# Turn, turn, turn

Does the letter "P", then, have rotational symmetry of order 1 or 0?

Your question has important implications from primary to degree level (geometry and group theory). The letter "N" has no line of symmetry as I cannot draw a line through the shape to make identical halves. But it does have rotational symmetry.

To understand the meaning of "order" of symmetry, try the following exercise: write the letter "N" and put a dot halfway along the diagonal of the letter to mark the point of rotation.

Put some tracing paper over the letter and trace it. Leave the tracing paper in its original position, covering the "N" exactly, and put a compass or sharp pencil on the dot to hold the paper in place at this point.

1. Rotate the tracing paper clockwise through 180x. As you see, the "N" covers the original exactly. But the traced "N" is in fact upside down to the original.

2.Rotate the paper again 180x clockwise. This returns the "N" to its original orientation, exactly covering the original.

This is what is meant by rotational symmetry: being able to exactly cover the original by performing rotations. In this case we are able to do this with two rotations, thus "N" has rotational symmetry of order 2.

In the case of "A", if we rotate about any point we find that there is only one way to cover the original exactly, and that is by turning the tracing through 360x - a complete turn. Thus the rotational symmetry of "A" is of order 1. At key stages 2-4, it is acceptable to say that a shape has rotational symmetry if the order of rotation is greater than 1; that is, that it can be turned through an angle smaller than 360x to lie exactly on itself.

By this definition, "A" does not have rotational symmetry. But a shape cannot have order 0, as the angle of rotation is found by dividing 360x by the order of rotation. So, for instance, a regular pentagon will rotate five times, each time through the angle: (360 V 5)x = 72x. In general terms, the angle of rotation = (360 V n)x, where n is the order of symmetry.

In the case of "A", if we said that the order was zero, the angle would be (360 V 0)x, which is undefined. We know that the only angle we can rotate "A" through to make it look the same is 360x, which can only be calculated if "A" has rotational symmetry of order 1: (360 V 1)x = 360x, a complete turn.

I would like to see questions like:"Does a shape have rotational symmetry?", rather than asking: "What is the order of rotational symmetry?"

It is important not only that we have a general consensus based on the verbal description of the order of rotational symmetry, but that this can actually be confirmed by the underlying maths. Rotational symmetry figures widely in all sorts of design, is an important part of maths, and is also fun. For the English teacher who contributed to the discussion on the maths forum I have created a poem: What are isometries?

The transformations - reflection, rotation, and translation - are called isometries, because when an object is transformed in any of these ways, the size of the shape remains unchanged.

Enlargement is not an isometry because the size changes. Isometries are great to demonstrate using dance. With some square tiles on the floor, you can even introduce vector notation for the movements in primary school, particularly for translation. They could then use this for investigating number patterns from movement.

Wendy Fortescue-Hubbard is a teacher and game inventor. She has been awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts (NESTA) www.nesta.org.uk to spread maths to the masses. Email your questions to Mathagony Aunt at teacher@tes.co.uk Or write to TES Teacher, Admiral House, 66-68 East Smithfield, London E1W 1BX