On two algebraic parametrizations for rational solutions of the cuboid   equations
  A rational perfect cuboid is a rectangular parallelepiped whose edges and face diagonals are given by rational numbers and whose space diagonal is equal to unity. Its existence is equivalent to the existence of a perfect cuboid with all integer edges and diagonals. Finding such a cuboid or proving its non-existence is an old unsolved problem. Recently, based on a symmetry approach, the equations of a perfect cuboid were transformed to factor equations. The factor equations turned out to be solvable and, being solved, have led to a pair of inverse problems. Our efforts in the present paper are toward solving these inverse problems. Algebraic parametrizations for their solutions using algebraic functions of two rational arguments are found. 
John Ramsden, Ruslan Sharipov
math.NT