3.1.1 Symmetry
Symmetry is a concept that describes the balance or proportion that exists between two or more parts of an object, shape, or pattern.
There are several types of symmetry, including:
Reflectional/Line symmetry
Where an object or shape can be divided into two equal parts by a mirror or plane
Rotational symmetry
Where an object or shape can be rotated around a central point or axis and still maintain its appearance
Translational symmetry
Where an object or shape can be shifted or translated in a particular direction and still maintain its appearance
Reflectional/Line symmetry:
Reflectional symmetry, also known as line symmetry or mirror symmetry, is a type of symmetry in which an object is identical to its mirror image. This means that if an imaginary line, called line of symmetry, is drawn down the center of the object, the two halves on either side of the line are mirror images of each other.
For example
A square has reflectional symmetry because it looks identical when reflected across vertical or horizontal or diagonal lines drawn through its center. It has \( 4 \) lines of symmetry.
An isosceles triangle is a triangle in which two of the sides have the same length. If a line of reflection is drawn through the midpoint of the base (the side with a different length), the two halves of the triangle on either side of the line will be mirror images of each other.
A rectangle has two lines of symmetry. The lines of symmetry of a rectangle are the lines that divide the rectangle into two congruent halves, which are mirror images of each other.
Lines of symmetry can also be considered folding lines.
When you fold a shape along a symmetry line, the two parts sit exactly on top of each other.
Rotational or radial symmetry:
Rotational symmetry is a property of an object that remains unchanged after a rotation by a certain angle about a fixed point called the center of rotation. In other words, if an object looks the same after rotating it by a certain angle around a fixed point, then it has rotational symmetry.
The number of times an object appears identical during a complete rotation is called its order of symmetry.
For example
A circle has an infinite order of rotational symmetry because it looks the same no matter how much it is rotated around its center.
An equilateral triangle has a rotational symmetry of order three because it looks the same after rotating it by \( 120 \) degrees around its center.
Rotational symmetry is an important concept in geometry, as it helps to identify and classify geometric shapes based on their symmetry properties.
The shape has rotational symmetry of order \( 2 \).
It is worth noting that returning to the original shape adds \( 1 \) to the order.
This implies that a shape can never have order \( 0 \).
A shape with rotational symmetry of order \( 1 \) is said to have no rotational symmetry.
Translational symmetry:
Translational symmetry is a property of an object or pattern that remains unchanged when shifted by a certain distance in a particular direction. In other words, if an object or pattern looks the same after being moved by a certain distance in a particular direction, then it has translational symmetry.
For example
A square has translational symmetry because it looks the same no matter where it is placed on a grid of equally spaced squares.
A striped pattern has translational symmetry because it looks the same no matter how much it is shifted horizontally or vertically.
It is used in the design of computer graphics and digital images, where it can be used to create repeating patterns and textures.
Planes of symmetry:
A plane of symmetry, also known as a mirror plane or reflection plane, is a flat surface that divides an object or shape into two identical halves that are mirror images of each other. When an object or shape has a plane of symmetry, any reflection of the object in that plane will produce an identical image.
For example
A cuboid has \( 3 \) planes of symmetry which are equidistant from the opposite faces of the cuboid.
One plane of symmetry divides the cuboid into two identical halves along its length, another plane divides it along its width, and the third plane divides it along its height. These planes of symmetry are located in the middle of the cuboid and are perpendicular to each other.
A cube has nine planes of symmetry, including one through its center that divides it into two identical halves, and six planes that intersect at its center and divide it into four identical quarters.
Planes of symmetry are important in the study of geometry and symmetry, as they provide a way to describe the ways in which an object or shape can be divided into identical halves or repeated in space.
Worked example:
The diagram shows a regular pentagon and a kite.
Complete the following statements.
The regular pentagon has ………………… lines of symmetry.
Draw on lines of symmetry to check how many times you could cut the pentagon exactly in half.
The regular pentagon has \( 5 \) lines of symmetry.
[1]
The kite has rotational symmetry of order ………………….. .
Imagine rotating the kite \( 360° \) about its center.
Count how many times the kite looks identical to its starting position.
The kite has rotational symmetry of order \( 1 \).
[1]