TITLE: Geometry or topology behind the "impossible staircase"
QUESTION [37 upvotes]: 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.

REPLY [7 votes]: It appears that the idea of impossible figures representing non-trivial cohomology classes has been considered by Penrose, who gives an interesting rigorization of the idea by taking the gluing data on the "overlaps" of non-impossible pieces to be how much you need to "zoom in/out" to make the pieces align.
Penrose, On the Cohomology of Impossible Figures.
Here is a freely available version (with French translation).
This article was brought to my attention by a recent MathOverflow post, which also cites an article by Chris Mortensen. Any further answers that are posted there are sure to be of interest.