Symmetry
Symmetry is defined as a proportionate and balanced similarity that is found in two halves of an object, that is, one-half is the mirror image of the other half. An object has symmetry if it can be divided into two identical pieces. The imaginary axis or line along which you can fold a figure to obtain the symmetrical halves is called the line of symmetry.
| 1. | What is Symmetry in Maths? |
| 2. | Line of Symmetry |
| 3. | Types of Symmetry |
| 4. | What is Point Symmetry? |
What is Symmetry in Maths?
Symmetry Definition
In Mathematics, symmetry states that one shape is identical to the other shape when it is moved, rotated, or flipped. When an object has a symmetry, we say that it is symmetrical. If an object does not have symmetry, we say that the object is asymmetrical. The concept of symmetry is commonly found in geometry.
Symmetry Examples
For example, when you are told to cut out a ‘heart’ from a piece of paper, you simply fold the paper, draw one-half of the heart at the fold and cut it out to find that the other half exactly matches the first half. The heart carved out is an example of symmetry. Similarly, a regular pentagon when divided as shown in the image below, has one part symmetrical to the other.
The definition of Symmetry, in Math, states that “symmetry is a mirror image”, i.e., when an image looks identical to the original image after the shape is being turned or flipped, then it is called symmetry. Symmetricity exists in patterns. It is a balanced and proportionate similarity found in two halves of an object, which means one-half is the mirror image of the other half. Symmetric objects are found all around us in day to day life, in art and architecture.
Line of Symmetry
The line of symmetry is a line that divides an object into two identical pieces. Here, we have a star and we can fold it into two equal halves. When a figure is folded in half, along its line of symmetry, both the halves match each other exactly. This line of symmetry is called the axis of symmetry.
The line of symmetry can be categorized based on its orientation as:
- Vertical Line of Symmetry
- Horizontal Line of Symmetry
- Diagonal Line of Symmetry
Vertical Line of Symmetry
A vertical line of symmetry is that line that runs down vertically, divides an image into two identical halves. For example, the following shape can be split into two identical halves by a standing straight line. In such a case, the line of symmetry is vertical.
Horizontal Line of Symmetry
The horizontal line of symmetry divides a shape into identical halves, when split horizontally, i.e., cut from right to left or vice-versa. For example, the following shape can be split into two equal halves when cut horizontally. In such a case, the line of symmetry is horizontal.
Diagonal Line of Symmetry
A diagonal line of symmetry divides a shape into identical halves when split across the diagonal corners. For example, we can split the following square shape across the corners to form two identical halves. In such a case, the line of symmetry is diagonal.
A line of symmetry is an axis along which an object when cut, will have identical halves. These objects might have one, two, or multiple lines of symmetry.
- One line of symmetry
- Two lines of symmetry
- Infinite lines of symmetry
One Line of Symmetry
Figures with one line of symmetry are symmetrical only about one axis. It may be horizontal, vertical, or diagonal. For example, the letter "A" has one line of symmetry, that is the vertical line of symmetry along its center.
Two Lines of Symmetry
Figures with two lines of symmetry are symmetrical only about two lines. The lines may vertical, horizontal, or diagonal lines. For example, the rectangle has two lines of symmetry, vertical and horizontal.
Infinite Lines of Symmetry
Figures with infinite lines of symmetry are symmetrical only about two lines. The lines may vertical, horizontal, or diagonal lines. For example, the rectangle has two lines of symmetry, vertical and horizontal.
The following table shows the examples for different shapes with the number of lines of symmetry that they have.
| Number of lines of symmetry | Examples of figures |
|---|---|
| No line of symmetry | Scalene triangle |
| Exactly one line of symmetry | Isosceles triangle |
| Exactly two lines of symmetry | Rectangle |
| Exactly three lines of symmetry | Equilateral triangle |
Types of Symmetry
Symmetry can be viewed when you flip, turn or slide an object. There are four types of symmetry that can be observed in various cases.
- Translational symmetry
- Rotational symmetry
- Reflexive symmetry
- Glide symmetry
Translation Symmetry
If an object is moved from one position to another, with the same orientation in the forward and backward motion, it is called translational symmetry. In other words, translation symmetry is defined as the sliding of an object about an axis. For example, the following figure, where the shape is moved forward and backward in the same orientation by keeping the fixed axis, depicts translational symmetry.
