In 2020, the 59 sporadic rational tetrahedra were identified. More recently, I found exact solutions for all of them. Most of them don't pair up well in terms of similar triangles that would allow them to be stuck together.
However, 18 of them are nice and all buildable in the Zome system. All dihedral angles are rational fractions of $\pi$. Blue and Green struts are multiples of 1/4, Yellow struts are multiples of 1/3 and Red struts are multiples of 1/5.
Theoretically, they could be rescaled to fit together into some nice shape. Question: with all of these in some shape, what is the maximum number of completely surrounded edges?