Tile Types and Matching Rules

There are three basic types of tiles in this tesselation:

, and.

These shapes are well known, see for example Steurer, W. and Deloudi,S. (2009): Crystallography of Quasicrystals. Springer-Verlag and are called star, boat and hexagon. The corresponding tiling is called HBS-tiling.

The faint lines show that the edges of all three tiles are bounded by half kites. So an obvious matching rule is to give the edges a direction which ensures that the half kites of adjacent tiles complement to full kites and not to parallelograms. In the Wikipedia list of aperiodic tile sets (the tiles are named starfish, ivy leaf and hex tile there) this method is used.

For our purpose, we use a different approach. Using another colour for the darts we observe that the corners of the tiles are the centers of three types of pentagons

This gives us three different types of stars while there still is only one boat and one hexagon. Appropriately coloured, we now have these five tiles with colour induced matching rules which force an aperiodic tiling:

, , ,

and .

We add some strapwork and create a basic set of "Girih tiles":

and.

As you can see in this picture there are characteristic structures of two hexagons and one boat which make up a decagon. Finally, we will add a decagonal tile to our tileset analogous to the decagon of the Girih tile set.

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