I've been thinking about the fact that an n×n square grid has as many cells as a triangular lattice with n triangles on each side.
Is there a nice continuous bijection between them? When n is odd, you can at least keep one line of symmetry throughout.

Here's the same thing for a 2×2 grid. The triangle in the middle has to go one way or the other, so you have to break the vertical line of symmetry

I've made a nice looping thing!
triangle-square.glitch.me