Square Truchet Tiles

We start, as always, with the simple Cartesian grid, and the simplest set of two Truchet tiles.

We of course are not limited to these two tiles. In fact, we can come up with as many tiles as we like, as long as they contain strokes that "exit" the tile at the four cardinal directions.

We can extend this to have two notches on each side of the square, parameterizing its position from the midpoint of the square edge. Since I think that non-crossing tilings look much better, and we have $4$ pairs to connect, this results in the Catalan number  $C_4=14$ tiles to consider.

Since a square has an even number of sides, we can put three notches on each side, resulting in $6$ pairs of notches to connect. We have $C_6=132$ tiles to consider.