Rotational Symmetry

The standard idea of symmetry is the lines through the shape, using folding or mirroring to split the shape precisely in half. However, as we have seen, some shapes (like parallelograms) are clearly symmetrical in some way but they just can't be split into two reflectively equal pieces. This is because their symmetry is rotational and is found in another way altogether.

If you were to take a parallelogram shape and push a pin through the middle, you could spin it round and the shape would look identical when you have spun it through 180°. It will then look the same once you reach 360° (as any shape would) so you can say that the shape has rotational symmetry of order 2. This means that the shape looks like the original shape twice during its 360° rotation.

Look at a square and how it turns:

As the square turns, point 'A' has moved to a different position. The square, of course, now looks identical. Continue to rotate the square and 'A' will move around and eventually be at the bottom right corner, then the bottom left corner, before returning to the top left corner. The square looks identical four times in the course of a full turn of 360°. It has rotational symmetry of order 4.

Regular shapes have rotational symmetry of the order of the number of sides. A heptagon will have rotational symmetry of order 7, for instance.

A question that uses rotational symmetry is one which looks at letters rather than shapes. I have seen this before:

Which of the following letters has rotational symmetry?

The key is not to look at the normal line symmetry but, mentally, rotate the letters around themselves. You can, of course, rotate the whole page and see whether the letters look the same upside-down or another way up. While 'W' and 'V' have line symmetry (flip them left-right and they are the same), they look totally different when rotated until they have gone through a full turn. However, 'H', 'I' and 'N' all look identical if rotated through 180° so they do have rotational symmetry. Incidentally, 'N' has no line symmetry.