A friend and I are having a bet. He states that there must be objects or mechanisms with 8 degrees of freedom (not counting translation} which have 3-fold symmetry (at least in some configurations). But we cannot find any.
He is thinking of objects like a deformable cubus with corners whose angles are not fixed. But such a cubus has
- three orientational degrees of freedom
- three internal angles which makes a total of only 6 degrees of freedom. A cubus has 3fold symmetry when seen along a diagonal, so that would fit; but 6 are not 8 degrees of freedom.
I brought up the idea of a tetrahedral skeleton, (like a methane molecule
On the other hand, I am not able to prove that the puzzle is impossible to solve.
Is there another solution? Where can one look for such objects or related theorems? Are there books or sites on these issues?
Thanks in advance!