# Introduction to Symmetry

Snow Crystals

Relevant examples from Escher's work:

## Symmetry

Symmetry is something all human beings look for and seem to intuitively understand. One way to describe symmetry is to say that it is harmony or beauty of form that results from balanced proportions.

More technically, symmetry is a correspondence between different parts of an object. For a geometric object, symmetry is a correspondence between pairs of points that are equally positioned about a point, line or plane.

## Reflection Symmetry

If points of a figure are equally positioned about a line, then we say the figure has reflection symmetry, or sometimes mirror symmetry. The line is called the reflection line, the mirror line, or the axis of symmetry. The axis of symmetry separates the figure into two parts, one of which is a mirror image of the other part.

The simplest case of reflection symmetry is known as bilateral symmetry. For example, each of the following figures exhibits bilateral symmetry:

The heart and smiley each have a vertical axis of symmetry, and the lobster has a horizontal axis of symmetry. The arrow has an axis of symmetry at an angle. If you draw the reflection line though any one of these figures, you will notice that for every point on one side of the line there is a corresponding point on the other side of the line. If you connect any two corresponding points with a segment, that segment will be perpendicular to the axis of symmetry and bisected by it (cut into two equal length segments):

Bilateral symmetry is the most common type of symmetry found in nature, occurring in almost all animals and many plants. Congnitive research has shown that the human mind is specially equipped to detect and prefer bilateral symmetry [[1]]. In fact, humans are especially good at detecting bilateral symmetry when the axis of symmetry is oriented vertically. As you proceed through this course, you will look for symmetry in all sorts of complicated images. Remember that your eyes are hard wired to do this well when the axis is vertical, and so it will be a tremendous help to rotate the images (or your head) as you look for symmetries.

Some objects or images can have more than one axis of reflection symmetry. Here are some examples, with the reflection axes shown as dotted red lines:

Pay special attention to the diagonal reflection axes in the cross. These are easy to overlook, and occur frequently.

## Rotational Symmetry

If points on a figure are equally positioned about a central point, then we say the object has rotational symmetry. A figure with rotational symmetry appears the same after rotating by some amount around the center point.

The angle of rotation of a symmetric figure is the smallest angle of rotation that preserves the figure. For example, the figure on the left can be turned by 180° (the same way you would turn an hourglass) and will look the same. The center (recycle) figure can be turned by 120°, and the star can be turned by 72°. For the star, where did 72° come from? The star has five points. To rotate it until it looks the same, you need to make $1/5$ of a complete 360° turn. Since $1/5 \times 360^\circ = 72^\circ$, this is a 72° angle rotation.

Using degrees to describe the rotation amount is inconvenient because the precise angle is not obvious from looking at the figure. Instead, we will almost always use the order of rotation to describe rotational symmetry:

Order of rotation
A figure has order $n$ rotational symmetry if $1/n$ of a complete turn leaves the figure unchanged.

As with the star, you can compute the angle of rotation from the order of rotation:

An order $n$ rotation corresponds to a $\frac{360^\circ}{n}$ angle of rotation.

In the three examples above, the hourglass has order 2 rotation symmetry, the recycle logo has order 3 rotation symmetry, and the star has order 5 rotation symmetry.

## Rosette Patterns

In this section, we begin our classification of the possible symmetries a plane figure may have. First notice that all the examples of reflection symmetry which contained more than one axis of reflection also contained rotational symmetry. You can explore this phenomenon in Composition Exploration.

A symmetry group is the collection of all symmetries of a plane figure. In the pictures we have seen so far, the symmetry groups have all been one of two types:

Rosette with C12 symmetry, St. Louis Cathedral Basilica
Cyclic symmetry group
Rotation symmetry only around a center point. If the rotation has order $n$, the group is called C$n$.
Dihedral symmetry group
Rotation symmetry around a center point with mirror lines through the center point. If the rotation has order $n$, there will be $n$ mirror lines and the group is called D$n$.

In both cyclic and dihedral groups, the case $n=1$ is a little special. A rotation of order 1 is not really a rotation, since the rotation angle of 360° does not move the figure. Then C1 symmetry is the same as no symmetry. D1 symmetry has no rotational symmetry and a single mirror line - what we previously called bilateral symmetry.

The table below gives examples of figures with all cyclic and dihedral symmetry groups for $n=1,2,3,4,5$.

Rot. order 1 2 3 4 5
Cyclic
Dihedral

Together, the cyclic and dihedral symmetry groups are known as rosette symmetry groups, and a pattern with rosette symmetry is known as a rosette pattern. Rosette patterns have been used as architectural and sculputural decoration for millenia — see wikipedia:Rosette (design) for details.

