M. gave us a lecture on how $S_4$ can be regarded as rigid transformation (rotations about the origin) of a cube centered at the origin. This was not straightforward for me to visualize. So ,after a long contemplation what to use: OpenGL/C++, Povray+Blender, Mathematica and its very useful animation features, I ended up deciding to use Mathematica. I had an intuition that Mathematica, albeit not being perfect in animation and visualization like Blender and Povray will provide for a very fast result (which at the moment was a priority for me). To visualize this you can use the following code:

g=Graphics3D[{Opacity[0.3], Cuboid[{-5, -5, -5}, {5, 5, 5}], Opacity[1.0], Thick, Green, Line[{{-5, -5, -5}, {5, 5, 5}}], Line[{{5, -5, -5}, {-5, 5, 5}}], Line[{{-5, 5, -5}, {5, -5, 5}}], Line[{{-5, -5, 5}, {5, 5, -5}}], Black, Text[Style[1, Large, Bold, Red], {5, 5, 5}], Text[Style[2, Large, Bold, Red], {5, -5, 5}], Text[Style[3, Large, Bold, Red], {-5, 5, 5}], Text[Style[4, Large, Bold, Red], {-5, -5, 5}]},Boxed->False,PlotRange->{{-9,9},{-9,9},{-9,9}}]; Animate[MapAt[Rotate[#, t,{0,5,5},{0,0,0}]&,g,{1}],{t,0,Pi}, AnimationRunning->False,AnimationRepetitions->1,AnimationDirection->ForwardBackward,FrameLabel->"Transposition(1,3)"] Animate[MapAt[Rotate[#,t,{5,5,5},{0,0,0}]&,g,{1}],{t,0,2Pi/3}, AnimationRunning->False,AnimationRepetitions->1,AnimationDirection->ForwardBackward,FrameLabel->"3-Cycle (2,3,4)"] Animate[MapAt[Rotate[#,t,{0,0,1},{0,0,0}]&,g,{1}],{t,0,Pi/2}, AnimationRunning->False,AnimationRepetitions->1,AnimationDirection->ForwardBackward,FrameLabel->"4-Cycle (1,3,4,2)"] Animate[MapAt[Rotate[#,t,{0,0,1},{0,0,0}]&,g,{1}],{t,0,Pi}, AnimationRunning->False,AnimationRepetitions->1,AnimationDirection->ForwardBackward,FrameLabel->"2 Transpositions (1,4)(2,3)"]

Yes combining the Animate in one single panel was rather a pain as I did not know how to change variables (axis and angle of rotation) all in one panel (I wasn’t able to do it with * Buttons* and

*in Mathematica).*

`Dynamic`

If you do not have Mathematica, you can still download the animation file here. Now I am proud to claim that I did a Mathematica animation on my own: Time from 0-knowledge to one animation = Roughly 1 hour. If this was Blender+Povray, although I have experience in working with them, I would still need much more time to model, texture and finally generate the animation. So my conclusion is: although the animations and features of Mathematica has many things left to be desired, it is probably the best choice if you want to animate a few mathematical object very fast on the fly. The 1hr work I invested, is now a 5min. work for any other remainder animation I want to make in Mathematica. If the animation I want to do would require much more time than this, e.g. complicated interactions with buttons, phong and radiosity etc., I would probably look at other options as well.

To save an avi you may use something like the following (notice: I have hidden sliders and panel for a better look of the video).

Export["C:/temp/output1.avi",

Manipulate[MapAt[Rotate[#, t,{0,5,5},{0,0,0}]&,g,{1}],{t,0,Pi,

AnimationRunning->True,AnimationDirection->ForwardBackward,AnimationRepetitions->1,Paneled->False,ControlType->None},

FrameLabel->"Transposition(1,3)"]

]

This results in a very poor video compression (quality of videos were OK, but for 4sec. I got 35Mb of video). But that is easily remedied by using Handbrake (I do this if I don’t have too much time to play around more complicated tools) or VirtualDub (which I only use if I have a lot of time to invest.. which is sadly less often the case). I combined the four different videos showing a permutation from each conjugacy class of $S_4$. Here is the final video after compression (also a link was given earlier for you to download the video in case you do not have HTML5 support in your browser):