /@ RandomVariate[NormalDistribution[], {20, 3}];

...and 2 random rotation axes...

n = Normalize /@ RandomVariate[NormalDistribution[], {2, 3}];

...and then rotate around both axes at different rates. Here's the code:

With[{cols = RGBColor /@ {"#fd5f00", "#76b39d", "#05004e"}},
 Manipulate[
  Graphics3D[
   {Thickness[.006],
    Line[#, 
       VertexColors -> 
        Table[Directive[Blend[cols[[;; 2]], t/33], Opacity[1 - t/50]], {t, 0, 50}]]
      & /@ Transpose[
      Table[
       spherepoints.RotationMatrix[2 ?, n[[1]]].RotationMatrix[3 ?, n[[2]]],
       {?, -s, ?/2 - s, ?/100}]]},
   Background -> cols[[-1]], ImageSize -> 540, Boxed -> False, 
   SphericalRegion -> True, PlotRange -> 1.1, ViewAngle -> ?/10],
  {s, 0, 2 ?}]
 ]

If you want to use the exact points and axes used to produce the animation, here they are:

n = {{0.3728141055805563, 0.6721175529008367, 0.6397402893071225}, 
     {-0.10208165334304724, -0.6120577359476601, -0.7841968272808646}};
spherepoints = {{-0.10799106985352425, -0.4160584039226773, 
     0.9029027264092216}, {0.15157900537807614, -0.5578647580382791, 
     0.8159722525107612}, {-0.3907113954330362, -0.5122140334828592, 
     0.764840760800567}, {0.9730729495569006, -0.21261435696902062, 
     0.089017807495373}, {0.05688278789414075, -0.9809950827315541, 
     0.18550740173346517}, {-0.4021137935203268, -0.3483613122969294, 
     0.8467283467296068}, {-0.6190979467900267, 0.6365973432675175, 
     0.45984949149163135}, {0.9284570780710303, -0.30562860674878534, 
     0.2110891018423282}, {-0.02111841597881698, -0.6807852829904154, 
     -0.7321785376328684}, {0.19282461280624952, -0.6222904436398903, 
     -0.7586654548947043}, {-0.47354875944631875, -0.8700494995707841, 
     0.13698700932380656}, {-0.49852600855636126, 0.8644855768934002, 
     0.06431567566424101}, {-0.3016842794402542, 0.23403999944393247, 
     -0.9242358325659629}, {0.5083619289119673, 0.06303537237490779, 
     -0.8588333313643925}, {0.16849837994835584, 0.19647359703572312, 
     0.9659225753768382}, {0.7430004401900321, 0.6651325623033352, 
     -0.07449174747056579}, {0.16245182018285645, 0.8928343037536511, 
     -0.4200670329364233}, {0.290783587893684, 0.39701839696616603, 
     0.870529320288577}, {-0.4909586596128248, -0.8633143801147486, 
     -0.1168241226727844}, {0.21102565308351864, 0.4447293257316856, 
     0.8704504584265061}};