Against rigidity in composition.
“To consult the rules of composition before making a picture is a little like consulting the law of gravitation before going for a walk.” – Edward Weston
The rule of thirds suggests that a well-composed photograph has a key point of interest at one intersection of an imaginary tic-tac-toe grid superimposed on our image. The general idea is that symmetry is boring, the eye delights in seeing some dynamic weighting in an image, and that by making one of these four points our “focus” we have created an optimal balance.
This is a good idea as a starting-point, but that’s about it. Once absorbed, it should be discarded as too mechanical to want to follow rigidly.
There are many factors of structure and tone that go into what most of us see as interesting composition. Each of those factors carries a “weight” that changes our sense of artistic balance.
Of course composition likes some sense of balance and offset, but the best artistic compositions are done by feel rather than formula.