Symmetries of a Tetrahedron video

[MrBeezer]

Hello,

I made this video with my iphone depicting the Symmetries of a Tetrahedron for a presentation I did recently:



I have been searching and trying to figure out if I have presented it correctly that a Tetrahedron has full S4 symmetry if we could reflect it in a "higher dimension."

I was basing this statement off of the fact that the rotational symmetry equilateral triangle is achieved through even permutations. However, the odd permutation suggest a reflection that requires the triangle to move through an additional dimension of space.

I am assuming we can say the same about a Tetrahedron where odd permutations can only be applied if we could move the shape through a higher spatial dimension. Is this stated correctly?

Thank you,

-Mike

 

[Ben Niehoff]

The rotational symmetry group of a regular tetrahedron in 3-space is $A_4$, the even permutations of 4 elements. If reflections are allowed, then you have $S_4$, the full group of permutations.