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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, 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-".

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 (sequence A000577 in OEIS):

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 OEISicon lightA070764; the number of free polyiamonds without holes is given by OEISicon lightA070765; the number of fixed polyiamonds is given by OEISicon lightA001420; the number of one-sided polyiamonds is given by OEISicon lightA006534.


Name Number of Forms Forms
Moniamond1 Polyiamond-1-1
Diamond1 Polyiamond-2-1
Triamond1 Polyiamond-3-1
Tetriamond3 Polyiamond-4-2Polyiamond-4-1Polyiamond-4-3
Pentiamond4 Polyiamond-5-1Polyiamond-5-2Polyiamond-5-3Polyiamond-5-4
Hexiamond12 Polyiamond-6-1Polyiamond-6-2Polyiamond-6-3Polyiamond-6-4Polyiamond-6-5Polyiamond-6-6Polyiamond-6-7Polyiamond-6-8Polyiamond-6-9Polyiamond-6-10Polyiamond-6-11Polyiamond-6-12

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).

Polyiamond-5-2 Polyiamond-4-1 (rotated) Polyiamond-3-1 Polyiamond-4-2 Polyiamond-2-1 (rotated)
Asymmetric Mirror, 0° Mirror, 30° Rotational, 2-Fold Mirror, 2-Fold
Polyiamond 3-fold rotational symmetry Polyiamond 3-fold mirror symmetry (0 deg) Polyiamond-1-1 Polyiamond 6-fold rotational symmetry Polyiamond-6-11
Rotational, 3-Fold Mirror, 0°, 3-fold Mirror, 30°, 3-fold Rotational, 6-Fold Mirror, 6-Fold

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.

See also

External links

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