Symmetrical rotating tetrahedron

Question

How to find the vertices of a regular tetrahedron? a dodecahedron?

I already get the coordinate of the tetrahedron, but i don't know how to rotate it just like the gif

Answer 1

tet = Tetrahedron[{{0, 0, 0}, {-(1/(2 Sqrt[3])), -(1/2), Sqrt[2/3]}, 
   {-(1/(2 Sqrt[3])), 1/2, Sqrt[2/3]}, {1/Sqrt[3], 0, Sqrt[2/3]}}]

frames = Table[Graphics3D[
    GeometricTransformation[{FaceForm[Opacity[.5, Blue], Opacity[.5, Yellow]], tet}, 
     RotationTransform[t 2 Pi, {0, 0, 1}, {0, 0, 0}]], 
    Lighting -> "Neutral", Boxed -> False, 
    PlotRange -> {{-1, 1}, {-1, 1}, {0, 1}}, 
    ViewPoint -> {3, 4, 1}], {t, -1, 1, .01}];

ListAnimate[frames]

To get some wobbling effect, you can use RotationTransform[t 2 Pi, {0, Abs[t]/5, 1}, {.1, .2, 0}] and use PlotRange -> {{-1, 1}, {-1, 1}, {-.1, 1}} 1.2:

Use Export["tets.gif", frames] to create an animated GIF file.

Answer 2

You can create a table of ViewPoint frames. The graphics option Viewpoint is the point in space where objects are viewed. For example I can create the polygon using the code above and then a table of the graphics with the viewpoint circling around a circle of radius 2 in the x-y plane:

 myAnimationFrames=Table[Show[myTetra,ViewPoint-> 
 {Re[2Exp[I t]],Im[2 Exp[I t]],0},PlotRange->2], 
 {t,0,2 \[Pi],0.2}];  

And then save it to a gif file:

 myAnimationName = NotebookDirectory[] <> 
 "myAnimationFile.gif";
 Export[myAnimationName, myAnimationFrames];