Pentomino - Symmetry

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.

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:

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.