Rotational and Reflection Symmetries Printer Friendly Version
Symmetry and its absence (asymmetry) play important roles in science. Symmetry means "same measure" and suggests balance or regularity of form. We are naturally familiar with symmetry from our experience of our own body, where it is clear that the right side more or less matches the left. Although symmetry is actually a rather complex topic, most of us have a certain intuitive sense regarding it.

A symmetry operation is an operation performed on an object or pattern which brings it into coincidence with itself. There are several classes of symmetry operations. Two basic ones will be considered in this introduction: rotation and reflection. It will be easier for us if we begin our exploration of symmetry by considering two-dimensional forms.

Rotation in Two Dimensions

A square is a simple example of a two-dimensional object having rotational symmetry. Such a square is shown in Figure 1. An axis of rotational symmetry passes through point P at the center of the square. The axis is perpendicular to the plane of the square. Each time the square is turned 90º about the axis, the new orientation coincides with the original. Four such equivalent orientations are found in bringing the square back to its original state. We therefore say that the square has rotational symmetry of order four. The order of rotational symmetry about a given axis is therefore found by counting how many times a "coincidence" occurs in a rotation of 360º.

Figure 1: Square centered on P.
The rotation axis is perpendicular to the plane at P.

Exploring 2-D Rotational Symmetries

 1. The capital letters of the alphabet are printed below. Make a list of all those that have rotational symmetry.   A B C D E F G H I J K L M N O P Q R S T U V W X Y Z   Letters having rotational symmetry:

 2. A pinwheel is an excellent example of rotational symmetry. Construct a pinwheel by cutting the diagonals of a paper square nearly to the center and bringing one corner of each triangular section to the center. Run a straight pin through the center.     What is the order of rotational symmetry of the pinwheel?

 3. The design of a Zanzibar mat is shown below.     Determine its order of rotational symmetry.

 4. Benzene is a planar molecule. What is the rotational symmetry of the benzene molecule as it is represented in the diagram that follows?

Refer to the following information for the next eight questions.

Eight figures are shown below. Indicate the order of rotational symmetry of each.

Reflection in Two Dimensions

An object enjoying reflection symmetry is able to be superimposed on its image as seen in a mirror. The order of symmetry is determined by how many mirror planes exist such that this condition is met. The square shown in Figure 1, for example, has four such planes: one through a horizontal axis, one through a vertical axis, and two through the diagonals. Two ways to test for reflection symmetry are the fold test and the mirror test. In the fold test, you check for symmetry by folding along a line. An axis of symmetry exists if the folding flips one part of the figure on top of the other so that the two parts match. In the mirror test, you try to place a mirror upright on the figure so that the reflection matches the part hidden by the mirror. The number of different positions for which such a relation exists determines the order of symmetry of the figure.

Exploring 2-D Reflection Symmetry

Refer to the following information for the next three questions.

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
 1. Which capital letters have reflection symmetry through a horizontal plane?

 2. Which letters have reflection symmetry through a vertical plane?

 3. Which letters exhibit neither rotation nor reflection symmetry?

 4. The sketch below is of the Kwele, an African mask.     Does it show any reflection symmetries? Please explain.

 5. Draw the axes of reflection symmetry of the benzene molecule.   How many total axes did you discover?

 6. Draw the axes of reflection symmetry of the Zanzibar mat.   How many total axes did you discover?

Refer to the following information for the next eight questions.

Determine the order of reflection symmetry of each figure below. Please sketch in the axes on each figure.

Fermilab
Topics in Modern Physics, May 1990
Mary Ethel Parrott, SND