# How do you prove you've found all symmetries of an object in 3D?

In group theory class we studied the example of rotational symmetries of the regular tetrahedon. The teacher showed us 12 symmetries and then said "if you stare long and hard you can convince yourself that those are all symmetries". Is there a way to rigorously prove this?

• What is the level of rigor required? – Kenny Lau Sep 14 '17 at 8:00
• It's just to satisfy personal curiosity, so as rigorous as possible such that I can understand (I have had 3 lessons of group theory, know basic linear algebra and proof methods). – Pel de Pinda Sep 14 '17 at 8:15
• There are 4 faces. Choose one to be the bottom face. Now, you can choose one of the 3 remaining faces to be the front face. This gives you $4\times3=12$ elements. Is this rigorous enough? – Kenny Lau Sep 14 '17 at 8:16
• That's rigourous enough to show that there are at least 12, but I was hoping you could prove that there cannot be more than 12. – Pel de Pinda Sep 14 '17 at 8:18

A linear transformation in $\mathbb R^3$ is uniquely determined by its action on any set of $3$ linearly independent vectors, or a superset of such a set. In particular, it is determined by its action on the vertices of the tetrahedron. So it suffices to consider permutations of the vertices.
There are $4!=24$ different permutations. Half of these reverse the orientation of the tetrahedron, which is not done by rotations. That leaves just $12$ permutations that could possibly be realized by rotations.