Symmetry around a Point in the Plane

Steven Dutch, Professor Emeritus, Natural and Applied Sciences, University of Wisconsin - Green Bay

Reflection and Rotation

Symmetry is the set of mathematical rules that describe the shape of an object. The two most common kinds are:

Reflection or Mirror symmetry

Two halves of an object are mirror images of each other. The figures below have reflection symmetry.

The letters A U V T Y all have reflection symmetry across a vertical plane. The letters D E C B have reflection symmetry across a horizontal plane. It is possible to have reflection symmetry in more than one direction, as in the object at right. The letters H I O X have both horizontal and vertical mirror symmetry. Note that:

People and most vertebrates basically have mirror symmetry (at least externally). In biology it is often called bilateral symmetry.

A Trivia Question

When you look in a mirror, why are your left and right sides reversed, but not your top and bottom?

See Answer

Rotational Symmetry

Parts of an object are related by rotation. As a general rule, you can rotate such an object through a certain angle and it will still have the same appearance.

The figures below have rotational symmetry:

Handedness (Chirality, Enantiomorphism)

Objects with rotational symmetry but no reflection symmetry exist in two forms that we might call "left-handed" and "right-handed". Objects with reflection symmetry are their own mirror image. Rotation symmetry without reflection is often used in graphic design to portray the idea of speed, power, or dynamic action.

In chemistry, the fact that some molecules have left- and right- handed versions is called chirality from the Greek word for hand. The same root appears in the word "chiropractor." In crystallography, shapes that lack mirror symmetry are called enantiomorphic.

symmetry and the Real World

Real objects never are perfectly symmetrical. Even in a crystal, there are impurity atoms and defects that spoil the perfect symmetry. We say an object has symmetry if it does as a whole, apart from minor defects. For example, we say people have bilateral symmetry even though they may have a mole on one side of their face but not the other.

When symmetry becomes imperfect enough to be important depends on the situation. A ding in a boat propeller is less critical than a nick on a high-speed turbine blade.

One-fold symmetry = No symmetry

Some objects have no symmetry at all: a computer keyboard, a left glove, the letters G Q d p J L. In a 360 degree rotation, the object matches its original appearance only once, after a full 360 degrees. Thus, we can say the object has one-fold symmetry. One-fold symmetry is the same as no symmetry at all.

This might seem like a roundabout description but actually all the mathematical formulas for symmetry work equally well for this definition of one-fold symmetry. It allows us to develop formulas for all symmetries, even no symmetry at all.

symmetry within symmetry

Objects of high symmetry also contain lower symmetries as well. Hexagon abcdef below has six-fold symmetry. However, we can rotate it by 120 degrees and preserve its appearance, so it has 3-fold symmetry. We can rotate it by 180 degrees and preserve its appearance, so it also has 2-fold symmetry, and of course we can rotate it by 360 degrees and preserve its appearance, so it has one-fold symmetry as well. Everything has one-fold symmetry.

Thus, 6-fold symmetry automatically includes 3-, 2-, and 1-fold symmetry.

In general, n-fold symmetry includes any lower number that divides n. An octagon has 8-, 4-, 2- and 1-fold symmetry.

Combining Symmetry Operations

Interesting things happen when we combine symmetries. For example, Figure a. below has reflection symmetry in two directions. However, it also has two-fold symmetry - it looks the same after a 180-degree rotation. The letters O H I X not only have mirror planes in two directions, but also two-fold symmetry as well.

To see what happens, look at the figure on the right. In the mirror plane, point a is reflected to b. If we add a two-fold symmetry axis, point a is rotated to c and b is rotated to d. Note that c is a mirror image of d. Also note, though, that we can now draw a mirror plane at right angles to the original plane, and d is a mirror image of a, and c is a mirror image of b. Put two mirror planes at right angles, and you get a two-fold symmetry axis. Put a two-fold axis on a mirror plane, and you get a second mirror plane. We see three principles here:

When you combine other rotations with mirror planes, there may be many symmetries. Consider hexagon abcdef below:

Combining Rotation and Reflection

A hexagon illustrates that an interesting thing happens when we combine rotation and reflection. Consider the following two cases:

In the case of the three-fold axis, the rotation rotates the mirror plane from a to b and then to c. symmetry operations act not only on patterns, but on other symmetry operations. The six-fold rotation rotates the mirror plane to b, c, d, e, and f, but a-d, b-e and c-f are the same: flipping a mirror plane 180 degrees merely flips it over onto itself. At first glance, the two situations look alike. So let's take a second glance.

In the case of a three-fold axis and a mirror plane, only three planes of symmetry result. But when a six-fold axis is involved, we get not only three mirror planes by direct rotation, but three unexpected ones as well. symmetry that is created by the interactions of other symmetries is called induced.

When n is odd, every rotation results in a new mirror plane, because the rotated left end of each plane falls between the right ends of two others. But every even-fold rotation results in a 180 degree rotation that simply flips the mirror planes over onto themselves, so that half the mirror planes are duplicates of each other. It's almost as if nature creates a second set to compensate.

Notation

We're now ready to set up some notation for symmetry operations:

                                                                
                              o 
                             / \                                                                        / \     
                        o---o   o---o             
                         \   \ /   /                  
                          o---o---o             
                         /   / \   \             
                        o---o   o---o    
                             \ /                        
                              o

Answer to Trivia Question:

A mirror doesn't reverse right and left: it reverses front and back. The mirror plane is perpendicular to your line of sight. The left side of your image is on your left side, and the right side of your image is on your right side. We only think the mirror reverses right and left because we'd have to turn around to face the way our image faces.

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Created 11 Feb. 1997, Last Update Sept. 22, 1999