One way to unveil a hidden structure of a undirected 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 undirected 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.
E.g. the skeleton graph of a Platonic solid will "spring" into the Platonic solid itself, thus unveiling its "true nature".
Do these ways of unveiling hidden structures have names? Which abstract mathematical principles do they follow? What do they have in common or - more generally - have to do with each other?