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Rotational Symmetry

Consider the following:

Spinner Paper windmill Giant wheel
In the above examples, each object rotates around a fixed point, either in a clockwise direction or
an anticlockwise direction. This movement is known as rotation. The fixed point about which the
object turns is called the centre of rotation and the angle through which the object can be rotated
to look exactly the same is referred as the angle of rotational symmetry or angle of symmetry.
When an object is rotated, its shape and size remain unchanged. A complete turn or full turn is a
rotation of 360°, a three-fourth turn is a rotation of 270°, a half turn is a rotation of 180°, and a
quarter turn is a rotation of 90°. After a full rotation of 360°, the object turns back to its original
position.

In a complete turn of 360°, the number of times an object Note: Angle of rotation = 360°
looks exactly the same is called its order of rotational Order of rotation
symmetry. Let us take the example of a multiplication symbol.

A B
D A C D B C A B
D C
C B B A A D D C
Initial position After 90° rotation After 180° After 270° After 360°
(90° + 90°) rotation (90° + 90° + 90°) (90° + 90° + 90° + 90°)
rotation rotation
In a complete turn, there are four positions, at 90°, at 180°, at 270°, and at 360° when the symbol
looks exactly the same. Hence, the symbol has a rotational symmetry of order 4.

Let us take another example of a regular hexagon.

A F F E E D D C

B E A D F C E B

C D B C A B F A
Original position It has been rotated by It has been rotated by It has been rotated by
60° anticlockwise from 120° anticlockwise from 180° anticlockwise from
first position. first position. first position.

C B B A A F

D A C F B E After rotation of 60°,
120°, 180°, 240°, 300° or
E F D E C D 360°, the hexagon looks
It has been rotated by It has been rotated by It has been rotated by the same. Hence, it has
240° anticlockwise from 300° anticlockwise from 360° anticlockwise from a rotational symmetry
first position first position first position of order 6.

263 Symmetry