Symmetry

Symmetry

Symmetry in mathematics refers to a property of an object or pattern in which the shape or form is unchanged by a specific transformation, such as reflection, rotation, or translation. In other words, the object or pattern is the same on both sides or after a certain operation is applied.

Examples of symmetrical shapes in math include squares, circles, and equilateral triangles.

For example, a square has four-fold rotational symmetry because it looks the same after being rotated 90 degrees four times.

Similarly, a line of symmetry is a line that divides a shape into two identical halves.

Examples of Symmetry

1. A square has symmetry because all of its sides are the same length, and its angles are 90^o.

2. A circle has symmetry because it is a perfect symmetrical shape with its center being the axis of symmetry.

3. An equilateral triangle has symmetry as all of its sides are the same length and its angles are equal.

4. A rectangle has symmetry as its opposite sides are equal in length and opposite angles are equal.

5. A star shape has symmetry as it has multiple points of symmetry, with the center of the star being the axis of symmetry.

6. A hexagon has symmetry as it has six lines of symmetry that pass through its center.

7. A pentagon has symmetry as it has five lines of symmetry that pass through its center.

8. An oval shape has symmetry as its vertical and horizontal diameters are equal.

9. A diamond shape has symmetry as it has two lines of symmetry that pass through its center.

10. An eight-pointed star shape has symmetry as it has eight lines of symmetry that pass through its center.

Line of Symmetry

A line of symmetry is a line that divides a shape or object into two identical halves. It is also known as reflective symmetry or bilateral symmetry. The shape or object is reflected across the line of symmetry, resulting in two identical images.

Examples of shapes with a line of symmetry include squares, rectangles, circles, and certain letters of the alphabet such as "H" and "X."

Based on its orientation, the line of symmetry can be characterized as follows:

Vertical Line of Symmetry 

It divides a shape or object into two equal halves vertically.

Example :

This alphabet has one line of the vertical line of symmetry.

Horizontal Line of Symmetry

It divides a shape or object into two equal halves horizontally.

Example:

These alphabets have one line a horizontal line of symmetry.

Diagonal Line of Symmetry

If a diagonal line divides the shape or an object into two equal halves then we call it a diagonal line of symmetry.

Example:

This shape has one diagonal line of symmetry.

Types of Symmetry

Symmetry may appear when you flip, slide, or turn an object. There are four different types of symmetry that can be seen in different situations. They are as follows:

Translation Symmetry

A translational symmetry is where an object is identical when moved a certain distance in a specific direction.

Example: In the figure below we can observe that the object is moved forward and backward in the same orientation by maintaining the constant axis.

Reflectional Symmetry

It refers to the property of an object or shape where it can be reflected over a line of symmetry and the reflected image will be identical to the original. This means that the object or shape has a mirror image that is identical to itself.

Examples: We can see many live examples in nature itself.

Rotational Symmetry

It refers to the property of an object or design where it remains unchanged after being rotated around a central point. This means that if an object or design is rotated by a certain angle, it will look the same as its original position.

Examples: Circles, squares, and rectangles are examples, which have rotational symmetry. Rotational symmetry can also be seen in nature, for example, in the petals of a flower.

The figure below shows the rotational symmetry of a square along with the degree of rotation.

Glide Symmetry

It is also known as a glide reflection and is a type of symmetry in which an object is reflected and then translated along a fixed axis. This results in a mirror image of the original object that is offset from the original by a certain distance.
Glide symmetry is a combination of a reflection and a translation.

Glide reflection is commutative, whether we glide first then reflect or we reflect first and then glide, the outcome remains the same.

Point Symmetry

It is also known as central symmetry or rotational symmetry. It refers to the geometric property of a shape or object where it can be rotated around a central point and still look identical. This means that if a shape is divided into two halves by a central point, the two halves will be mirror images of each other. For example, a circle has point symmetry because it can be rotated around its center and still look the same.

Example: A circle has point symmetry. If a line is drawn from the center of the circle to any point on the circumference, a matching point can be found on the opposite side of the center that is the same distance away.