Symmetry

Meaning of symmetry

Most likely, you have already encountered the concept of symmetry outside of math. For example, when you think of symmetry, you may think of a mirror image. Indeed, a mirror is an excellent example of symmetry in the real world. Mirrors showcase a type of symmetry called reflective symmetry, in which one half of the object is a mirror or reflection of the other. Reflective symmetry is only one type of symmetry, however. The definition of symmetry in mathematics is more precise than the common usage of symmetry in everyday life. A formal definition is as follows:

Types of symmetry

In this section, we will cover four different types of symmetry:

• Translational symmetry
• Rotational symmetry
• Reflective symmetry
• Glide symmetry

Translational symmetry

Translational symmetry occurs when an object has undergone a shift or translation (i.e., it has had its location changed), but there have been no other transformations performed on it. In other words, translational symmetry only involves the object moving position, such as up or down as well as left or right. In translational symmetry, the object will remain the same size and shape, and it won't rotate in any way.

Rotational symmetry

Rotational symmetry applies if a shape can be partially rotated about its center and still look the same. In other words, for any object to be rotationally symmetrical, it must have at least two positions within the full turn of rotation (360 degrees) where it looks identical.

To describe rotational symmetry, we may refer to the order of rotational symmetry, which describes the number of times that the object is identical to its original position within the full rotation of 360 degrees. For example, an equilateral triangle has a rotational symmetry of order 3 because it can be partially rotated 3 times and still appear identical. To find out the order of rotational symmetry of a shape, you can use the following equation:

$orderofrotationalsymmetry=\frac{360°}{AngleofRotation}$

Point symmetry

Point symmetry occurs when there is a common point of reflection for every point on a shape. That common point is called a point of symmetry. Note that the reflection for each point is in the opposite direction, so that it looks the same from the top as it does from the bottom. An example of point symmetry is the letter H.

Reflective symmetry

Reflective symmetry is a type of symmetry in which one half of the object reflects the other half. Reflective symmetry is known by several different names, including line symmetry and mirror symmetry. All three terms have the same meaning: one half of the object is identical to the other.

To check if an object exhibits reflective symmetry, imagine folding it in half along the line of symmetry (an imaginary line that divides the two reflective halves). If both halves match, then the shape has reflective symmetry along the line that it was folded over.

Line of symmetry

The line of symmetry is a line that can be drawn on an object whereby both sides will reflect one another, cutting the object in half. For example, if you place a line of symmetry down the middle of a square, both sides will be the same.

Different shapes can have varying amounts of lines of symmetry:

• A square has 4 lines of symmetry
• A equilateral triangle has 3 lines of symmetry
• A trapezium doesn't have any lines of symmetry

Axis of symmetry

Symmetry isn't reserved for shapes and patterns: it can also be seen in graphs. When graphs have symmetry, it can be a useful property because it allows us to predict and better understand the symmetric portions. Generally, graphs are symmetric about an axis of symmetry.

Like the line of symmetry, the axis of symmetry is a straight line over which an object is reflected in order to obtain two equal and mirror parts. If a graph was symmetrical for both negative and positive values of x, then it would have an axis of symmetry along the y-axis. As a straight line, we can describe the axis of symmetry with the equation of a line: $y=mx+b$. Let's take a look at an example that demonstrates the axis of symmetry.

Glide reflection

Glide reflection is a combination of two different transformations, a translation and a reflection, although not necessarily in that order. The translation is always parallel to the line of reflection in a glide reflection. If the reflected image moves further away or closer to the line of reflection, then it is not a case of glide reflection.

Glide reflections can create symmetry. When two glide reflections happen, the original image returns back to the same position, creating a reflection and a line of symmetry between the two images.

A real-life example of glide symmetry is footprints in the sand. When we compare the left footprint with the right footprint in the image below, we see that the left footprint can be reflected over a line of reflection and translated up along that line to obtain the print of the right foot.

An example of glide symmetry, unsplash.com