A fresh look

I have been working for a few days to understand the following passage in a mathematical paper: "A nontrivial Alexandroff angle can be defined. Armed with the concept of Alexandroff angle..." Great! The Alexandroff angle can be defined! So about that definition? What do you mean you didn't bother to include it?!? Ack! So I've been alternately staring at this paper and drawing 2-cell embeddings, all the while trying to understand this stupid angle. In the middle of this, a friend called and I chatted for awhile. After hanging up I stared down at the paper and suddenly understood what the Alexandroff angle had to be. Why does it work this way?