It seems that this is the first question on Truchet tiles on MO.

Shown above is a picture of a random tile, which you can see the resulting configuration is much like many membranes of cells.

I wonder if there may be some interesting and deep math behind random Truchet tiles, but I do not know much about this topic. As I can guess, there might be questions like these:

Given a random tile in finite grid, what is the expectation length of a closed curve?

Questions like percolation on this graph, for example, give an infinite tiling of the plane, what is the probability that there is an infinite connected area? (2-color the basic pattern with blue and red, then the red area is like cells.)

Connections with the $O(1)$ loop model. (I'm sorry, I know very little about this, but I do think it is in the picture.)

Of course this list is very incomplete, and I hope someone can fill it and show me the connections with other branches of mathematics. Thanks!

Ising model on a square lattice, see Discrete Complex Analysis and Probability by Stanislav Smirnov. $\endgroup$ – john mangual Jan 19 '15 at 14:58