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A polyiamond (also polyamond or simply iamond) is a polyform in which the base form is an equilateral triangle. The word polyiamond is a back-formation from diamond, motivated by the fact that this word is often used to describe the shape of a pair of equilateral triangles placed base to base.

Counting polyiamonds

The basic combinatorial question is how many different polyiamonds with a given number of triangles exist. If mirror images are considered identical, the number of possible n-iamonds for n = 1, 2, 3, … is :

:1, 1, 1, 3, 4, 12, 24, 66, 160, …

As with polyominoes, fixed polyiamonds (where different orientations count as distinct) and one-sided polyiamonds (where mirror images count as distinct but rotations count as identical) may also be defined. The number of free polyiamonds with holes is given by ; the number of free polyiamonds without holes is given by ; the number of fixed polyiamonds is given by ; the number of one-sided polyiamonds is given by .

the moniamond:
the diamond:
the triamond:
the 3 tetriamonds:
the 4 pentiamonds:

Symmetries

Possible symmetries are mirror symmetry, 2-, 3-, and 6-fold rotational symmetry, and each combined with mirror symmetry.

2-fold rotational symmetry with and without mirror symmetry requires at least 2 and 4 triangles, respectively. 6-fold rotational symmetry with and without mirror symmetry requires at least 6 and 18 triangles, respectively. No symmetry requires at least 5 triangles. Only 3-fold rotational symmetry requires at least 12 triangles.

In the case of only mirror symmetry we can distinguish having the symmetry axis aligned with the grid or rotated 30° (requires at least 4 and 3 triangles, respectively); ditto for 3-fold rotational symmetry, combined with mirror symmetry (requires at least 18 and 1 triangles, respectively).

Generalizations

Like polyominoes, but unlike polyhexes, polyiamonds have three-dimensional counterparts, formed by aggregating tetrahedra. However, polytetrahedra do not tile 3-space in the way polyiamonds can tile 2-space, so they are of little mathematical interest.

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