Geometry or topology behind the “impossible staircase”

Question

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.

Answer 1

I have seen the impossible staircase used to give an intuitive picture of cohomology. One has a bunch of local pictures (the individual sides of the staircase) and one wants to patch them together into a global picture (the entire staircase), but there's an obstruction to doing this (the implied heights don't match up), and this obstruction is in some sense a nonzero element of some cohomology group. In fact this is more or less how sheaf cohomology is defined, although I'm not the best person to explain in-depth about that.

In fact it seems to me like the impossible staircase represents in some way a nonzero element of the first cohomology of the circle. But I'm not sure how to be more precise about this.

Actually, I think I do know: the impossible staircase can be used to think about the analytic continuation of the logarithm $\log z$ along the path $z = e^{i \theta}$. After traveling along this path from $0 \le \theta \le 2 \pi$ we start from a height of $\log 1 = 0$ and end up at a height of $\log 1 = 2 \pi$. This reflects the fact that $\frac{dz}{z}$ is nonzero in the de Rham cohomology $H^1(\mathbb{C} - \{ 0 \}, \mathbb{R})$ (its antiderivative exists locally but not globally).

Answer 2

Monodromy doesn't literally work like a paradoxical staircase, but more like a completely possible spiral staircase that goes infinitely upwards and infinitely downwards. As illustration, see this graphic of the natural logarithm over multiple branches (cut off at top and bottom of course):

$\hskip 2.3 in$

(Source: Wikipedia. Bigger version.)

The paradoxical staircase has two mathematically relevant defining features: (1) it wraps back around to itself, and (2) it (at least ostensibly) increases in height along the way. Keeping to the case of complex numbers $\mathbb{C}$, it's possible both features could be captured if you identified points a certain height apart with each other, e.g. considered the two points $z$ and $z+\omega$ equal to each other for all $z\in\mathbb{C}$ and some fixed $\omega$. If you let $\omega = 2\pi i$, then you turn the complex plane in to some kind of tube that can be represented in the horizontal strip $0 \le \mathrm{Im}(s) < 2\pi$ (wrapping around top to bottom). Then the infinite staircase associated with $ \ln (z)$ (for example) is actually a single set of stairs that has both of the paradoxical staircase's defining features.

Answer 3

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.

Answer 4

David Huffman, though trained as an engineer, did lots of mathematical work. He is famous for the codes named after him (Huffman codes) that were a jumping off point for the mathematics of data compression.

http://www.huffmancoding.com/my-family/my-uncle/scientific-american

However, he also developed a coding system for taking drawings of polyhedra and checking they were "legal." The idea was to be able to have a computer "report" that something like the impossible staircase could not be built. Sugihara's book reports on this work and goes beyond it:

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.103.2279&rep=rep1&type=pdf