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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-'. The name was suggested by recreational mathematics writer Thomas H. O'Beirne in ''New Scientist'' 1961 number 1, page 164. ==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 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, … . 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 . |- |Triamond |align=center|1 | |- |Tetriamond |align=center|3 | |- |Pentiamond |align=center|4 | |- |Hexiamond |align=center|12 | |- |} ==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). Polyiamond Symmetries 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Polyiamond」の詳細全文を読む スポンサード リンク
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