Now, it should matter which way they're flipped. Is there more than one way to arrange the flips of the common edges? Or there is only the one?

And feel free to reply with questions – I'll do my best to answer them here.

Come check them out if you're in the neighborhood!

Of course I wrote an academic paper with all of the gory details – look for it in the upcoming Bridges 2026 conference proceedings. Here's a figure that gives a taste of what the software does.

FEM solvers tend to throw up their hands and quit when you point them at non-manifold geometry like the folded and glued square, but I hollered at mine until it started behaving.

How many unique tile designs do you see here? (Hint, it's the same in both images.)

If you look closely at the wooden tiles you'll see there are three distinct tile designs. And here are two different icosahedron tilings, one with a single tile design, and one with every tile unique.

BTW when solving the PDEs, we don't consider the effects of the folds, just the "glue".

The central trick is to solve the Gray-Scott PDEs on an unusual topological domain that collapses all of the edges into a single segment.

PBCs are like the old-school Pac-Man arcade game: crossing the left side of the "screen" warps you to the right side, and ditto for top/bottom.

(BTW if these orbs look familiar, maybe you saw the thread I posted about a year ago about similar work I did.)

To get this you need the tiles' edges to all be identical and mirror-symmetric. This is not obvious at first glance when looking at the overall tiling, IMO.