Symmetry

Learning Objectives

• Understand the meaning of symmetry.
• Determine all the symmetries for a given plane figure.
• Draw or complete a figure with a given symmetry.
• Identify planes of symmetry for three-dimensional figures.

Introduction

You know a lot about symmetry, even if you haven’t studied it before. Symmetry is found throughout our world—both the natural world and the human-made world that we live in. You may have studied symmetry in math classes, or even in other classes such as biology and art, where symmetry is a basic principle.

The transformations we developed in earlier work have counterparts in symmetry. We will focus here on three plane symmetries and a three-dimensional symmetry.

• line symmetry
• rotational symmetry
• point symmetry
• planes of symmetry

A plane of symmetry may be a new concept, as it applies to three-dimensional objects.

Line Symmetry

Line symmetry is very familiar. It could be called “left-right” symmetry.

A plane (two-dimensional) figure has a line of symmetry if the figure can be reflected over the line and the image of every point of the figure is a point on the original figure.

In effect, this says that the reflection is the original figure itself.

Another way to express this is to say that:

• the line of symmetry divides the figure into two congruent halves
• each half can be flipped (reflected) over the line
• and when it is flipped each half is identical to the other half

Many figures have line symmetry, but some do not have line symmetry. Some figures have more than one line of symmetry.

In biology line symmetry is called bilateral symmetry. The plane representation of a leaf, for example, may have bilateral symmetry—it can be split down the middle into two halves that are reflections of each other.

Rotational Symmetry

A plane (two-dimensional) figure has rotational symmetry if the figure can be rotated and the image of every point of the figure is a point on the original figure.

In effect, this says that the rotated image is the original figure itself.

Another way to express this is to say: After being rotated, the figure looks exactly as it did before the rotation.

Note that the center of rotation is the “center” of the figure.

In biology rotational symmetry is called radial symmetry. The plane representation of a starfish, for example, may have radial symmetry—it can be turned (rotated) and it will look the same before, and after, being turned. The photographs below show how sea stars (commonly called starfish) demonstrate 5-fold radial symmetry.

Point Symmetry

We need to define some terms before point symmetry can be defined.

Reflection in a point: Points and are reflections of each other in point if and are collinear and .

In the diagram:

• is the reflection of in point (and vice versa).
• is the reflection of in point (and vice versa).

A plane (two-dimensional) figure has point symmetry if the reflection (in the center) of every point on the figure is also a point on the figure.

A figure with point symmetry looks the same right side up and upside down; it looks the same from the left and from the right.

The figures below have point symmetry.

Note that all segments connecting a point of the figure to its image intersect at a common point called the center.

Point symmetry is a special case of rotational symmetry.

• If a figure has point symmetry it has rotational symmetry.
• The converse is not true. If a figure has rotational symmetry it may, or may not, have point symmetry.

Many flowers have petals that are arranged in point symmetry. (Keep in mind that some flowers have petals. They do not have point symmetry. See the next example.)

Here is a figure that has rotational symmetry but not point symmetry.

Planes of Symmetry

Three-dimensional (3-D) figures also have symmetry. They can have line or point symmetry, just as two-dimensional figures can.

A 3-D figure can also have one or more planes of symmetry.

A plane of symmetry divides a 3-D figure into two parts that are reflections of each other in the plane.

The plane cuts through the cylinder exactly halfway up the cylinder. It is a plane of symmetry for the cylinder.

Notice that this cylinder has many more planes of symmetry. Every plane that is perpendicular to the top base of the cylinder and contains the center of the base is a plane of symmetry.

The plane in the diagram above is the only plane of symmetry of the cylinder, that is parallel to the base.

Example 1

How many planes of symmetry does the rectangular prism below have?

There are three planes of symmetry: one parallel to and halfway between each pair of parallel faces.

Lesson Summary

In this lesson we brought together our earlier concepts of transformations and our knowledge about different kinds of shapes and figures. These were combined to enable us to describe the symmetry of an object.

For two-dimensional figures we worked with:

• line symmetry
• rotational symmetry
• point symmetry

For three-dimensional figures we defined one additional symmetry, which is a plane of symmetry.

Points to Consider

Symmetry seems to be the preferred format for objects in the real world. Think about animals and plants, and about microbes and planets. There is a reason why nearly all built objects are symmetric too. Think about buildings, and tires, and light bulbs, and much more.

As you go through your daily life, be alert and aware of the symmetry you encounter.

The following questions are for your own review. The answers are listed below for you to check your work and understanding.

Review Question

True or false?

-1. Every triangle has line symmetry.
0. Some triangles have line symmetry.
1. Every rectangle has line symmetry.
2. Every rectangle has exactly two lines of symmetry.
3. Every parallelogram has line symmetry.
4. Some parallelograms have line symmetry.
5. No rhombus has more than two lines of symmetry.
6. No right triangle has a line of symmetry.
7. Every regular polygon has more than two lines of symmetry.
8. Every sector of a circle has a line of symmetry.
9. Every parallelogram has rotational symmetry.
10. Every pentagon has rotational symmetry.
11. No pentagon has point symmetry.
12. Every plane that contains the center is a plane of symmetry of a sphere.
13. A football shape has a line of symmetry.
14. Add a line of symmetry the drawing.

    

15. Draw a quadrilateral that has two pairs of congruent sides and exactly one line of symmetry.
16. Which of the following pictures has point symmetry?

    

17. How many planes of symmetry does a cube have?

Review Answers

-1. False
0. True
1. True
2. False
3. False
4. True
5. False
6. False
7. True
8. True
9. True
10. False
11. True
12. True
13. True

14. 

15. Any kite that is not a rhombus. Two examples are shown.

    

16. The four of hearts
17. Nine