Rotational Symmetry

Symmetry is everywhere. When we say that mathematics is a subject that is all around us, we actually mean it because no matter what you look at, you can find something related to math in it. Observe the things around you like the Television set that you have in your house,  the positioning of the table, the chair, the refrigerator and things that are kept inside a kitchen or any other things that are kept near you. If you actually notice that there is some kind of logic behind the positioning of these items inside your home. 

We seek patterns in their day to day lives. This is why buildings, cars and everything is made in a specific structure to make sure that this important idea of symmetry is something that continues to stay in our surroundings. 

An introduction to Rotational Symmetry

Rotational Symmetry of shape states that an object looks the same when it is rotated on its axis. Most of the geometrical shapes seem to appear as a symmetry when they are rotated clockwise, anticlockwise or rotated with some angle such as 180°,360°, etc. Rotational Symmetry is an interesting topic that can be understood by taking some real-life examples from your surroundings. Some of the examples of geometrical shapes that appear as symmetry are square, hexagon and circle. A scalene triangle does not appear to be symmetrical when rotated. Hence, it is asymmetrical in shape.

There should be at least two similar orders to have symmetry as the word symmetry is a combination of two words ‘sync+metry’. The objects which do not appear to be symmetrical when you flip, slide, or turn are considered asymmetrical in shape. Symmetry is found all around us, in nature, in architecture, and in art. There are various types of symmetry. These are,

• Reflection Symmetry 

• Translational Symmetry 

• Rotational Symmetry 

What is Rotational Symmetry?

Any figure or shape that rotates around a center point and looks exactly similar as it was before the rotation, is said to have rotational symmetry. Some shapes which have rotational symmetry are squares, circles, hexagons, etc.

In Geometry, many shapes have rotational symmetry. These are:

• Equilateral triangles

• Squares

• Rectangles

• Circles

• Regular Polygons

How to Determine The Order of Rotational Symmetry of Any Shape?

The order of rotational symmetry is the number of times any shape or an object is rotated and still looks similar to it was before the rotation.

The order of rotational symmetry of a regular hexagon is equivalent to the number of sides a polygon has.

The order of rotational symmetry can also be found by determining the smallest angle you can rotate any shape so that it looks the same as the original figure.  

80° = Order 2

120° = Order 3,

90° = Order 4.

The product of the angle and the order will be equal to 360°.

Center of An Object

The center of any shape or object with rotational symmetry is the point around which rotation appears. For example, if a person spins the basketball on the tip of his finger, then the tip of his finger will be considered as rotational symmetry. If any object has a rotational symmetry then the center of an object will also be its center of mass.

Order of Rotational Symmetry

The number of times any shape or an object that can be rotated and yet looks similar as it was before the rotation, is known as the order of rotational symmetry. For example, a star can be rotated 5 times along its tip and looks similar each time. Hence, the order of rotational symmetry of the star is 5.

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If we examine the order of rotational symmetry for a regular hexagon then we will find that it is equal to 6. A regular hexagon has 6 equal sides and can be rotated at an angle of 60 degrees.

The angle of Rotational Symmetry

The angle of rotation is the smallest angle a shape is turned or flipped to make it look similar to its original shape. 

• A quarter turn means a rotation of 90°

• A half-turn indicates a rotation of 180°

• A complete turn indicates a rotation of 360°

• An object is considered as a  rotational symmetry if it strings along more than once during a complete rotation, i.e.360°

Rotational Symmetry Letter

There are various English alphabets that have rotational symmetry when they are rotated clockwise or anticlockwise about an axis. Some of the English alphabets which have rotational symmetry are: Z, H, S, N, and O.These alphabets will exactly look similar to the original when it will be rotated 180 degrees clockwise or anticlockwise

Some of the  examples of rotational symmetry are given below:

Rotational Symmetry Examples

Solved Examples

1. Which of the following figures have rotational symmetry of more than order 1?

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Solution: 

In the above figure, a,b,d,e, and f have rotational symmetry of more than order 1.

Figure (a) has rotational symmetry of order 4, figures (b) and (e) have rotational symmetry of order 3, figure (d) has rotational symmetry of order 2, and figure (f) has rotational symmetry of order 4.

2. Determine the order of rotational symmetry of a square and the angles of such rotation.

If the square is rotated either by 90°, 180°, 270°, or by 360° then the shape of the square will look exactly similar to its original shape. Hence the square has rotational symmetry of order 4.

3. Determine the order of rotational symmetry of a rhombus and the angles of such rotation.

If the square is rotated either by 180 or by 360, then the shape of the rhombus will look exactly similar to its original shape. Hence the rhombus has rotational symmetry of order 2.

Quiz Time

1. Which of the figures given below does not have a line of symmetry but has rotational symmetry?

1. H

2. I

3. Z

4. X

2. Determine the smallest angle of rotation that maps the image to itself.

1. 72

2. 180

3. 144

4. 45

Facts

• The World’s largest Ferris wheel London eye has rotational symmetry of order 32.

Conclusion

Every single chapter in math can be easily related to life. Rotational symmetry is another one of those topics that can be studied well by taking real-life examples and finding out ways and methods to associate the knowledge learned to your everyday life. There is no doubt that by getting to solve all the problems from your textbook, you will be solidifying the idea and concept behind the things that you learn in a chapter, but by real-life application of things, you will be able to score even better!