Polyiamond
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, motivated by the fact that this word is often used to describe the shape of a pair of equilateral triangles placed base to base.
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 :
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 ; 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 .
| the moniamond: |  | The moniamond |
|
| the diamond: |  | The diamond |
|
| the triamond: |  | The triamond |
|
the 3 tetriamonds: |  | The 3 tetriamonds |
|
the 4 pentiamonds: |  | The 4 pentiamonds |
|
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. No symmetry requires at least 5 triangles. Only 3-fold rotational symmetry requires at least 12 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).
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, so they are of little mathematical interest.
*
Polyiamond at MathWorld
