This question on the topology of Escher games reminded me of a question I've had in my head for a little while now.

Is there anything interesting geometric or topological that can be said about the so-called "impossible staircase"?

**Motivation:** The other day, I found myself explaining the concept of monodromy (which I myself admittedly do not fully understand) to a layman friend of mine. She took my explanation to mean something like the impossible staircase, which I'm not sure is the best analogy. But it got me thinking nevertheless, and so I'm curious to see if there's anything interesting to say on the matter.