Rotational Symmetry
When an object is rotated in a particular direction, around a point, then it is known as rotational symmetry, also known as radial symmetry. Rotational symmetry exists when a shape is turned, and the shape is identical to the origin. The angle of rotational symmetry is the smallest angle at which the figure can be rotated to coincide with itself and the order of symmetry is how the object coincides with itself when it is in rotation.
In geometry, there are many shapes that depict rotational symmetry. For example, figures such as circle, square, rectangle depict rotational symmetry. The following image shows how the structure of a starfish follows rotational symmetry. If you turn or rotate the starfish about point P, it will still look the same from all directions. The famous Ferris wheel, the London Eye, is an example of rotational symmetry. You can find many objects in real life that have rotational symmetry like wheels, windmills, road-signs, ceiling fans, and so on.
Reflexive Symmetry
Reflective symmetry, also called mirror symmetry, is a type of symmetry where one half of the object reflects the other half of the object. For example, in general, human faces are identical on the left and right sides.
Glide Symmetry
Glide symmetry is the combination of both translation and reflection transformations. A glide reflection is commutative in nature and the change in combination’s order does not alter the output of the glide reflection.
Fun Facts on Symmetry
- A kaleidoscope has mirrors inside it that produce images that have multiple lines of symmetry. The angle between the mirror decides the number of lines of symmetry.
- We may have observed several symmetrical objects in our daily life like rangolis or kolams. The striking aspect of symmetry can be observed in rangoli designs. These designs are famous in India for their unique and symmetrical patterns. They depict the colorful science of symmetry.
What is Point Symmetry?
An object has a point symmetry if every part of the object has a matching part. Many letters of the English alphabet have point symmetry. The point O is the central point and the matching parts are in opposite directions.
If an object looks the same when you turn it upside down, then it is said to have point symmetry. The shape and the matching parts must be in opposite directions.
Important Notes
Given below are some important points related to the concept of symmetry:
- All regular polygons are symmetrical in shape. The number of lines of symmetry is the same as the number of its sides.
- An object and its image are symmetrical with reference to its mirror line.
- If a figure has rotational symmetry of 180º, then it has point symmetry.
Solved Examples on Symmetry
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Example 1: If the following figure shows a reflexive line of symmetry, complete the figure.
Solution:
It is given that the figure has a reflexive line of symmetry. That means the second half or the missing part of the figure will be exactly the same as given on the other side.
Thus, the complete figure is:
-
Example 2: Identify which of the following figures is symmetrical about the given line l?
Solution:
As we know, when an object is exactly the same when you turn it or flip it, that object has symmetry.
Thus, from the above-given figures, only figure (c) has symmetry.
FAQs on Symmetry
What is Symmetry in Math?
Symmetry is defined as a proportionate and balanced similarity that is found in two halves of an object, that is, one-half is the mirror image of the other half. For example, different shapes like square, rectangle, circle are symmetric along their respective lines of symmetry.
What is a Symmetrical Shape?
A 2D shape can be called symmetrical if a line can be drawn through it and either side is a reflection of the other. For example, a square has a symmetrical shape.
What are the 4 Types of Symmetry?
Symmetry can be categorized into four types:
- Translational symmetry: If the object is moved or translated from one position to another, the same orientation in the forward and backward motion is called translational symmetry.
- Rotational symmetry: When an object is rotated in a particular direction, around a point, then it is known as rotational symmetry.
- Reflexive symmetry: Reflective symmetry, also called mirror symmetry, is a type of symmetry where one half of the object reflects the other half of the object.
- Glide symmetry: Glide symmetry is the combination of both translation and reflection transformations.
What is Symmetry? Explain With an Example.
When an object is exactly the same when you turn it or flip it, that object has symmetry. Symmetrical objects are of the same size and shape. Nature has plenty of objects having symmetry. For example, the petals in a flower, a butterfly, etc.
What is a Symmetric Pattern?
All patterns having symmetry are called symmetric patterns. The leaves of plants have various patterns and shapes. Most of these leaves depict symmetric patterns if we take the middle vein as the line of vertical symmetry.
What Do you Mean by a Line of Symmetry?
The line of symmetry is a line that divides an object into two identical pieces. For example, the diagonal of a square divides it into two equal halves, this is referred to as the line of symmetry for a square.
Can a line of symmetry be parallel?
No, the line of symmetry cannot be parallel. All lines of symmetry drawn for any shape will always coincide with each other.