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# Heptomino

A heptomino (or 7-omino) is a polyomino of order 7, that is, a polygon in the plane made of 7 equal-sized squares connected edge-to-edge. As with other polyominoes, rotations and reflections of a heptomino are not considered to be distinct shapes and with this convention, there are 108 different "free" heptominoes.

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

• 84 heptominoes (coloured grey) have no symmetry. Their symmetry groups consist only of the identity mapping
• 9 heptominoes (coloured red) have an axis of reflection 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.

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• 7 heptominoes (coloured green) have an axis of reflection symmetry at 45° to the gridlines. Their symmetry groups have two elements, the identity and a diagonal reflection.

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• 4 heptominoes (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.

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• 3 heptominoes (coloured purple) have two axes of reflection symmetry, both aligned with the gridlines. Their symmetry groups have four elements.

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• 1 heptomino (also coloured purple) has two axes of reflection symmetry, both aligned with the diagonals. Its symmetry groups has four elements.

If reflections of a heptomino were to be considered distinct, as they are with one-sided heptominoes, then the first and fourth categories above would each double in size, resulting in an extra 88 heptominoes for a total of 196 distinct one-sided heptominoes.

## Packing and tiling

Although a complete set of 108 heptominoes has a total of 756 squares, it is not possible to pack them into a rectangle. The proof of this is trivial, since there is one heptomino which has a hole.

Not all heptominoes are capable of tiling the plane; the one with a hole is one such example. In fact, under some definitions, figures such as this are not considered to be polyominoes because they are not topological disks.

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