A polyiamond (also polyamond or simply iamond) is a polyform whose base form is an equilateral triangle. The word polyiamond is a back-formation from diamond, because this word is often used to describe the shape of a pair of equilateral triangles placed base to base, and the initial "di-" looked like a Greek prefix meaning "two-".
[edit] Counting polyiamonds
The basic combinatorial question is how many different polyiamonds exist with a given number of cells. Like polyominoes, polyiamonds may be free or one-sided. Free polyiamonds are invariant under reflection as well as translation and rotation. One-sided polyiamonds distinguish reflections.
The number of free n-iamonds for n = 1, 2, 3, … is:
- 1, 1, 1, 3, 4, 12, 24, 66, 160, … (sequence A000577 in OEIS).
The number of free polyiamonds with holes is given by 
A070764; the number of free polyiamonds without holes is given by A070765; the number of fixed polyiamonds is given by A001420; the number of one-sided polyiamonds is given by A006534.
| Name |
Number of forms |
Forms |
| Moniamond |
1 |
|
| Diamond |
1 |
|
| Triamond |
1 |
|
| Tetriamond |
3 |
|
| Pentiamond |
4 |
|
| Hexiamond |
12 |
|
[edit] 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. Asymmetry requires at least 5 triangles. 3-fold rotational symmetry without mirror symmetry requires at least 7 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).
 |
 |
 |
 |
 |
| Asymmetric |
Mirror, 0° |
Mirror, 30° |
Rotational, 2-Fold |
Mirror, 2-Fold |
 |
 |
 |
 |
 |
| Rotational, 3-Fold |
Mirror, 0°, 3-fold |
Mirror, 30°, 3-fold |
Rotational, 6-Fold |
Mirror, 6-Fold |
[edit] 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.
[edit] Tessellations
Every polyiamond of order 6 or less tiles the plane. All but one of the heptiamonds tile the plane.[1]
[edit] Correspondence with Polyhexes
Pentiamond with corresponding pentahex superimposed.
Every polyiamond corresponds to a polyhex, as illustrated at right. Conversely, every polyhex is also a polyiamond, because each hexagonal cell of a polyhex is the union of six adjacent equilateral triangles.
[edit] See also
[edit] External links
[edit] References
A070764; the number of free polyiamonds without holes is given by 
.svg)
.svg)
.svg)
