The Four Types of Symmetry in the Plane
written by Dr. Susan Addington
California Math Show
formatted and edited by Suzanne Alejandre
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A pattern is symmetric if there is at least one symmetry
(rotation, translation, reflection, glide reflection)
that leaves the pattern unchanged.
To rotate an object means to turn it around. Every rotation has a center and an angle.
To translate an object means to move it without rotating or reflecting it.
Every translation has a direction and a distance.
To reflect an object means to produce its mirror image. Every reflection has a mirror line. A reflection of an "R" is a backwards "R".
A glide reflection combines a reflection with a translation along the direction of the mirror line. Glide reflections are the only type of symmetry that involve more than one step.
Symmetries create patterns that help us organize our world conceptually. Symmetric patterns occur in nature, and are invented by artists, craftspeople, musicians, choreographers, and mathematicians.
In mathematics, the idea of symmetry gives us a precise way to think about this subject. We will talk about plane symmetries, those that take place on a flat plane, but the ideas generalize to spatial symmetries too.
Plane symmetry involves moving all points around the plane so that their positions relative to each other remain the same, although their absolute positions may change. Symmetries preserve distances, angles, sizes, and shapes.
- For example, rotation by 90 degrees about a fixed point is an example of a plane symmetry.
- Another basic type of symmetry is a reflection. The reflection of a figure in the plane about a line moves its reflected image to where it would appear if you viewed it using a mirror placed on the line. Another way to make a reflection is to fold a piece of paper and trace the figure onto the other side of the fold.
- A third type of symmetry is translation. Translating an object means moving it without rotating or reflecting it. You can describe a translation by stating how far it moves an object, and in what direction.
- The fourth (and last) type of symmetry is a glide reflection.
A glide reflection combines a reflection with a translation along the direction of the mirror line.
A figure, picture, or pattern is said to be symmetric if there is at least one symmetry that leaves the figure unchanged. For example, the letters in
form a symmetric pattern: if you draw a vertical line through the center of the "Y" and then reflect the entire phrase across the line, the left side becomes the right side and vice versa. The picture doesn't change.
If you draw the figure of a person walking and copy it to make a line of walkers going infinitely in both directions, you have made a symmetric pattern. You can translate the whole group ahead one person, and the procession will look the same. This pattern has an infinite number of symmetries, since you can translate forward by one person, two people, or three people, or backwards by the same numbers, or even by no people. There is one symmetry of this pattern for each integer (positive, negative, and zero whole numbers).
- Classify all the capital letters in English (in their simplest forms) according to their symmetries. For example, "A" has a reflection in a vertical line, and "R" has no symmetry (except rotation by 0 degrees).
- Make a symmetric pattern by starting with an asymmetric shape (a letter is fine) and repeating a single translation over and over (also translate it backwards). That is, decide on a direction and distance for your translation (for example, 5 cm to the right). Translate your letter 5 cm to the right, then translate the new letter 5 cm to the right, etc. Also translate the original letter 5 cm. to the left, etc. Did you get any other types of symmetries (reflections, glide reflections, or rotations) in the process?
- Make a symmetric pattern by starting with an asymmetric shape and repeating a single glide reflection over and over (also glide it backwards). That is, pick a reflection line and a translation in a direction parallel to the reflection line. Keep applying the same glide reflection to the new shapes that you generate until you run out of paper. Did you get any other types of symmetries (reflections, translations, or rotations) in the process?
- On a piece of paper, draw a letter R (call it R1) and two parallel lines about an inch apart. Call the lines L1 and L2. Reflect the R across the first line, L1, and call it R2. Reflect R2 across the line L2 and call it R3.
a. How is R3 related to R1? (by which type of symmetry?)
b. Continue your pattern by reflecting the new Rs across L1 and L2. Keep going until you run out of paper or until you don't get anything new. Would this be an infinite pattern if you had an infinitely large piece of paper?
c. What symmetries does your pattern have besides reflections across L1 and L2?
- Repeat the previous activity using lines that are not parallel but intersect at a 45-degree angle.
a. Now how is R3 related to R1?
b. Continue your pattern by reflecting the new Rs across L1 and L2.
Keep going until you run out of paper or until you don't get anything new. Would this be an infinite pattern if you had an infinitely large piece of paper?
c.What symmetries does your pattern have besides reflections across L1 and L2?
- Remember making strings of paper dolls or snowflakes by cutting a strip
of folded paper? Adapt one of these activities so that it explicitly talks about symmetries. Invent some questions about it appropriate for children.
- (Extension) Consider patterns generated by reflections across two
intersecting lines, but use other angles. For example, what if the lines formed an angle of 111 degrees, or 60 degrees? If you can, find a general pattern that will predict results for any angle.