Tuesday, July 31, 2012

Girih tile math

Girih tile angles.All angles that appear in the girih tiles are multiples of 36 degrees, an angle that appears in a regular decagon. The girih pieces are strongly related to Penrose tiles: each girih tile can be decomposed into dart and kite Penrose tiles. The Penrose tiles are famous for creating aperiodic tilings, patterns that do not repeat themselves. Aperiodic patterns are possible to create with the girih tiles as well - for example, girih may be laid in a pattern of fivefold rotational symmetry. Fivefold symmetry is impossible in periodic tilings.

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)

Girih tiles with puzzle tabs.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 a girih tile.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.

Two decagon solutions.

The girih drawings for laser cutting and more pictures of the tiles and of the laser cutting process.
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