Information AboutPolyiamond |
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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 '', 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 POLYIAMONDS 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 Polyomino es, ''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 .
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° (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). GENERALIZATIONS Like Polyomino es, but unlike Polyhex es, polyiamonds have three- Dimension al counterparts, formed by aggregating Tetrahedra . However, Polytetrahedra do not tile 3-space in the way polyiamonds can tile 2-space. SEE ALSO EXTERNAL LINKS |
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