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# Tesselating with Ammann-Beenker tiles

Most tilings we see around us are periodic tesselations. They are tesselations because there are no gaps, and periodic because they repeat like wallpaper. In fact, until the 1960s, people thought that any finite set of tile shapes that could tesselate could be used to tesselate periodically. Then researchers discovered tiles that only tesselate non-periodically. Such tiles are called aperiodic.

Robert Ammann, an amateur mathematician, found sets of aperiodic tiles. One set has two simple shapes: a square, and a rhombus.

Of course, you can tile the plane like wallpaper with rhombi and squares. But Ammann added edge-matching features to these tiles. These features must form one of two shapes where tiles meet.

We’ve made those features look like hearts and arrows. To ensure that you connect all edges correctly, we provide the edge markings as red and black key tiles, that fit into the original tiles.

Each of Ammann’s tiles can be flipped over. The square–based tile has reflective symmetry, meaning it is identical to its mirror image. So flipping it over does not affect where you can place it.

The rhombus–based tile, on the other hand, is different from its mirror image. Since our tile set is magnetic and the magnet is on one side of the tile, we provide both orientations.

As you put the tiles together, you will often see that there are spaces that can’t be filled by a key tile. Here, the arrow won’t fit.

However, you can rotate the right tile 180°, or replace the right tile with its rotated mirror image:

The tiles can tesselate the plane, so a natural objective is to tile outward from the center in all directions without leaving any gaps. Or, you can deliberately leave gaps, or negative spaces, in which case you have a packing.

You can find the Ammann-Beenker tiles in our store.