A polyiamond is a shape formed by a finite number of equilateral triangles.

Three members of the family admit a periodic tiling:

: the tetriamond (four equilateral triangles)

: the octiamond (eight equilateral triangles)

: a new shape ([1, Figure 6.2])

Six kites are formed by connecting the center of a regular hexagon to the midpoints of its edges. A shape made from multiple connected kites is known as a polykite. The shapes and are both polykites.

Both the aperiodic tiling and the proof of aperiodicity are complicated [2, 3].

Remarkably, the family may provide a simpler proof sketched here as:

1. Suppose the large tiling is periodic. Then there is a minimal periodic patch with translation vector .

2. The polyiamond shapes and would each have translation vectors.

3. Since the area of is 4 and the area of is 8, a term is needed in one of the translation vectors.

4. A translation vector of does not work on the triangular grid.

The paper [1] is still under review, so this sketch is conjecture.