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# Rotational Symmetry|Definition & Meaning

## Definition

A figure which becomes identical to itself after rotation through some angel is said to have rotational symmetry. For some figures, the rotation must be through a certain angle e.g., a 90 degrees rotation for a Square produces identical figure. However, for a circle and ellipse – rotation through any angle produces a identical figure and thus have rotational symmetry for all.

In your day-to-day life, you have probably come across the word “**symmetrical**” very frequently. It is a symmetrical and comparable resemblance seen in **two** halves of a **figure. **That is, one part is the replica of the other part. In other words, it is a **similarity** that is discovered in an object’s **halves**.

A shape is said to be **asymmetrical** if it does not exhibit symmetrical **balance** across its entirety. There are many examples of symmetry all around us, including in **nature**, **architecture**, and **art**. It is possible to investigate it when you **move**, **rotate**, or **twist** an object. There are many possible varieties of symmetry, including the following:

- Reflection symmetry
- Translational symmetry
- Rotational symmetry

## Rotational Symmetry

A figure is said to have **rotational** symmetry of **order** “n” if it can be turned by 360°/n and maintain the exact same **appearance** as the figure it started out as after the **revolution**. In this context, **n** is a positive integer that stands for the number of times the image can be **spun** while maintaining its original appearance in its **entirety**.

An image with rotational symmetry of order **4** has, for instance, **four** unique **orientations** that become indistinguishable from one another after the figure is rotated by 360 degrees plus four quarters, which equals 90 degrees.

Utilizing transformation theory allows for the definition of rotational symmetry from a purely mathematical perspective. A rotational transformation is a linear transition that maps one set of points in a two-dimensional surface towards another collection of points by turning the first group of points around a fixed point, which is referred to as the center of rotation.

A figure is said to have rotational symmetry if it is unaffected by a rotational transformation—that is, if it is unaltered after the transformation—and if it can be rotated without changing its shape.

## Types of Rotational Symmetry

There are **three** different kinds of rotational symmetry, which are as follows:

### Point Symmetry

Point symmetry, often referred to as **central** symmetry or **origin** symmetry, is a kind of rotational symmetry in which a **figure** retains its appearance even after being rotated around a **central** point by an angle of **180 degrees**. The term “center of rotational symmetry” refers to this central location in the **structure**.

One example of **point** symmetry is the shape of a **heart**, which retains its appearance even after being rotated about its **center** by an angle of **180 degrees**.

### Line Symmetry

Line symmetry also referred to as **axial** symmetry or reflectional symmetry, is a kind of rotational symmetry in which a **figure** retains its appearance after being reflected along a **line**. Other names for line symmetry include **reflectional** symmetry and **axial** symmetry. The axis of symmetry refers to this particular line in the diagram.

For instance, the letter **“X”** has line symmetry since it retains its appearance regardless of which of its **diagonals** it is reflected across.

### Plane Symmetry

Plane symmetry also referred to as **translational** symmetry or complete symmetry, is a kind of rotational symmetry in which a figure retains its appearance even after being turned **360 degrees** around a **central** point and, if required, translated back to its **initial** position.

Other names for this type of symmetry are entire symmetry and full **translational** symmetry. The term “**center** of rotational symmetry” refers to this central location in the **structure**.

For instance, a **circle** possesses plane symmetry because it retains its **shape** even after being rotated around its center by a full 360 degrees.

## Measuring Rotational Symmetry

The measurement of **rotational** symmetry requires an **understanding** of two fundamental ideas: The order of rotational symmetry and the angle of rotation.

### Order of Rotational Symmetry

The **number of times** that a shape may be rotated while maintaining an appearance that is identical to the **original** form is referred to as the order of rotational symmetry.

The ratio of 360 degrees to the **angle** of rotation is what constitutes the order of rotational symmetry; more specifically, the order of rotational symmetry is equal to 360 degrees divided by the **angle of revolution**.

A **figure** with rotational symmetry of order **4** has, for instance, four unique **orientations** that become indistinguishable from one another after the figure is rotated by 360 degrees plus **four quarters**, which equals **90 degrees**.

### Angle of Rotation

The **degree** to which a figure must be turned in order to produce an image that is identical to the figure’s initial representation is referred to as the **angle of rotation**. Because it establishes the **order** of rotational symmetry, the angle of rotation is an essential **parameter** in the process of determining rotational **symmetry** and measuring it.

A circle, for instance, exhibits plane symmetry since it retains its appearance even after being turned by an angle of **360 degrees** about its center, and thus angle of rotation for a **circle** is 360 degrees. The following procedures can be carried out in order to determine whether or not a figure possesses rotational symmetry:

- First,
**sketch out**the figure and then locate the center of rotational symmetry for it. - Give the figure a particular amount of
**rotation**and then examine how it differs from the first figure. - Continue rotating the figure at a
**similar angle**until it has the exact same appearance as the first version of the figure. - Make a note of the
**number of times**the figure needs to be rotated before it yields the same result as the first time. - Determine the
**angle of rotation**by taking the requisite number of rotations and dividing that number by 360 degrees.

## Applications of Rotational Symmetry

There are a number of **essential** uses for rotational symmetry in a variety of domains, including the following:

### Art and Design

In the fields of art and design, rotational symmetry is frequently employed as a method for developing **compositions** that are both aesthetically **beautiful** and balanced. Designs and layouts that have a feeling of **equilibrium** and symmetry can be created by artists and designers through the utilization of rotational symmetry.

For instance, a **mandala** is a spherical design with rotational symmetry that is employed in a variety of artistic expressions and **meditation** practices.

### Biology

In the field of biology, the structure of **organisms** and the parts that make up their bodies is often described using a concept called rotational symmetry. In biology, rotational symmetry can be seen in the form of things like the spiral structure of **seashells** and the radial symmetry of **sea anemones**, for instance.

### Physics

In the field of physics, rotational symmetry is a way to characterize the symmetries that are present in **physical systems** as well as the **physical laws** themselves. For instance, the rotational symmetry of a **molecule** is what decides the properties of the molecule, both chemically and physically.

Rotational symmetry is the foundation for many of the principles that **govern physics**, including the principle of conservation of **angular momentum**.

### Mathematics

In the field of mathematics, rotational symmetry is applied in **geometry** and **topology** to investigate the characteristics of geometric figures and the ways in which they can be transformed. The study of groups, symmetries, and various other complicated **mathematical** concepts necessitates the utilization of rotational symmetry.

### Computer Science

In the field of computer science, **image analysis,** and **computer animation** are two applications that make utilization of rotational symmetry. The rotational symmetry principle is used, for instance, in the computation of **algorithms** for spinning and resizing images.

## Example of Rotational Symmetry

Find the order of symmetry of an **equilateral triangle**.

### Solution

As we have discussed above that the **order** of symmetry of any geometrical figure is the **number** of times that a shape may be **rotated** while maintaining an appearance that is **identical** to the original form. Now we will calculate the order of symmetry of an equilateral triangle by considering the following diagram.

As we can clearly see from the above picture that the equilateral triangle can be rotated **thrice** while **retaining** its **original** shape and form.

Thus, the order of symmetry of an equilateral triangle is **three**.

*All images/mathematical drawings were created with GeoGebra.*