Rotational Symmetry

Objective:

Thid topic gives an overview of; 

  • Rotational Symmetry
  • Centre of rotation
  • Angle of rotation.
  • Rotational symmetry order

Rotational Symmetry

When the hands of a clock go round, we could say that they rotate. The hands of a clock rotate in only one direction, about a fixed point, the centre of the clock-face. Rotation, like movement of the hands of a clock, is called a clockwise rotation; otherwise it is said to be anticlockwise. If you spin the wheel of a bicycle, it rotates. It can rotate in either way: both clockwise and anticlockwise. 

When an object rotates, its shape and size do not change. The rotation turns an object about a fixed point. This fixed point is the centre of rotation

The angle of turning during rotation is called the angle of rotation. A full turn, you know, means a rotation of 360°. A half-turn means rotation by 180°; a quarter-turn is rotation by 90°.

When it is 12 O’clock, the hands of a clock are together. By 3 O’clock, the minute hand would have made three complete turns; but the hour hand would have made only a quarter-turn. 

The Paper windmill in the picture looks symmetrical ; but you do not find any line of symmetry. No folding can help you to have coincident halves. However if you rotate it by 90° about the fixed point, the windmill will look exactly the same. We say the windmill has a rotational symmetry.

Any object or shape is said to have rotational symmetry if it looks exactly the same at least once during a complete rotation through three hundred and sixty degrees.

In a full turn, there are precisely four positions (on rotation through the angles 90°, 180°, 270° and 360°) when the windmill looks exactly the same. Because of this, we say it has a rotational symmetry of order 4.

Here is one more example for rotational symmetry. Consider a square with P as one of its corners. Let us perform quarter-turns about the centre of the square marked x .

(i) is the initial position. Rotation by 90° about the centre leads to (ii). Note the position of P now. Rotate again through 90° and you get (iii). In this way, when you complete four quarter-turns, the square reaches its original position. It now looks the same as (i). This can be seen with the help of the positions taken by P.

Thus a square has a rotational symmetry of order 4 about its centre. Observe that in this case,

  1. The centre of rotation is the centre of the square.
  2. The angle of rotation is 90°.
  3. The direction of rotation is clockwise.
  4. The order of rotational symmetry is 4.

Draw two identical parallelograms, one-ABCD on a piece of paper and the other A' B' C' D' on a transparent sheet. Mark the points of intersection of their diagonals, O and O' respectively .

Place the parallelograms such that A' lies on A, B' lies on B and so on. O' then falls on O. Stick a pin into the shapes at the point O. Now turn the transparent shape in the clockwise direction. The point where we have the pin is the centre of rotation. It is the intersecting point of the diagonals in this case.

Every object has a rotational symmetry of order 1, as it occupies same position after a rotation of 360° (i.e., one complete revolution).Such cases have no interest for us.You have around you many shapes, which possess rotational symmetry .

For example, when you slice certain fruits, the cross-sections are shapes with rotational symmetry. This might surprise you when you notice them .Then there are many road signs that exhibit rotational symmetry. Next time when you walk along a busy road, try to identify such road signs and find about the order of rotational symmetry.

Summary:

  • Rotation turns an object about a fixed point.This fixed point is the centre of rotation.The angle by which the object rotates is the angle of rotation. A half-turn means rotation by 180° a quarter-turn means rotation by 90°. Rotation may be clockwise or anticlockwise.
  •  If, after a rotation, an object looks exactly the same, we say that it has a rotational symmetry.
  • In a complete turn (of 360°), the number of times an object looks exactly the same is called the order of rotational symmetry. The order of symmetry of a square, for example, is 4 while, for an equilateral triangle, it is 3.
  •  Some shapes have only one line of symmetry, like the letter E; some have only rotational symmetry, like the letter S; and some have both symmetries like the letter H. 
  • The study of symmetry is important because of its frequent use in day-to-day life and more because of the beautiful designs it can provide us.

 

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