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A pentomino is a plane geometric figure formed by joining five equal squares edge to edge. It is a polyomino with five cells. There are twelve pentominoes, not counting rotations and reflections as distinct. They are used chiefly in recreational mathematics for puzzles and problems.〔(Eric Harshbarger - Pentominoes )〕 Pentominoes were formally defined by American professor Solomon W. Golomb starting in 1953 and later in his 1965 book ''Polyominoes: Puzzles, Patterns, Problems, and Packings''.〔(Eric Harshbarger - Pentominoes )〕〔(people.rit.edu - Introduction - polyomino and pentomino )〕 Golomb coined the term "pentomino" from the Ancient Greek / ''pénte'', "five", and the -omino of domino, fancifully interpreting the "d-" of "domino" as if it were a form of the Greek prefix "di-" (two). Golomb named the 12 ''free'' pentominoes after letters of the Latin alphabet that they resemble. Ordinarily, the pentomino obtained by reflecting or rotating a pentomino does not count as a different pentomino. The F, L, N, P, Y, and Z pentominoes are chiral; adding their reflections (F', J, N', Q, Y', S) brings the number of ''one-sided'' pentominoes to 18. Pentominoes I, T, U, V, W, and X, remain the same when reflected. This matters in some video games in which the pieces may not be reflected, such as Tetris imitations and Rampart. Each of the twelve pentominoes satisfies the Conway criterion; hence every pentomino is capable of tiling the plane. Each chiral pentomino can tile the plane without reflecting it. John Horton Conway proposed an alternate labeling scheme for pentominoes, using O instead of I, Q instead of L, R instead of F, and S instead of N. The resemblance to the letters is more strained, especially for the O pentomino, but this scheme has the advantage of using 12 consecutive letters of the alphabet. It is used by convention in discussing Conway's Game of Life, where, for example, one speaks of the R-pentomino instead of the F-pentomino. ==Symmetry== Pentominoes have the following categories of symmetry: *F, L, N, P, and Y can be oriented in 8 ways: 4 by rotation, and 4 more for the mirror image. Their symmetry group consists only of the identity mapping. *T, and U can be oriented in 4 ways by rotation. They have an axis of reflection aligned with the gridlines. Their symmetry group has two elements, the identity and the reflection in a line parallel to the sides of the squares. *V and W also can be oriented in 4 ways by rotation. They have an axis of reflection symmetry at 45° to the gridlines. Their symmetry group has two elements, the identity and a diagonal reflection. *Z can be oriented in 4 ways: 2 by rotation, and 2 more for the mirror image. It has point symmetry, also known as rotational symmetry of order 2. Its symmetry group has two elements, the identity and the 180° rotation. *I can be oriented in 2 ways by rotation. It has two axes of reflection symmetry, both aligned with the gridlines. Its symmetry group has four elements, the identity, two reflections and the 180° rotation. It is the dihedral group of order 2, also known as the Klein four-group. *X can be oriented in only one way. It has four axes of reflection symmetry, aligned with the gridlines and the diagonals, and rotational symmetry of order 4. Its symmetry group, the dihedral group of order 4, has eight elements. If reflections of a pentomino are considered distinct, as they are with one-sided pentominoes, then the first and fourth categories above double in size, resulting in an extra 6 pentominoes for a total of 18. If rotations are also considered distinct, then the pentominoes from the first category count eightfold, the ones from the next three categories (T, U, V, W, Z) count fourfold, I counts twice, and X counts only once. This results in 5×8 + 5×4 + 2 + 1 = 63 ''fixed'' pentominoes. For example, the eight possible orientations of the L, F, N, P, and Y pentominoes are as follows: 150px150px 150px 150px150px For 2D figures in general there are two more categories: *Being orientable in 2 ways by a rotation of 90°, with two axes of reflection symmetry, both aligned with the diagonals. This type of symmetry requires at least a heptomino. *Being orientable in 2 ways, which are each other's mirror images, for example a swastika. This type of symmetry requires at least an octomino. 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「pentomino」の詳細全文を読む スポンサード リンク
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