The Four Types of Symmetry in the Plane

A pattern is symmetric if there is at least one symmetry
(rotation, translation, reflection, glide reflection)
that leaves the pattern unchanged.

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.

(examples below)

 

1. For example, rotation by 90 degrees about a fixed point is an example of a plane symmetry.

    

2. 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.

    

3. 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.

    

4. 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.

Rotation

To rotate an object means to turn it around. Every rotation has a center and an angle.

Translation

To translate an object means to move it without rotating or reflecting it. Every translation has a direction and a distance.

Reflection

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".

Glide Reflection

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.

Just remember the golden rule: A fiigure, picture, or pattern is said to be symmetric if there is at least one symmetry that leaves the figure unchanged