Today I rant about the two methods I have come up with to do what is in the following image.  

One involves subdividing the edges that appear to be cut in half, using [alt] + J to make quads then doing the steps covered here (if applicable) to add a vertex to the corners, and make them quads.  

The other method (which involves less singles-vertex selecting) is to select the area on one side of where detail is to be added, extrude by a multiple of 1 unit (hold the control key), move the vertices on the edge, and move what was selected, and extruded, back to where it was before.  

If nothing crashes I will attempt to present this in some sort of animation

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Suzanne, but something isn't right. Let's split around the eye.
	Image of .

I hope that made sense.  There is no good tool, but I like the extrude method (at the moment, maybe the novelty hasn't worn off yet) because it allows using different selection tools (subdividing all of the edges at once for the previous method  will increase detail in both directions, not just one).  

I don't think that made sense.  Oh well.