Finite Symmetry Groups

Dissecting the Term
"Finite Symmetry Groups"

There are three words that need to be clarified here:
  • Finite
  • Symmetry
  • Group

What Does Finite Refer To?

The word finite, in this context, refers to the image we're working with. We assume all images are finite, meaning that we can draw a boundary around them. The patterns do not repeat indefinitely in any direction.

What Does Symmetry Refer To?

A symmetry of a picture is an action that leaves the image apparently unchanged.

Every symmetry is a rigid motion, but not every rigid motion is a symmetry of every image.

Recall: The Rigid Motions

Rotations

Translations

Reflections

What Does Symmetry Refer To?

Click the images on the right to reflect them. If they look different from their original on the left, it means that particular reflection is NOT a symmetry of that image.
Reflecting across a vertical line is not a symmetry for the yin yang, but it is a symmetry of the smiley face.

What Does Symmetry Refer To?

Some images have more than one reflection symmetry. This star can be reflected across 5 different lines:

What Does Symmetry Refer To?

Finite images also have rotational symmetry, meaning the image can be turned by a certain angle and left unchanged.
The smiley face has to be rotated all the way around before it looks the same, but the pentagon can be rotated by other angles and we wouldn't see a difference.

What Does Symmetry Refer To?

In summary, a symmetry of a picture is something we can do to it that leaves it unchanged. For finite images these symmetries will always be rotations or reflections.

What Is A Group?

In mathematics, a group refers to a set of things, together with a binary operation on those things, that have
  1. an identity
  2. inverses
  3. an associative property
Most operations have the associative property, so it's the first two we'll really be worried about.

What Is A Group?

An example of a group are the integers (whole numbers) together with addition.
  1. The identity is 0, because adding 0 to any other whole number leaves it unchanged: \(0 + 3 = 3\)
  2. Inverses are negatives, because adding -x "undoes" the process of adding x. If we add 3 to 7, then add -3, we're back to 7.
  3. The associative property holds: \( x + (y + z) = (x+y)+z\)

What Is A Group?

Another example of a group is the set of positions of a Rubik's Cube:
  1. The identity is the way it comes out of the box: all sides are the same color.
  2. Inverses are the reverse of whatever we do: the inverse of rotating one side is rotating it back to where it was.
  3. (We'll ignore associativity, but it holds too!)

Finite Symmetry Groups

The symmetries of an image also form a group.
  1. The identity is doing nothing, but we'll count this as a 0 degree rotation.
  2. The inverse of each symmetry is whatever undoes the motion we apply. Reflections are their own inverse, because reflecting an image across the same line brings the image back to where it was. The inverse of a rotation of x degrees is a rotation of -x degrees (equivalent to 360-x degrees).
  3. Associativity means that we can expect to always get the same result when we apply three symmetries to an image.

Cyclic Groups

If an image only has rotational symmetries, its symmetry group is called a Cyclic Group, denoted \(C_n\) where \(n\) is the largest number of times we can rotate the image before it gets back to where it originally was.

Dihedral Groups

If an image has rotations and reflections that leave it unchanged, its symmetry group is called a dihedral group, denoted \(D_n\). Here, \(n\) is also the largest number of times we can rotate the image before it gets back to where it originally was. This number will also be equal to the number of reflections the image has.

Identifying Finite Symmetry Groups

  1. Look for reflections. If there are any reflective symmetries, count them. An image with \(n\) reflective symmetries has symmetry group \(D_n\).
  2. If there are no reflections, look for the smallest angle it can be rotated, then count how many times we'd have to apply that rotation to get it back to where we started. If we can rotate \(n\) times before it gets back to the starting position, the image has symmetry group \(C_n\).

Some Examples

What can we do to a square that leaves it apparently unchanged?

Some Examples

What can we do to a square that leaves it apparently unchanged?



We can rotate it four times (by an angle of 90 degrees each time). It's easier to see this if we label a corner:

Some Examples

What can we do to a square that leaves it apparently unchanged?



We can also reflect it across four axes:

Some Examples

What can we do to a square that leaves it apparently unchanged?



So a square has 4 rotations and 4 reflections: its symmetry group is \(D_4\).

Some Examples

What about this fan shape?

Some Examples

If we reflect it, it looks different, because of the curve in the "blades".


So we have no reflective symmetries.

Some Examples

But we can rotate it three times before it gets back to the beginning.


The symmetry group of the fan would be \(C_3\).

Some Examples

This cactus also has no reflections, but we could rotate it 5 times:


The symmetry group of the cactus would be \(C_5\).

Some Examples

This starfish has 5 axes of reflection and 5 rotations.


The symmetry group of the starfish would be \(D_5\).

Some Examples

The yin yang has no reflections, and can only be rotated completely around.


This still counts as a symmetry! Its symmetry group is \(C_1\).

Some Examples

This butterfly has one axis of reflection, and only the "trivial" rotation all the way around.


Its symmetry group is \(D_1\).

Summary

Every image has at least one symmetry, but it might just be the "trivial" full rotation. In this case, its group is \(C_1\).

It's easier to count reflections. If there are \(n\) reflections, the group is \(D_n\).

If there are no reflections, the group is \(C_n\), where \(n\) is the maximum number of times it can be rotated until it's back to the beginning.

One Last Issue

Some images can be rotated an infinite number of ways without changing them, like a circle. They'll also have an infinite number of reflections. These images have symmetry group \(D_\infty\).