Symmetry

Most people will have a general idea about when something is symmetrical. However, in mathematics, the definition of something having symmetry is much more exact.

The shape must have a point or line which is the centre of symmetry and every point on one side of this should have an exact opposite, the same distance from the centre, on the other side. There are different types of symmetry which we will look at here.

Line symmetry

Line symmetry is where we draw a line and every point on one side has a matching point on the other side. Different shapes can have differing lines of symmetry and can also have any number of them.

The shape above has one line of symmetry, which is shown in the diagram, and no others. This is because the part on the left, if folded through the centre line, would fit perfectly onto the right-hand side. There are no other ways of doing this with the shape, therefore it must have only one line of symmetry.

Some shapes will have many lines of symmetry; a square for example has four.

A shape can have any number of lines of symmetry, the most being an infinite amount, which is seen with a circle. Any line through a circle that passes through the exact centre is a line of symmetry, so there are an infinite amount! However, for this course we will look at shapes that have less lines of symmetry than this. A shape can also have no lines of symmetry, the most basic example of this being a parallelogram.

With the parallelogram above, there is no possible way to draw a line through it that would be a line of symmetry. This is because, in no way could half of this shape be folded onto the other to cover it completely.

Drawing a shape using symmetry

When it comes to drawing a shape using a line of symmetry, we say that we are looking to reflect the shape in the axis, which is the symmetry line.

To do this we must take every point that is already given and draw another point that is the same distance from the symmetry line (sometimes called the mirror line), then we join these new points up and we have an exact copy of the original that is flipped in the mirror line.

Example

Reflect the shape below in the symmetry line.

To do this we must draw new points that are at right angles to the symmetry line. They must be the exact distance away from the line as their opposing point that is already on the diagram. By doing this we get the shape below.

Clearly, all the new points and lines are exactly the same length and position as the original, just reflected in the line. If you pictured folding this shape along the mirror line then the original would flip onto and cover all of the new shape that we have drawn.

Example

Reflect the following shape in the mirror line.

Doing the exact same method as before, so that each new point is an equal distance from the line as its corresponding point, we get the shape below.

Clearly, each point has a direct opposite that is a reflection in the line. The coloured dots show each original point and its new, reflected point. The dots with the same colours are at equal distances from the line and if you were to draw a line between them, it would create a 90˚ angle with the mirror line. This is a great way of checking your work if you are unsure about your answer.

Rotational symmetry

Rotational symmetry is where we can rotate a shape around a point and it will be the exact same. The easiest way to keep track of the shape when doing this is to label the vertices (another name for the corners of a shape).

The square above can be rotated by  around the middle point so that the point A will be in the position where B was originally, B will be where C is etc. This is shown below.

By doing this the shape moves back onto itself and would cover the original perfectly. This means that the square therefore has rotational symmetry about the centre point.

Order of rotational symmetry

The ‘order’ of the symmetry is basically how many different angles that we could rotate the shape so that it would still be the exact same. So, before, we rotated by 90˚ but we could have also rotated by  or  to get the same shape.

Every shape will have at least one order of rotational symmetry because when a shape is rotated by  it is the same as leaving it unchanged, as every point will do a full rotation and will end up in the same place as the start.

How to find the order of rotational symmetry

To find the order of symmetry that a shape has we must find the smallest possible angle that gives symmetry. Then the order of the shape is how many times this angle will go into . So, if the smallest angle is  of a full rotation then the order of rotational symmetry for that shape is 6. So for our square we had earlier, the smallest angle that made the shape symmetrical was, which goes into  four times. Therefore, a square has an order of rotational symmetry of 4.

Example

What is the order of symmetry for the shape below?

The shape is what’s called a ‘regular’ hexagon, meaning all of its sides are of equal length. If we were to rotate the shape about the centre , it would look exactly the same as it does now. And, since there are six lots of  in , we know that the shape has a rotational symmetry order of 6.