[adeledeweylopez]

Terry Tao has a really great post where he depicts group actions (every Rubik's cube move sequence is a group action) as arrows in a graph, and with this picture, commutators can be seen as arrows that make a relatively small change: https://terrytao.wordpress.com/2012/05/11/cayley-graphs-and-... (see figure 14).

If you want to take the arrows very literally, you can think of the commutator as measuring the amount of curvature in the space the arrows lie in. The less of a difference the commutator makes, the less curved - with a commutator that doesn't do anything meaning the space is flat. Expressed in these terms, the two-part algorithm technique amounts to finding places in Rubik's cube algorithm space that are almost – but not quite – flat.