We study the algebraic and geometric structure of the space of minimal trajectories in the discrete hypercube Q₄. We show that the set of minimal trajectories between opposite vertices can be naturally identified with the symmetric group S₄, where each trajectory corresponds to a permutation describing the order of elementary coordinate changes. Endowed with its natural adjacency relation, this space coincides with the Cayley graph of S₄ with respect to adjacent transpositions, and is isomorphic to the edge graph of the permutahedron P₄. This establishes a structural correspondence between minimal trajectories, permutations, and vertices of this polytope, providing a unified framework connecting discrete geometry, algebraic combinatorics, and group theory.
Christian Perez Puig (Wed,) studied this question.