Page 256 - Ganit Kaushal

9.4 ROTATIONAL SYMMETRY

Look at the following figures and try to answer these questions.

Windmill Pinwheel Swastik Random Shape
Is there any line of symmetry in any one of them?

Good! You are right. None of these figures have any line of symmetry.
Yet they look symmetrical at the first look, don’t they?
Yes, they do. Great answer again!
Then, what kind of symmetry is this? Can you explain?

No answers, you said! No worries. Let us together try to explore it all.
Look at the windmill first. If we rotate this windmill by an angle of 90° about its centre ‘O’ (the red point in the
figure), the whole shape overlaps on itself again and it starts looking exactly the same again. So, the windmill
is said to have rotational symmetry because the shape overlaps on itself after rotating it by an angle of 90°
about its centre. The angle 90° is called the angle of rotational symmetry.

A shape is said to have ‘rotational symmetry’ if after rotating it by an angle θ (theta) about a point, the shape
overlaps completely onto itself. The angle ‘θ’ is called the angle of rotational symmetry, and the point is
called the centre of rotational symmetry.
Now, it is notable that if the windmill is rotated at an angle of 180°, it overlaps onto itself, and if it is rotated
at an angle of 270° it overlaps onto itself again. So, 180° and 270° are also the angles of rotational symmetry.
Thus, we define 90° as the ‘least angle of rotational symmetry’.
The angle of rotational symmetry is also called the angle of symmetry for ease, and may be written as AoS
for short. The Least Angle of Symmetry will be written as LAoS for short.

For the windmill, the angles of symmetry are: 90° (quarter-turn), 180° (half-turn), 270° (three-quarter turn)
and 360° (full turn). Thus, we see that the windmill has 4 angles of symmetry (see figure) and out of them, 90°
is the least angle of symmetry.

A B D A C D B C A B

360°
90°

180° 270°

D C C B B A A D D C
Initial position After 90° rotation After 180° rotation After 270° rotation After 360° rotation
It is interesting to note that all multiples of the least angle of symmetry up to 360° are angles of symmetry.
In other words, the least angle of symmetry is a divisor of 360.

Observe that when any figure is rotated by 360°, it comes back to its original position. So 360° is always an
angle of symmetry for any shape.

254 Mathematics-6