Girih tile math

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)

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.

File for laser cutting girih tiles

The laser cutter reads vector graphics; a red line means 'cut' and a black surface means 'engrave'. I made an svg file with the girih tiles placed side by side. You can download the file, visit your local Fab Lab, and make your own girih tiles! There is some room for improvement in the file - each side is cut twice, which is a waste of time and possibly burns the acrylic more than necessary. This file works fine, but it would be even better if one would remove those double lines.


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Contents of the file:



At the sides of each piece, there is a 'teeth' pattern, which I put there to make the pieces align better. Another 'innovation' is the double black line that forms the outline of a rope tied in an infinite knot, with a crossing at the sides of each girih piece. It turned out that it is possible to design the tiles so that the rope regularly passes above, then below, then above... for any pattern that one builds with them.

More pictures of the tiles and of the laser cutting process.

Laser cutting Girih tiles

The laser cutter at Fab Lab Groningen.


The machine can both engrave and cut. It is almost magical to see one's design gradually appear as a physical object. Here the laser is cutting our girih tiles from a 3 mm acrylic sheet. The machine does the engraving first, one sees the knot pattern formed by the pieces appear. This is how girih patterns typically look when they are used for decoration, you see the knot pattern but not the borders between the pieces. Then the pieces are cut. The cut lines are quite different from the lines drawn on the tiles. Probably this is part of the reason for the complexity and beauty of girih patterns.


I find the Fab Lab concept fantastic, giving anyone the chance to use this kind of professional fabrication machines. They had 3D printers and a CNC mill as well. Not to mention the nice people at the Fab Lab, guiding me through the process of using the laser cutter!

Someone else also made a set of  laser cut girih tiles, at the Fab Lab in Lille. Some more pictures of our tiles, and the svg file for the laser cutter.

Platform game, level 4

After many months, we finally have a new version of the platform game! As always, it comes with a new level.

Download the game here: for Windows, Linux(32) and Linux(64). See these instructions for installing it on different platforms.

Acrylic Girih Tiles

We stumbled across the local Fablab on our holiday in Groningen, the Netherlands. We wanted to try their laser cutter, so I designed some girih tiles in Inkscape. These things have so many wonderful features, so more posts to are sure to follow - the laser cutting process, and the svg file for the laser cutting machine.



A nice collection of girih cut out of paper.