Polyiamond
Wikimedia Commons has media related to Polyiamonds. 
A polyiamond (also polyamond or simply iamond) is a polyform whose base form is an equilateral triangle. The word polyiamond is a backformation 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".
Contents
Counting
The basic combinatorial question is, How many different polyiamonds exist with a given number of cells? Like polyominoes, polyiamonds may be either free or onesided. Free polyiamonds are invariant under reflection as well as translation and rotation. Onesided polyiamonds distinguish reflections.
The number of free niamonds for n = 1, 2, 3, … is:
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 onesided polyiamonds is given by A006534.
Name  Number of forms  Forms  

Moniamond  1 


Diamond  1 


Triamond  1 


Tetriamond  3 


Pentiamond  4 


Hexiamond  12 

Symmetries
Possible symmetries are mirror symmetry, 2, 3, and 6fold rotational symmetry, and each combined with mirror symmetry.
2fold rotational symmetry with and without mirror symmetry requires at least 2 and 4 triangles, respectively. 6fold rotational symmetry with and without mirror symmetry requires at least 6 and 18 triangles, respectively. Asymmetry requires at least 5 triangles. 3fold 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 3fold rotational symmetry, combined with mirror symmetry (requires at least 18 and 1 triangles, respectively).
Generalizations
Like polyominoes, but unlike polyhexes, polyiamonds have threedimensional counterparts, formed by aggregating tetrahedra. However, polytetrahedra do not tile 3space in the way polyiamonds can tile 2space.
Tessellations
Every polyiamond of order 6 or less tiles the plane. All but one of the heptiamonds tile the plane.^{1}
Correspondence with polyhexes
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.
See also
External links
 Weisstein, Eric W., "Polyiamond", MathWorld.
 Polyiamonds at The Poly Pages. Polyiamond tilings.
 VERHEXT — a 1960s puzzle game by Heinz Haber based on hexiamonds
References

HPTS  Area Progetti  EduSoft  JavaEdu  N.Saperi  Ass.Scuola..  TS BCTV  TS VideoRes  TSODP  TRTWE  