Callum wrote:
I'm pretty sure that the small edge pieces hang out too much and would need magnets to hold them in.
On a regular tetrahedron yes, but on a reuleaux tetrahedron it is possible. I did a quick sketch in solidworks:


As you can see, the rail is small on the little edge piece, but I think it is enough stable.
Interesting wavy cut pattern when looking at the edges :
