Rotational Symmetry: Definition, Types, Examples, Problems

Rotational Symmetry: If a geometric shape or object can be divided into two or more identical pieces and put in an orderly manner, it is said to be symmetric. The property of rotational symmetry is that a shape looks the same after a partial or full rotation.

The number of possible orientations in which an object appears the same for each full rotation is the degree of rotational symmetry. In this article, we will discuss symmetry especially rotational symmetry, real-life examples, and lines of symmetry.

What is Symmetry?

Symmetry is a term used in the study of geometry that derives from a Greek phrase that means to measure together. Two objects must be the same size and shape to be symmetrical, with one object oriented differently than the other. A single object, such as a face, can have symmetry.

“Symmetry is the mirror image,” according to the mathematical definition. Symmetrical images, shapes, or objects are those that are identical to one another. 

Line Symmetry

A symmetry line is a line that splits a shape into two halves perfectly. If you fold the shape along the line of symmetry, both sides of the object or shape will match exactly. The curvature of the line would not alter if a mirror were placed alongside it.

An axis of symmetry is a line that divides or slices a form or object into two equal portions. Symmetry axis is also known as a symmetrical line. A line of symmetry divides a shape or item into two portions that are mirror reflections of one another.

In the illustration below, the object star is folded along the symmetry line. Each part on either side of the symmetry line is the same.

Based on orientation, the line of symmetry can be split into three categories:

1. Vertical Line of Symmetry: A vertical line of symmetry separates or slices objects, shapes, or images into two identical portions, either downwards or upwards. The vertical symmetry line separates or slices the shapes in two directions: downwards and above.

Example: Cut the below shape vertically downwards as shown. On the left and right sides of the symmetry line, the components of the forms are identical. As a result, the symmetry line is formed.

2. Horizontal Line of Symmetry: The components generated when you divide or cut any item or shape horizontally, from left to right or right to left, are identical. They are called symmetrical figures, and the symmetry line is called a horizontal line of symmetry.

Example: Cut the shape below horizontally from left to right as an example. The shape’s components above and below the symmetry line are identical.

3. Diagonal Line of Symmetry: It is formed when we divide or cut an object or shape into two identical parts on the diagonal. The object is cut diagonally by the diagonal line of symmetry.

Example: When the object below is split diagonally, both sides of the diagonal (line of symmetry) are mirror reflections of each other. The diagonal line of symmetry is the name given to the line of symmetry.

Rotational Symmetry

The rotational symmetry of a shape describes how an object’s shape remains the same when rotated on its own axis. When turned \({180^{\rm{o}}}\) or with certain angles, clockwise or anticlockwise, many geometrical shapes appear to be symmetrical. Square, circle, hexagon, and other shapes are examples. When a scalene triangle is rotated, it loses its symmetry since the shape is asymmetric.

Rotational Symmetry Examples

1. The symmetry order of the paper windmill is \(4.\)

2. The recycling logo has a symmetry order of \(3.\)

2. The roundabout road sign has a symmetry order of \(3.\)

Line Symmetry and Rotational Symmetry

Line Symmetry: Multiple lines of symmetry can be found in shapes or patterns, depending on how many times they can be folded in half and still look the same on both sides.

Rotational Symmetry: When a shape or pattern can be rotated or turned around a central point while remaining the same, it is said to have rotational symmetry. A shape can be said to have rotational symmetry of order \(X\) if it can be rotated about a central point \(X\) time while remaining the same.

Because a regular pentagon has five equal sides and five lines of symmetry, its rotational symmetry order will also be five. Because of its rotational symmetry, the pentagon can be rotated \(5\) times before returning to its initial position. Each time it is turned, it will look exactly like the original position.

Centre of Rotation

A rotation is a transformation that causes the preimage figure to rotate or spin to the image figure’s place. Everything else spins around a single fixed point called the centre of rotation in all rotations.

This point could be inside the figure, in which case the figure will remain stationary and will just spin. Alternatively, the point could be outside the figure, causing the figure to move.

Angle of Rotational Symmetry

The angle of rotation is the angle of turning during rotation.

1. A quarter turn is a \({90^{\rm{o}}}\) rotation.
2. A \({180^{\rm{o}}}\) rotation is referred to as a half turn.
3. A full turn is defined as a \({360^{\rm{o}}}\) rotation.

