Hexiamonds

Hexiamonds are polyiamonds composed of six equilateral triangles. Wolfram MathWorld gives them the names

Eisenstein integers

The Eisenstein integers are numbers of the form $a + b\omega$, where $\omega = \frac{-1 + i\sqrt{3}}{2}$. $\mathbb{Z}[\omega]$ is a ring whose elements form a triangular lattice.

A triangle in the lattice can be uniquely defined as a set of three Eisenstein integers, and a set of six connected triangles corresponds to a unique hexiamond position in the lattice.

Exact cover

Knuth writes in Dancing Links

In the late 1950s, T. H. O'Beirne introduced a pleasant variation on polyominoes by substituting triangles for squares. ... O’Beirne was particularly fascinated by the fact that seven of the twelve hexiamonds have different shapes when they are flipped over, and that the resulting 19 one-sided hexiamonds [i.e., those distinct up to reflection; see pictures/orientations.pdf for which hexiamonds are chiral] have the correct number of triangles to form a hexagon: a hexagon of hexiamonds.

Here is one such hexagon of hexiamonds.