Pentomino

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A pentomino is a polyomino composed of five (Ancient Greek πέντε / pénte) congruent squares, connected along their edges (which sometimes is said to be an orthogonal connection).

There are 12 different free pentominoes, often named after the letters of the Latin alphabet that they vaguely resemble. Ordinarily, the pentomino obtained by reflection or rotation of a pentomino does not count as a different pentomino.

The F, L, N, P, Y, and Z pentominoes are chiral in two dimensions; adding their reflections (F', J, N', Q, Y', S) brings the number of one-sided pentominoes to 18. The others, lettered I, T, U, V, W, and X, are equivalent to some rotation of their mirror images. This matters in some computer games, where mirror image moves are not allowed, such as Tetris-clones and Rampart.

Each of the twelve pentominoes can be tiled to fill the plane. In addition, each chiral pentomino can be tiled without using its reflection.

John Horton Conway proposed an alternate labeling scheme. He uses O instead of I, Q instead of L, R instead of F, and S instead of N. The resemblance to the letters is a bit more strained (most notably that the "O," a straight line, bears no resemblance to an actual letter O), but this scheme has the advantage that it uses 12 consecutive letters of the alphabet. This scheme is used in connection with Conway's Game of Life, so it talks about the R-pentomino instead of the F-pentomino.

Contents

Symmetry

Considering rotations of multiples of 90 degrees only, there are the following symmetry categories:

  • L, N, P, F 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 symmetry 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.

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