Angles of symmetry

Look at figures with radial arms: equal arms that go out from a centre, like a plus sign. With 4 arms and 90° between neighbouring arms, the figure has 4 angles of symmetry: 90°, 180°, 270° and 360°.

Can 3 radial arms give exactly 3 angles of symmetry? The book first shows 3 arms that are NOT evenly spaced. Only a full 360° turn brings that figure back onto itself, so it has no rotational symmetry. For a turn to fit, each angle between arms must land on the next one, so all three angles must be equal. A full turn is 360°, shared equally among 3 angles: 360° ÷ 3 = 120°. With 120° between the arms, the angles of symmetry are 120°, 240° (120° + 120°) and 360°.

The same idea works for any number of evenly spaced equal arms: the smallest angle of symmetry is 360° divided by the number of arms. A starfish with 5 equal arms: 360° ÷ 5 = 72°. 6 arms: 360° ÷ 6 = 60°. 7 arms: 360° ÷ 7, which is not a whole number of degrees; as a mixed fraction it is 51 3/7°, because 7 × 51 = 357 and 3 degrees are left over.

Look at the lists.

• With 2 angles of symmetry they are 180° and 360°.
• With 3, they are 120°, 240° and 360°.
• With 4, they are 90°, 180°, 270° and 360°.

In each case, all the angles of symmetry are multiples of the smallest one. So if the smallest angle of symmetry is 60°, the angles are 60°, 120°, 180°, 240°, 300° and 360°.