Truchet tiles

A square with two arcs in it, rotated at random, is enough to draw curves that wander across the whole floor and never cross.

In 1704 Sébastien Truchet published a study of square tiles split along a diagonal into two colours, and counted the patterns that their four orientations could make. The version most people now call a Truchet tile is a later variant, described by Cyril Stanley Smith in 1987: each square carries two quarter-circle arcs, each joining the midpoints of two neighbouring sides.

Tile B is tile A turned by 90°. Every arc ends at the midpoint of a side.

Because every arc ends exactly at a midpoint, any arrangement of the two tiles joins up. Arcs from neighbouring squares meet at the shared midpoint with the same tangent, so the result is a set of smooth curves, and since the two arcs in one tile never touch, curves never cross each other. Each curve either closes into a loop or runs off the edge of the grid.

Fourteen by eight tiles, each orientation chosen by a coin flip (seed 1704).

The regions between the curves can always be coloured with two colours so that neighbours differ. The quick way to see it: colour the corners of the square grid like a chessboard. Each arc cuts off exactly one corner of its tile, so every region contains grid corners of a single colour, and regions on the two sides of an arc get opposite colours.

Things to try:

Bias the coin. With probability 0.9 for one tile the curves become long diagonal stripes; at 0.5 they look like a maze.
Replace the quarter circles by straight segments between the same midpoints. The curves turn into zig-zag paths with corners.
Use hexagons with three arcs per tile, each joining the midpoints of two neighbouring edges. There are again two ways to place a tile, and the curves still never cross.