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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.## 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.## Polyhedral nets for the Cube

## References and external links

The figure shows all possible hexominoes, coloured according to their symmetry groups:

- 20 hexominoes (coloured grey) 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 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.

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 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.

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.

- Page by Jürgen Köller on hexominos, including symmetry, packing and other aspects
- Polyomino page of David Eppstein's Geometry Junkyard
- French Eleven animations showing the patterns of the cube
- Polypolygon tilings, Steven Dutch.

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Last updated on Friday September 12, 2008 at 17:19:54 PDT (GMT -0700)

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Last updated on Friday September 12, 2008 at 17:19:54 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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