Consider the following three basic questions about tetrahedra:
- Does a given tetrahedron tile space?
- Which tetrahedra are scissors-congruent to a cube?
- Can one describe the tetrahedra all of whose six dihedral angles are a rational number of degrees?
The first question goes back to Aristotle, the second is from Hilbert’s list of problems, and the third one was asked by Conway and Jones.
These questions are related by the following implications: any tetrahedron that tiles space is scissors-congruent to a cube, and also any tetrahedron with all six dihedral angles a rational number of degrees is scissors-congruent to a cube.
In a recent paper (arXiv:2011.14232) Kedlaya, Kolpakov, Poonen and Rubinstein solve the third question completely!
I recommend reading the press release of MIT regarding their result (link). It’s only three pages and contains some nice pictures!