# SYMMETRY

### 1. SYMMETRY AS A TRANSFORMATION

A symmetry is a transformation …

that leaves an object looking unchanged.

This activity was designed when Grades 3-6 teachers asked for ideas for teaching about symmetry and transformation. Typically, symmetry and transformation are taught as distinct concepts. However, to a mathematician, a symmetry is a transformation: a transformation that leaves an object looking unchanged.

### 2. ROTATION SYMMETRIES OF THE SQUARE

How many rotation symmetries does a square have?

The 4 rotation symmetries of the square (using 0, 1/4, 1/2 and 3/4 turns):

The 4 rotation symmetries of the square (using 0, 90, 180 and 270 degrees):

### 3. SYMMETRY AS A PERMUTATION

A symmetry can be a permutation?

When we label the vertices of the square with 1, 2, 3, and 4 dots, we can find permutations that are equivalent to symmetries.

A permutation is an arrangement of a set or subset of objects.

Here are the 4 rotation symmetries of the square as permutations.

### 4. GROUP THEORY

Symmetry as a transformation is at the heart of Group Theory.

In fact, Group Theory is the study of symmetries, which are transformations that leave an object appearing unchanged.

#### What is a group?

The 4 rotation symmetries of the square, shown below, are an example of a group.

The 4 rotation symmetries of the square are a group because they satisfy the following conditions:

1. They are a closed set. If you combine 2 or more rotations symmetries of the square (as 1/4-turn followed by 1/2-0turn) the result (3/4-turn) is always one of the 4 rotation symmetries of the square. An example of a set that is not closed would be the numbers {1, 2, 3, 4} with the operation of addition. If you add any 2 or more of these numbers, the result is not always part of the original set.
2. The set contains an identity element. The 0-turn rotation symmetry is the identity element. If you apply a 0-turn to one of the other rotation symmetries, its identity does not change. For example, applying a 0-turn to a 1/2-turn results in 1/2 turn. Do numbers have identity elements? Yes. For addition, the identity element is the number 0. Adding 0 to a number does not change its identity. What are the identity elements for subtraction, multiplication and division?
3. Each element of the set has an inverse. For example, the inverse of 1/4-turn (clockwise) would be negative 1/4-turn (counterclockwise).
4. The operation used to combine symmetries is associative. That is, if we combine 3 symmetries, say 1/4-turn + 1/2-turn + 3/4-turn, the result is the same even if we combine them in different ways. For example: (1/4-turn + 1/2-turn) + 3/4-turn = 1/4-turn + (1/2-turn + 3/4-turn) = 6/4-turn or 1/2-turn. Is addition associative for numbers? What about subtraction, multiplication and division?

### 5. REFLECTION SYMMETRIES OF THE SQUARE

What are the reflection symmetries of the square?

#### The reflection symmetries of the square

Here is one of the reflection symmetries of the square:

There are 3 more. Can you find them? You may use this PDF handout to record them.

#### Group?

Do the reflection symmetries of the square form a group?

Or, do they like to meet new symmetries when they transform one another?

See 4. Group Theory (above) for the 4 necessary conditions to form a group. And, read the Grad e 2 student experience below.

#### “Math is so cool!” (Grade 2) Use code to combine 2 reflections at mathsurprise.ca/apps/sym/rotation-reflectionWhat is the result?

In a Grades 2/3 classroom, while using a coding environment to reflect a square, a Grade 2 student noticed something peculiar. Here is the conversation:

• Something is wrong.
• What is it?
• I put 2 reflection symmetries together and I got a rotation symmetry.
• That’s interesting. Try it again with 2 different reflection symmetries.
• It’s still a rotation.
• It looks like you discovered something new.
• Math is so cool!

### 6. BUMPER SYMMETRIES

What happens when you imagine the 4 rotation symmetries of the square as bumper cars, where they transform one another as they bump?

Grades 2/3 students used sticky notes to simulate symmetries as bumper cars.

### 7. BUMPER SYMMETRIES WITH CODE

Investigate bumper symmetries with code at this link: https://mathsurprise.ca/apps/sym/bumper-squares/

#### In a Grade 6 classroom

Before setting off the 4 rotation symmetries of the square as bumper cars, we asked a class of Grade 6 students to predict what might happen.

Three volunteers put their hands up and we asked them to come up and tell us their predictions quietly (so no one else would hear). All 3 predictions were different, and all 3 were correct. How is that possible?

Here are the 3 predictions:

1. All squares will be red
2. All squares will be 0-degree rotation
3. All squares will be 360-degree rotation