A figure is considered to have rotational symmetry if it rotates in the same direction more than once throughout a full \({360^{\rm{o}}}\) rotation.

Order of Rotational Symmetry

The number of times you can spin a geometric object to make it seem exactly like the original figure is called its order of rotational symmetry.

All you must do is rotate the figure \({360^{\rm{o}}}.\) You will be returned to the original figure once you have rotated the figure \({360^{\rm{o}}}.\)

Rotational Symmetry of Equilateral Triangle

Equal sides and angles define an equilateral triangle.

When you spin an equilateral triangle \(\frac{{{{360}^{\rm{o}}}}}{3} = {120^{\rm{o}}},\) it always looks the same (symmetric). As a result, spinning \({120^{\rm{o}}}\) clockwise or counterclockwise will produce the same triangle.An equilateral triangle’s order of rotational symmetry is \(3,\) which means that each \({120^{\rm{o}}}\) rotation returns the original equilateral triangle. It is worth noting that three times \({120^{\rm{o}}}\) equals \({360^{\rm{o}}}\)

Rotational Symmetry of Alphabets

In English alphabets, rotational symmetry refers to the fact that the letter retains its appearance after being rotated. The letters \({\rm{H,I,N,S,X}}\) and \(Z\) are capital letters with rotational symmetry.

Solved Examples – Rotational Symmetry

Q.1. What is the order of rotation of an equilateral triangle?
Ans: An equilateral triangle’s order of rotational symmetry is \(3,\) which means that each \({120^{\rm{o}}}\) rotation returns the original equilateral triangle. It is worth noting that three times \({120^{\rm{o}}}\) equals \({360^{\rm{o}}}.\)

Q.2. Name the quadrilaterals which have both line and rotational symmetry of order more than 1.
Ans: Square is the quadrilaterals that have both line and rotational symmetry of order of more than \(1.\)

Q.3. Write any 5 English alphabets with the vertical line of symmetry.
Ans: \(5\) English alphabets with the vertical line of symmetry are \({\rm{H,I,M,O,T}}.\)

Q.4. What is the order of symmetry of the Swastik symbol.

Ans: \(4\) is the order of symmetry of the Swastik symbol.

Q.5. What is the order of rotation of the square?
Ans: \(4\) is the order of rotation of the square. Because each \({90^{\rm{o}}}\) rotation of a square returns it to its original shape, it has a rotational symmetry order of four. It is worth noting that \(4\) times \({90^{\rm{o}}}\) equals \({360^{\rm{o}}}.\)

Summary

When a shape or pattern can be rotated or turned around a central point while remaining the same, it is said to have rotational symmetry. The symmetry line is a line that splits a shape in half perfectly. If you fold the shape along the line of symmetry, both sides of the object or shape will match exactly. This article includes an explanation about symmetry, line symmetry, rotational symmetry, order of rotational symmetry, angle of rotation, etc. 

It helps for a good understanding of rotational symmetry. The outcome of this article helps in solving the different problems based on rotational symmetry.

FAQs on Rotational Symmetry

Q.1. What is rotational symmetry?
Ans: The rotational symmetry of a shape describes how an object’s shape remains the same when rotated on its axis.

Q.2. What is the order of rotational symmetry of a square?
Ans: \(4\) is the order of rotational symmetry of a square.

Q.3. Which has no rotation of symmetry?
Ans: A triclinic unit cell has no rotational symmetry.

Q.4. How do you find the rotational symmetry?
Ans: A figure has rotational symmetry if the image agrees with the preimage when rotated by an angle between \({0^{\rm{o}}}\) and \({360^{\rm{o}}}.\)

Q.5. Which image has reflectional rotational and point symmetry?
Ans: Point of symmetry states that every part has a matching part, the same distance from the central point but that in the opposite direction. From the image, the only circle followed the reflectional, rotational, and point symmetry.

Q.6. What is the line of symmetry?
Ans: The symmetrical line is a line that splits a shape in half perfectly. If you fold the shape along the line of symmetry, both sides of the object or shape will match exactly. The curvature of the line would not alter if a mirror were placed along with it.