Hexomino
A hexomino is a polyomino of order 6, that is, a polygon in the plane made of 6 equal-sized squares connected edge-to-edge. As with other polyominoes, rotations and reflections of a hexomino are not considered to be distinct shapes and with this convention, there are thirty-five different hexominoes.
The figure shows all possible hexominoes, coloured according to their symmetry groups:
- 20 hexominoes (coloured black) have no symmetry. Their symmetry groups consist only of the identity mapping
- 6 hexominoes (coloured red) have an axis of mirror symmetry aligned with the gridlines. Their symmetry groups have two elements, the identity and a reflection in a line parallel to the sides of the squares.
- 2 hexominoes (coloured green) have an axis of mirror symmetry at 45° to the gridlines. Their symmetry groups have two elements, the identity and a diagonal reflection.
- 5 hexominoes (coloured blue) have point symmetry, also known as rotational symmetry of order 2. Their symmetry groups have has two elements, the identity and a 180° rotation.
- 2 hexominoes (coloured purple) have two axes of mirror symmetry, both aligned with the gridlines. Their symmetry groups have four elements.
If reflections of a hexomino were to be considered distinct, as they are with one-sided hexominoes, then the first and fourth categories above would each double in size, resulting in an extra 25 hexominoes for a total of 60 distinct one-sided hexominoes.
Packing and tiling
Although a complete set of 35 hexominoes has a total of 210 squares, it is not possible to pack them into a rectangle. (Such an arrangement is possible with the 12 pentominoes which can be packed into any of the rectangles 3 × 20, 4 × 15, 5 × 12 and 6 × 10.) A simple way to demonstrate that such a packing of hexominoes is not possible is via a parity argument. If the hexominoes are placed on a checkerboard pattern, then 11 of the hexominoes will cover an even number of black squares (either 2 white and 4 black or vice-versa) and 24 of the hexominoes will cover an odd number of black squares (3 white and 3 black). Overall, an even number of black squares will be covered in any arrangement. However, any rectangle of 210 squares will have 105 black squares and 105 white squares.
However, there are other simple figures of 210 squares that can be packed with the hexominoes. For example, a 15 × 15 square with a 3 × 5 rectangle removed from the centre has 210 squares. With checkerboard colouring, it has 106 white and 104 black squares (or vice versa), so parity does not prevent a packing, and a packing is indeed possible -- see [1]. Also, it is possible for two sets of pieces to fit a rectangle of size 420.
Each of the 35 hexominos is capable of tiling the plane.
Polyhedral nets for the Cube
A polyhedral net for the cube is necessarily a hexomino, with 11 hexominos actually being nets. They appear on the right, again coloured according to their symmetry groups.