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Symmetry

Symmetry can be seen in nature all around us, such as when a mountain range is reflected on the water of a lake.

An example of reflection in nature, pixabay.com

In this article, we will cover the concept of symmetry, differentiating between different types. We will also discuss how the principles of symmetry can be applied to figures on graphs.

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:

In mathematics, symmetry is when two or more objects are identical after a transformation has been preformed, such as a flip, a slide, or a turn. The mathematical object could be a shape within a plane or a line on a graph, for example.

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.

Let's take the figure below as an example and see how translational symmetry looks:

Translational symmetry example, StudySmarter Originals

Now, let's translate the figure up and to the left:

Translational symmetry example, StudySmarter Originals

You can see that the figure has not changed size or shape, and it hasn't been rotated. It has simply moved position. For these reasons, it is an example of translational symmetry.

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:

A triangle showcasing rotational symmetry - StudySmarter Originals

As you can see in the diagram above, point A is moving with each rotation, but the triangle itself looks exactly the same. In the case of this triangle, we say that it has rotational symmetry of order 3 because there are 3 positions that the triangle can be rotated to where it will have symmetry. Let's check this by looking at the formula to find the order of rotational symmetry:

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.

An important thing to note is that point symmetry is the same as a shape having rotational symmetry of order 2, or in other words, the object looks identical after you have rotated it 180 degrees about its center.

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.

An example of reflective symmetry - StudySmarter Originals

In this example, the original square is labelled CDEF, and the line of symmetry is the y-axis. As you can see, the reflected image has the same y-value coordinates but has negative x-values.

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: . Let's take a look at an example that demonstrates the axis of symmetry.

Let's consider the graph below and determine if there is any symmetry.

A parabola on a graph - StudySmarter Originals

In this graph of a parabola, there is a vertical axis of symmetry, as the equation of the parabola is. We can see that the lowest point of the graph is at y = -2. This is also the point on the graph where the parabola curve crosses over the y-axis, and, therefore, the location of the axis of symmetry must be the y-axis, or x = 0.

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

Symmetry - Key takeaways

• Symmetry occurs when two or more objects are identical after a transformation has been performed.
• There are 4 main types of symmetry: translational symmetry, rotational symmetry, reflective symmetry, and glide symmetry.
• Translational symmetry is where an object has been moved (translated) but still looks the same.
• Rotational symmetry is where an object has been rotated around its center and looks the same in more than one position.
• Reflective symmetry is where an object is reflected over an axis or a line, resulting in one half looking exactly the same as the other.
• Glide symmetry combines two steps of transformations, a reflection and then a translation, although not necessarily in that order.

In mathematics, symmetry is when two or more objects are identical after a transformation has been preformed, such as a flip, a slide, or a turn.

The axis of symmetry is a straight line over which an object is reflected in order to obtain two equal and mirror parts. As a straight line, we can describe the axis of symmetry with the equation of a line: .

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.

An example of symmetry is a flower. Most flowers exhibit a form of symmetry called rotational symmetry. That is, they can be rotated around their center and still look the same as before they were rotated.

Transformations occur when an object is moved or changed (shifted up, rotated, etc.). Symmetry deals with transformations, but in order to be symmetrical, the objects need to be identical after the transformations occur.

Final Symmetry Quiz

Question

What types of symmetry are there?

There are 4 different types of symmetry: Translational symmetry, rotational symmetry, reflective symmetry and glide symmetry.

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Question

What is translational symmetry?

Translational symmetry is where an object has been moved laterally but not rotated or resized.

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Question

What is rotational symmetry?

Rotational symmetry is where an object can be rotated around its centre and still look identical to its original position.

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Question

What is the order of rotational symmetry?

The order of rotational symmetry is the number of times a shape can be rotated and still be identical within a 360o rotation. For example, an equilateral triangle has rotational symmetry of order 3 because it can be rotated to 3 different orientations and still be identical.

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Question

What is point symmetry?

Point symmetry is identical to the rotational symmetry of order 2. In point symmetry, there is a single common point of reflection for each point on the shape.

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Question

What is reflective symmetry?

Reflective symmetry is where one half of an object reflects the other half of the object.

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Question

What is glide symmetry?

Glide symmetry is a combination of a reflection and a translation. The translation done in glide symmetry is always perpendicular to the axis of reflection.

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Question

What is an axis of symmetry?

An axis of symmetry is the straight line that a graph or shape is reflected through. The axis of symmetry is never curved, so will always have an equation of .

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Question

What are some examples of symmetry in nature?

Flowers are often a very good example of rotational symmetry.

A great example of reflective symmetry is a still lake, reflecting a landscape on the surface.

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Question

Reflect the square ABCD in the line y = -x

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Question

What is the axis of symmetry for the equation of the parabola x = y2

x = 0

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Question

How many orders of rotational symmetry does a circle have?

Infinite

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