Polyiamond

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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-" looked like a Greek prefix meaning "two-".

Counting[edit]

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 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
Polyiamond-1-1.svg
Diamond 1
Polyiamond-2-1.svg
Triamond 1
Polyiamond-3-1.svg
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

Symmetries[edit]

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.png

Generalizations[edit]

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.

Tessellations[edit]

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.

See also[edit]

External links[edit]

References[edit]