I had assumed animating Islamic tiles would be a big complicated endeavor but it only took a few hours of messing around in Flash to figure it out. I just made a half-equilateral-triangle mask and tiled it, then made a fairly simple animation underneath:
Obviously all kinds of variations can be made from here. If only I had tried this earlier!
The reason this is a Big Deal for me, is I didn’t know how easy it would be to tile squares with hexagons. I thought if I made a hex-based system like this I’d just get hexagons and equilateral triangles, not squares. But look – there are squares all over the place!