One way to unveil a hidden structure of a symmetric graph - given as an adjacency matrix - is to permute the rows and columns until a pattern with a maximal geometrical symmetry is found. (The maximum may be local or global.) The pattern in question is a 2-dimensional one.

Another way to unveil a hidden structure of a symmetric graph is to replace its edges by identical springs (that may - unphysically - cross each other in the course of relaxation) and look at the 3-dimensional figure in which the system has reached minimal energy. (This figure may be unique or not. It may have locally or globally minimal energy.) The pattern in question is a 3-dimensional one.

Do these ways of unveiling hidden structures have names? Which abstract mathematical principles do they follow? What do they have in common?