Rational tetrahedra with edges in arithmetic progression (Q2565544)

This paper discusses tetrahedra with rational edges forming an arithmetic progression, focussing on whether they can have rational volume or rational face areas. Several infinite families are found that have rational volume; a face can have rational area only if its edges themselves are in arithmetic progression and a tetrahedron can have at most one such rational face area. The proofs proceed by considering various cases of possible edge configurations utilizing Heron's formula, Mordell's birational transformations and cubic elliptic curves. In particular, the following questions are discussed: Q1 -- Can any of the faces of a tetrahedron have rational area? Q2 -- Can two or more faces have rational area simultaneously? Q3 -- Can all faces have rational area simultaneously? The paper ends with the following conjecture: A tetrahedron with a certain configuration of edges \{faces: 7, 8, 9; 6, 7, 11; 9, 10, 11; 6, 8, 10\} is the only tetrahedron with edges in arithmetic progression having a rational volume and a rational face area.