Rotational symmetry

The book shows a paper windmill with four blades. It looks symmetrical, but it has no line of symmetry: fold it any way and the halves do not overlap. Yet if you turn it by 90° about the point at its centre, it looks exactly the same! We say the windmill has rotational symmetry.

• The fixed point about which a figure is turned is called the centre of rotation.
• An angle through which a figure can be turned to look exactly the same is called an angle of rotational symmetry, or just an angle of symmetry.

The windmill does not look the same after any turn of less than 90°. Its angles of symmetry are 90° (a quarter turn), 180° (a half turn), 270° (a three-quarter turn) and 360° (a full turn), so it has 4 angles of symmetry. (Some books call this number the order of rotational symmetry.)

Any figure turned through 360° comes back to where it started, just as you look the same after turning all the way round on the spot. So 360° is an angle of symmetry of every figure. A square also has 4 angles of symmetry: a 90° turn about its centre takes corner A to B's place, B to C's, C to D's and D back to A's, and 180°, 270° and 360° work too.

The book's example is a strip whose top edge is shorter than its bottom edge, with both ends slanting. Turned by 180° about its centre, the short edge ends up at the bottom, so it does not match. It comes back to its own shape only after a full 360° turn. A figure like this does not have rotational symmetry; a drawing of a smiley face is another. A figure has rotational symmetry only when it has an angle of symmetry that is more than 0° and less than 360°.