The rosette symmetry groups are a first step towards a classification of all possible symmetry groups for plane figures. But consider this figure:

The "target" can be rotated by any angle without changing its appearance. Because of this, the order of rotation is undefined, and the symmetry group of the figure is not one of the cyclic or dihedral groups described above. On the other hand, this figure has infinitely many symmetries - it has reflection symmetry across any line through its center. We will see other examples of pictures with infinitely many symmetries in the next section.

However, we can make the following statement, a complete classification of finite symmetry groups for planar figures:

For a figure in the plane with finitely many symmetries, the cyclic groups C1, C2, C3, ... and the dihedral groups D1, D2, D3, ... are the only possible symmetry groups.

Proving this statement is lengthy and somewhat technical. However, we will make other statements like this one and it is important to have some understanding of why no other symmetry groups can occur, so we give a sketch of the argument. The approach is to rule out possibilities until only the cyclic and dihedral groups are left:

Step 1: Rule out two parallel axes of reflection: If a figure has two parallel axes of reflection, then a "hall of mirrors" effect quickly forces it to have infinitely many mirror symmetries. Since we assumed our figure has finitely many symmetries, it cannot have parallel axes of reflection.

Step 2: Rule out two separate centers of rotation. This is the most technical part of the argument, and too difficult to go into here. You can get a feeling for why two rotation centers lead to infinte symmetries by working Rosette Exercises#Two rotations.

Step 3: Show that axes of reflection pass through rotation centers, if there are any. If a rotation center is off of a mirror line, there must be a second rotation center which is its mirror image. By Step 2, this cannot happen.

Step 4: Show that two intersecting axes of reflection force a rotation center at their intersection. You should have convinced yourself of this with the Composition Exploration.

Conclusion:

• If there are no reflection lines, then from Step 2 there is only one rotation center and the group is cyclic.
• If there is only one reflection line and no rotation, the group is D1 for bilateral symmetry.
• If there are multiple reflection lines, they must intersect, by Step 1. They must all intersect at the same point, because each point of intersection gives rise to a rotation center (by Step 4) and there can be at most one rotation center (by Step 2). Then the group is a dihedral group.

## Colors and Symmetry

So far all of our examples have been single color figures - either a black outline around a white region, or a black region. Colors introduce some complications. Consider the yin-yang symbol:

 Yin-yang Rotated 180° Outline

Clearly this symbol has symmetry. However, a 180° rotation of the figure looks different. If the figure is simply the set of black points, then the 180° rotation is not a symmetry. In this case, it is better to think of the figure as an outline and the two regions as colored black and white, respectively. Then the 180° rotation preserves the outline and interchanges the colors.

Shield of Valais

For another example, consider the shield of the Swiss canton of Valais, which (according to Escher's son George[1]) partly inspired Escher's print Day and Night. The outline of the shield has bilateral symmetry, but the shield does not - the red and white colors are interchanged.

Remember that a symmetry gives a correspondence between pairs of points of an image. When dealing with colored images, there are actually three reasonable kinds of symmetry:

Color preserving symmetry
Corresponding points are the same color.
Color symmetry
Each color has a "target" color. Each point is sent to a point which has its target color. We say that the colors are permuted.
Symmetry of the outline
A symmetry you would get by completely ignoring the colors.

For example, the yin-yang has no color preserving symmetry. The 180° rotation is a color symmetry, and also a symmetry of the outline. Similarly, the shield of Valais has no color preserving symmetry, but a reflection which is a color symmetry and a symmetry of the outline.

Since there are three kinds of symmetry, there are also three kinds of symmetry group: The color preserving symmetry group, the color symmetry group, and the symmetry group of the outline. Thus the yin-yang has color symmetry group C2, and the shield of Valais has color symmetry group D1.

Compass Rose

Let's look at a more interesting example, the compass rose. The outline has symmetry group D4. However, none of these symmetries permute the colors — one grey arrow will become red and the others will remain grey. For a permutation of colors, all the grey arrows should become red or they should all stay grey. The figure has no color symmetry.

Again look at the compass rose, but now suppose the red arrow was grey like the other directions. Then the resulting image would have color symmetry group D4 with all reflections interchanging light and dark grey. The group of color preserving symmetries would be C4.

Although issues with color could complicate our study of symmetry, they almost never do. Because humans equate symmetry with beauty, most artistic patterns are colored in such a way that the colors emphasize the symmetry of the pattern, rather than destroying it. The mathematical term for this is perfect coloring:

Perfect coloring
A colored pattern whose color symmetry group is the same as the symmetry group of its outline.

The upshot is that you can usually look for symmetry simply by "ignoring" colors.

Almost all of Escher's symmetric patterns are perfectly colored. The only exceptions are two of his earliest sketches, Sketch #3 (weightlifters) and Sketch #14 (lizards). It takes no mathematical training to look at these two sketches and feel that something is slightly amiss — your brain is very good at picking up a lack of symmetry. You can learn more about these two sketches by doing Wallpaper Exercises#imperfect colors later in the course.

## Notes

1. Visions of Symmetry, pg. 238