Introduction
Symmetry is all around us — in nature, in the objects we use, and even in the letters we write. It makes things look balanced and beautiful. From the wings of a butterfly to the patterns in rangoli, symmetry helps create harmony in shapes and designs. In this chapter, we will learn what symmetry is, explore different types of lines of symmetry, and discover how to create symmetrical shapes and patterns through fun activities.
Symmetry
Meaning:
Symmetry means that when we divide a shape into two halves, one half is the mirror image of the other.
Let us look at the types of Lines of Symmetry using the Alphabet Symmetry Example:
Prem and Manu want to paste ‘Happy Birthday’ cutouts on a wall for Lali’s birthday. While preparing cutouts of letters, they observe that some letters can be cut out in an easy way. They remember that they learnt about reflection symmetry and lines of symmetry in Grade 4.
Let us recall:
Reflection Symmetry: A shape has reflection symmetry if one half is the mirror image of the other half. Imagine folding it along a line — both sides match exactly.
Line of Symmetry: The line that divides a shape into two equal, mirror-image halves is called the line of symmetry.
- Vertical Line of Symmetry – The line goes up and down.
- Horizontal Line of Symmetry – The line goes left to right.
- Some shapes have more than one line of symmetry.
They used their knowledge of lines of symmetry to make the cutouts. The letter A has a vertical line of symmetry. So, to cut out the letter ‘A’:
Similarly, for: The letter H has two lines of symmetry.
Let us try one more example:
Which of the following alphabet cutouts can be made by just drawing half (1/2 ) or quarter (1/4 ) of the letter? You can do it by drawing lines of symmetry on the letters.
Which of the letters have a horizontal line of symmetry? _________________
Which of the letters has a vertical line of symmetry? ____________________
Which letters have both vertical and horizontal lines of symmetry?________
Ans: Letters that can be made by drawing half (1/2) or quarter (1/4) of the letter: E, X, T, O (because they have at least one line of symmetry to fold along)
Letters with a horizontal line of symmetry: E, X, O
Letters with a vertical line of symmetry: X, T, O, V
Letters with both vertical and horizontal lines of symmetry: X, O
So far, we have seen shapes that can be divided into equal mirror-image halves using lines of symmetry. But what if instead of folding them, we turn them? This brings us to our next concept, known as the Rotational Symmetry.
Rotational Symmetry
Meaning:
A shape has rotational symmetry if it looks the same after being turned (rotated) around its centre by a certain angle (less than a full turn).
Order of Rotational Symmetry:
The number of times a shape matches itself during one complete turn (360°).
Let Us Make a Windmill Firki
To see rotational symmetry in action, let’s make a windmill firki! When the firki spins around its centre, its blades look the same in many positions during one full turn.
Lali makes firkis for her friends. Follow the steps given below to make your firki:
1. Take a square paper.
2. Fold the paper in half diagonally to make two triangles.
3. Open and fold it the other way to make two more triangles.
4. Open it again. You will see an ‘X’ shape on the paper.
5. Use scissors to cut along the four lines of the ‘X’. Stop cutting about halfway to the centre.
6. Take one corner of each triangle and fold it gently towards the centre of the paper. Do not press it flat.
7. Fold every other corner towards the centre.
8. Push a pin through the folded corners and the centre of the paper.
9. Push the pin through a stick or straw
Observe the dot in the firki. Does the firki look the same after 1/4, 1/2, 3/4, and a full turn?
Ans: Yes, the firki looks the same after 1/4, 1/2, 3/4, and a full turn because it has rotational symmetry of order 4.
Shapes with Both Reflection and Rotational Symmetry
Now that we know about reflection symmetry (folding to get mirror images) and rotational symmetry (turning to get the same shape), let us look at shapes that have both.
Some shapes, like a square or certain patterns in rangoli, can be folded along lines of symmetry and still look the same when turned. For example, a square has 4 lines of symmetry and also rotational symmetry of order 4.
Let us explore more examples having both reflection and rotational symmetry:
1. Find symmetry in the digits:
Which digit(s) have reflection symmetry? ___________________________
Which digit(s) have rotational symmetry? ___________________________
Which digit(s) have both rotational and reflection symmetries? ________
Now, let us look at the following numbers: ||, |00|
Do these have (a) rotational symmetry, (b) reflection symmetry or (c) both symmetries?
Give examples of 2-, 3-, and 4-digit numbers which have rotational symmetry, reflection symmetry, or both.
Ans:
Which digit(s) have reflection symmetry?
0, 1, 8
Which digit(s) have rotational symmetry?
0, 1, 8
Which digit(s) have both rotational and reflection symmetries?
0, 1, 8
For || and |00|: Both have both symmetries. Examples:
- Rotational symmetry:
2-digit → 69, 11, 88
3-digit → 101, 609, 818
4-digit → 1001, 1881 - Reflection symmetry:
2-digit → 11, 88
3-digit → 181, 101
4-digit → 1001, 8008 - Both symmetries:
2-digit → 11, 88
3-digit → 101, 808
4-digit → 1001, 1881
2. Making Designs
(a) Does the design have rotational symmetry? Yes/No.
(b) Try to change the design by adding some shape(s) so that the new design looks the same after a 12 turn. Draw the new design in your notebook.
(c) Now try to modify or add more shapes so that the new design looks the same after 14 turns. Draw the new design in your notebook.
(d) Do the new designs have reflection symmetry? If yes, draw the lines of symmetry
Ans:
(a) No. The design does not look the same after turning it ½ turn (180°) or ¼ turn (90°) because the right side has an extra circle that breaks the balance.
(b) Add another identical circle on the left side, opposite to the existing one. Now the design will look the same after a ½ turn.
(c) To get ¼ turn symmetry, you will need to place identical shapes in all four directions (top, bottom, left, and right) so that after turning 90°, the design remains unchanged. This means adding circles and triangles in the missing positions.
(d) 1. The modified design with ½ turn symmetry will have one vertical line of symmetry.
2. The modified design with ¼ turn symmetry will have two lines of symmetry — vertical and horizontal (and possibly diagonal if shapes are perfectly aligned).