Symmetry Chapter Notes | Mathematics Olympiad Class 4 PDF Download

Introduction

In geometry, symmetry is defined as a balanced and proportionate similarity that is found in two halves of an object. It means one-half is the mirror image of the other half. The imaginary line or axis along which you can fold a figure to obtain the symmetrical halves is called the line of symmetry.
If an object is symmetrical, it means that it is equal on both sides. Suppose, if we fold a paper such that half of the paper coincides with the other half of the paper, then the paper has symmetry.
Symmetry can be defined for both regular and irregular shapes. For example, a square is a regular (all sides are equal) and a rectangle is an irregular shape (since only opposite sides are equal). The symmetries for both shapes are different. Check different figures with symmetry here.

Symmetry in Mathematics

In Mathematics, a meaning of symmetry defines that one shape is exactly like the other shape when it is moved, rotated, or flipped. Consider an example, when you are told to cut out a ‘heart’ from a piece of paper, don’t 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.

Symmetry Math definition states that “symmetry is a mirror image”. When an image looks identical to the original image after the shape is being turned or flipped, then it is called symmetry. It exists in patterns. You may have often heard of the term ‘symmetry’ in day to day life. It is a balanced and proportionate similarity found in two halves of an object, that is, one-half is the mirror image of the other half. And a shape that is not symmetrical is referred to as asymmetrical. Symmetric objects are found all around us, in nature, architecture, and art.

Symmetrical Figures

Symmetrical shapes or figures are the objects where we can place a line such that the images on both sides of the line mirror each other.  The below set of figures form symmetrical shapes when we place a plane or draw the lines.

For example, figure (b) has the symmetrical figures when we draw two lines of symmetry as shown below.

Line of Symmetry

The imaginary line or axis along which you fold a figure to obtain the symmetrical halves is called the line of symmetry. It basically divides an object into two mirror-image halves. The line of symmetry can be vertical, horizontal or diagonal. There may be one or more lines of symmetry.

One Line Symmetry

Figure is symmetrical only about one axis. It may be horizontal or vertical. The word ATOYOTA has one axis of symmetry along the axis passing through Y.

Two lines Symmetry

Figure is symmetrical with only about two lines. The lines may be vertical and horizontal lines as viewed in the letters H and X. Thus, we can see here two lines symmetry.

Three Lines Symmetry

An example of three lines of symmetry is an equilateral triangle. Here, the mirror line passes from the vertex to the opposite side dividing the triangle into two equal right triangles.

Four Lines Symmetry 

Four lines of symmetry can be seen in a square, that has all the sides equal. 

Infinite lines

Some figures have not one or two, but infinite lines passing through the centre, and the figure is still symmetrical. Example: a circle.

Lines of Symmetry for Different Shapes

• Lines Of Symmetry in a Parallelogram
• Lines of Symmetry in a Rectangle
• Rhombus Lines of Symmetry

Types of Symmetry

Symmetry may be viewed when you flip, slide or turn an object. There are four types of symmetry that can be observed in various situations, they are:

• Translation Symmetry
• Rotational Symmetry
• Reflection Symmetry
• Glide Symmetry

Translation Symmetry

If the object is translated or moved from one position to another, the same orientation in the forward and backward motion is called translational symmetry. In other words, it is defined as the sliding of an object about an axis. This can be observed clearly from the figure given below, where the shape is moved forward and backward in the same orientation by keeping the fixed axis.

Rotational Symmetry

When an object is rotated in a particular direction, around a point, then it is known as rotational symmetry or radial symmetry. Rotational symmetry existed when a shape 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. The order of symmetry is how the object coincides with itself when it is in rotation.
In geometry, many shapes consist of rotational symmetry. For example, the figures such as circle, square, rectangle have rotational symmetry. Rotational symmetry can also be found in nature, for instance, in the petals of a flower.
Below figure shows the rotational symmetry of a square along with the degree of rotation.

Reflexive Symmetry 

Reflection symmetry is a type of symmetry in which one half of the object reflects the other half of the object. It is also called mirror symmetry or line of symmetry. A classic example of reflection symmetry can be observed in nature, as represented in the below figure.

Glide Symmetry

The combination of both translation and reflection transformations is defined as the glide reflection. A glide reflection is commutative in nature. If we change the combination’s order, it will not alter the output of the glide reflection.

Symmetrical Shapes

The symmetry of shapes can be identified whether it is a line of symmetry, reflection or rotational based on the appearance of the shape.
The shapes can be regular or irregular. Based on their regularity, the shapes can have symmetry in different ways.  Also, it is possible that some shapes does not have symmetry. For example, a tree may or may not have symmetry.