Each girih tile can be constructed of smaller girih tiles. The Penrose tiles have the same property. If this subdivision is repeated, it may lead to an aperiodic tiling - depending on the rules for replacing large tiles with smaller ones. More on this in this article by Raymond Tennant (pdf).
|From the paper in Science.|
We first heard about these tiles in a paper in Science (here without subscription)
We wanted to give our pieces some jigsaw-puzzle-like tabs to keep the tiles aligned when building, but the pentagon creates a parity problem. Instead we drew a zigzag shape on each side. Now all the sides are identical - no parity problems - and the sides align nicely.
The knot pattern on each piece is two straight lines in from the middle of each edge, at 54 degree angles. Where these lines meet inside the tile, they are joined. For the other tiles, the rules are unambiguous, but for the ten-sided pieces, Wikipedia mentions that there should be two solutions (but all pictures I've seen show only one) Well, after some thinking, we found another solution (probably 'the' other solution), so we're happy to show our ten-sided pieces with their different knot patterns.
EDIT: Some more decagon solutions.
The girih drawings for laser cutting and more pictures of the tiles and of the laser cutting process.