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"Triamond" redirects here. For the crystal structure, see Laves graph.

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-' looks like a Greek prefix meaning 'two-' (though diamond actually derives from Greek adamant). The name was suggested by recreational mathematics writer Thomas H. O'Beirne in New Scientist 1961 number 1, page 164.


The basic combinatorial question is, How many different polyiamonds exist with a given number of cells? Like polyominoes, polyiamonds may be either 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 the OEIS).

The number of free polyiamonds with holes is given by OEISA070764; the number of free polyiamonds without holes is given by OEISA070765; the number of fixed polyiamonds is given by OEISA001420; the number of one-sided polyiamonds is given by OEISA006534.

Name Number of forms Forms
Moniamond 1
Diamond 1
Triamond 1
Tetriamond 3
Polyiamond-4-2.svg Polyiamond-4-1.svg Polyiamond-4-3.svg
Pentiamond 4
Polyiamond-5-1.svg Polyiamond-5-2.svg Polyiamond-5-3.svg Polyiamond-5-4.svg
Hexiamond 12
Polyiamond-6-1.svg Polyiamond-6-2.svg Polyiamond-6-3.svg Polyiamond-6-4.svg Polyiamond-6-5.svg Polyiamond-6-6.svg Polyiamond-6-7.svg Polyiamond-6-8.svg Polyiamond-6-9.svg Polyiamond-6-10.svg Polyiamond-6-11.svg Polyiamond-6-12.svg


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 Symmetries


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.


Every polyiamond of order 6 or less tiles the plane. All but one of the heptiamonds tile the plane.[1]

Correspondence with polyhexes[edit]

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. (Note, however, that neither correspondence is one-to-one.)

See also[edit]

External links[edit]