Have you wondered why your mirror reflection appears symmetrical while a few objects do not? Or could you guess what the similarity between two marine animals – a starfish and an octopus are? If you guessed they have a symmetrical body, then you are correct. A symmetrical body is an object or thing that can be cut along a particular axis, producing similar shapes. For example, if a starfish is cut across its limbs, you will get similar shapes. Or, if an octopus is cut along its head, it will also produce similar shapes. Or, look at your body in the mirror. Doesn’t it look symmetrical from either side if you draw an imaginary axis along your face? Now, let us understand what a symmetrical body or simply, symmetry means.
Line of Symmetry Definition
A line of symmetry is an imaginary line or axis which passes through the center of a body or an object. If you fold the body along this axis, you will get two or more similar figures. This axis is known as the axis of symmetry.
The term symmetry comes from a Greek word ‘sun + metron’, which later transformed into Latin ‘symmetria’, meaning ‘with measure’. Hence, the term symmetry means the state of having two halves that match each other exactly in size, shape, and other parameters.
As seen in the above starfish and octopus example, you will get similar shapes if you cut them along their axis of symmetry.
Let us take another example and understand the line of symmetry. If you see the figure, a square is made initially. You will get two similar small rectangles if you fold the square horizontally along the straight line. If you further fold the square along the vertical line of symmetry, you will get four small squares. And if you further fold the square along the diagonal lines of symmetry, you will get more triangles. Hence, with every fold along the line of symmetry, you will get a similar shape or object.
This line of symmetry is also known as the mirror line because it presents two reflections of an image with the same dimensions that can coincide. Hence, it is also called reflection symmetry. An object can have more than one line of symmetry depending on the object’s geometry.
‘What is a line of symmetry’ has been discussed. Now let’s come to how many lines of symmetry are there in a particular object.
Understanding how many lines of symmetry a body has?
Various shapes and figures have different lines of symmetry. Each shape can either have one, two, three, or any specific number lines of symmetry. However, various shapes have infinite lines of symmetry. A few of the shapes’ lines of symmetry are discussed below.
A kite has only one line of symmetry. The shape can be cut only vertically, producing mirror images.
If you wonder how many lines of symmetry a rectangle has, you should know it has two lines of symmetry, i.e., one horizontally and another vertically.
An equilateral triangle has three lines of symmetry. Unlike other triangles, such as scalene, isosceles, or right-angled, an equilateral triangle is the one that has maximum lines of symmetry. These lines of symmetry pass through the center of the shape and the mid-way of its sides towards the corners, as shown in the figure.
A circle has infinite lines of symmetry. Since the shape is symmetrical along all its infinite axes; hence, it has infinite lines of symmetry.
No lines of symmetry
Various objects do not have any lines of symmetry. These objects are known to have zero lines of symmetry. This is due to the fact that they do not have any symmetrical axes. The given shapes below do not have any axis of symmetry.
The point to be noted here is that though these objects do not have any lines of symmetry, as can be seen in the figure, they will somehow be similar. If you see these figures in 2D, they will look asymmetrical. However, if you view these shapes in 3D, like a real key, and see them from the top, they will have one line of symmetry and their thickness.
Hence, every 3D body will have at least one line of symmetry if its thickness is the same along its length. If the thickness is not similar, the objects will not have any line of symmetry.
Now, coming to how to determine which line of symmetry is which, let’s look below!
Types of Line of Symmetry
There are various types of symmetry. The main ones are:
If the object has symmetry along its forward and backward paths, it is said to have translation symmetry. In simpler terms, if an object can slide symmetrically, then it is translation symmetry. As you can see in the figure, the image has translation symmetry as it slides from one position to another.
If the object’s shape remains the same when rotated about an axis, then that object is said to have rotational symmetry. The rotation can occur along any axis. However, the result after rotation must have the same image as before rotation. For example, see the image below. The image shows rotational symmetry about the center axis that comes out of the paper.
When cut through an axis, the object with mirror images is known to have reflection symmetry. For example, a butterfly. It has perfect reflection symmetry. One side of the image can overlap or coincide with the other side in this symmetry.
The combination of both reflection and translation symmetry is known as glide reflection. It is commutative in nature. This means that if you change the order of the combination, it will not change the output of the glide reflection. The perfect example of a glide symmetry is the orientation of leaves on a branch. Not all the orientations of leaves are glide symmetric; however, the one shown in the figure has glide symmetry. If you rotate the branch, the leaves will move and come back to their original shape as they were before. That’s the property of glide symmetry.
Properties of Line of Symmetry
Below are a few properties that you must remember to grasp the concept of symmetry effectively:
- If a body has no line of symmetry, then that implies the figure is asymmetrical.
- A shape or object can have Infinite lines of symmetry. For example, in circles.
- An object can have one line of symmetry only. For example, a butterfly is symmetrical along the y-axis only.
- A few objects can have two lines of symmetry only. For example, the shape shown in the image below:
- Some bodies do have multiple (more than two) lines of symmetry. For example, the shape given below.
Facts About Line of Symmetry
Here are a few facts about lines of symmetry that will help you memorize the concept better:
- A triangle will have three, one or no lines of symmetry.
- A quadrilateral will have four or two or even no lines of symmetry.
- A regular pentagon has 5 lines of symmetry.
- A hexagon has six lines of symmetry.
- A heptagon has seven lines of symmetry.
- A scalene triangle has no line of symmetry.
- An isosceles triangle has only one line of symmetry.
- A rectangle has two lines of symmetry.
- An equilateral triangle has three lines of symmetry.
- A circle has an infinite number of lines of symmetry.
Example 1: Identify how many lines of symmetry does the given figure have?
Solution: As you can see from the figure above, the image looks symmetrical from the left and right sides. If you overlap the left and right images, they will coincide perfectly. Hence, the above figure has only one line of symmetry, i.e., the vertical line passing through the center, cutting the image into two halves.
Example 2: Identify how many lines of symmetry do the given images have?
Solution: To find the lines of symmetry in the given shapes, let us study them one at a time.
- The first image has only one line of symmetry along the vertical axis.
- The second image has two lines of symmetry, i.e., one horizontally and another vertically.
- The third image also has only one line of symmetry along the vertical axis.
Example 3: How many lines of symmetry do the given figure have?
Solution: The given image has only one line of symmetry. That symmetry is glide symmetry. If you cut the image horizontally and rotate it, then the image will look similar to what it was earlier. Hence, it only has one line of symmetry.