One of the kits I used in the classroom was a set of parallelograms. We had a few kits left over, and when we sold them, we decided at the time not to make any more. They didn’t seem as mathematically dramatic as our new fractal tiles, although they are very interesting to play with. Recently, one of our customers commented on how her son spent a lot of time playing with his parallelogram tiles. It was all we needed to hear to bring them back.

A single, simple shape allows for a remarkable amount of exploration.

If you join pieces at the corners, you can put six together to get a star:

Note that pieces can be reflected (i.e. flipped over). By flipping every other piece, the star becomes:

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You can have 3, 4, 5, or 6 tiles meet at a point, and since each tile can have one of two orientations, there are many ways to connect these. Here are three of the several ways you can connect 4 pieces:

All of that flexibility is sufficient to create countless design variations. However, this parallelogram happens to belong to a special set of objects known as polyiamonds, which are polygons that are formed by joining the edges of equilateral triangles.

If a polyiamond is composed of 4 triangles, then it is called a *tetriamond*. Besides our parallelogram, there are two other tetriamonds, the *v* and the triangle:

Here are some designs we’ve created.