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: 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.
Name
Number of forms
Forms
Moniamond
1
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° ; ditto for 3-fold rotational symmetry, combined with mirror symmetry.
Every polyiamond of order 6 or less tilesthe plane. All but one of the heptiamonds tile the plane, the exception being the V-heptiamond.
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.
The set of 22 polyiamonds, from order 1 up to order 6, constitutes the shape of the playing pieces in the board gameBlokus Trigon, where players attempt to tile a plane with as many polyiamonds as possible, subject to the game rules.
