"Finite Symmetry Groups"

- Finite
- Symmetry
- Group

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

Translations

Reflections

Reflecting across a vertical line is not a symmetry for the yin yang, but it is a symmetry of the smiley face.

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.

- an identity
- inverses
- an associative property

- The identity is 0, because adding 0 to any other whole number leaves it unchanged: \(0 + 3 = 3\)
- 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.
- The associative property holds: \( x + (y + z) = (x+y)+z\)

- The identity is the way it comes out of the box: all sides are the same color.
- Inverses are the reverse of whatever we do: the inverse of rotating one side is rotating it back to where it was.
- (We'll ignore associativity, but it holds too!)

- The identity is doing nothing, but we'll count this as a 0 degree rotation.
- 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).
- Associativity means that we can expect to always get the same result when we apply three symmetries to an image.

- Look for reflections. If there are any reflective symmetries, count them. An image with \(n\) reflective symmetries has symmetry group \(D_n\).
- 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 \(C_n\).

We can

We can also

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

So we have no reflective symmetries.

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

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

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

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

Its symmetry group is \(D_1\).

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.