# Topological surface thought experiment

Imagine a two-dimensional version of your lives on some compact, connected surface (orientable or non-orientable). How would you figure out on which surface you are living? Are there experiments you could conduct to determine the orientability and genus of the surface?

• "Time is a flat circle"? Apr 7, 2014 at 3:36

Alternatively, you can count for each $$\ell\geq0$$ the number of simple closed geodesics of length at most $$\ell$$. As $$\ell$$ goes to infinity, this grows like $$L^{6g-6}$$ with $$g$$ the genus of the surface (provided the genus is at least $$2$$ and the curvature is constant and equal to $$-1$$). This is a theorem of Mirzakhani.
• What's the capital $L$? Just a constant? Apr 10, 2014 at 0:53