Symmetry-Based Approach to the Problem of a Perfect Cuboid

A perfect cuboid is a rectangular parallelepiped in which the lengths of all edges, the lengths of all face diagonals, and also the lengths of spatial diagonals are integers. No such cuboid has yet been found, but their nonexistence have also not been proved. The problem of a perfect cuboid is among the unsolved mathematical problems. The problem has a natural $S_3$-symmetry connected to the permutations of edges of the cuboid and the corresponding permutations of face diagonals. In this paper, we give a survey of author's results and results of J. R. Ramsden on using the $S_3$ symmetry for the reduction and analysis of the Diophantine equations for a perfect cuboid.