Both kinds of symmetry

The chapter ends by comparing the two kinds of symmetry.

• Some figures have a line of symmetry but no rotational symmetry.
• Others have rotational symmetry but no line of symmetry.
• And some have both.

A square has both: it has 4 lines of symmetry, and its angles of symmetry are 90°, 180°, 270° and 360°. A triangle with all three sides equal also has both: 3 lines of symmetry, and angles of symmetry 120°, 240° and 360°. The circle is the most symmetric shape of all. Turn a circle about its centre by any angle, and it covers itself, so every angle is an angle of symmetry. And every diameter (a line through the centre from one side of the circle to the other) is a line of symmetry.

A triangle with two equal sides and a third side of a different length has exactly one line of symmetry, but no rotational symmetry: only a full 360° turn brings it back onto itself. So it has line symmetry only.

The paper windmill is the opposite. Fold it any way and the halves do not overlap, so it has no line of symmetry, but a 90° turn makes it look the same. It has rotational symmetry only. The book also asks you to try drawing a four-sided figure with rotational symmetry but no reflection symmetry, like the windmill.