SYMMETRY

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Symmetry. The more it changes, the more it stays the same.
Symmetry. The more it changes, the more it stays the same. Click To Tweet

1. SYMMETRY AS A TRANSFORMATION

A symmetry is a transformation …

that leaves an object looking unchanged.

For millennia, symmetry was primarily used to describe the properties of shapes (Stewart, 2013), which is what most elementary school children learn about symmetry today (Healy, 2003). Over the last couple of centuries, mathematicians have come to view symmetry not as an attribute but as a transformation that leaves an object apparently unchanged (or invariant, in mathematics language).

Gadanidis, G, Clements, E. & Yiu, C. (2018). Group theory, computational thinking and young mathematicians. Mathematical Thinking and Learning 20(1), 32-53. 

Grades 3-6

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.

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?

A zero rotation, or 1234

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.

It is unclear how symmetry as a transformation and group theory evolved historically. Stewart (2013) suggests that it grew out of the work of Galois, whose study of solutions of algebraic equations used the idea that “An algebraic formula has symmetry if its variables can be interchanged without altering its value” (p. 2). On the other hand, Wussing (2007/1984) suggests that the development of group theory “had three equally important historical roots, namely, the theory of algebraic equations, number theory, and geometry. Abstract group theory was the result of a gradual process of abstraction from implicit and explicit group-theoretic methods and concepts involving the interaction of its three historical roots” (p. 16).

Gadanidis, G, Clements, E. & Yiu, C. (2018). Group theory, computational thinking and young mathematicians. Mathematical Thinking and Learning 20(1), 32-53. 

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:

identity element
  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-reflection
What 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

Symmetries as bumper cars.

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

background

Looking back to the first year of this project, we realize that the progress we have made has been in part due to our research into different ways of communicating about group theory, by talking with mathematicians, reading books about symmetry and group theory, and accessing online resources. Just as important has been the CT tools we work and think with. Being able to create our own coding environments has offered us direct and vicarious personal experience with our design principles […]. We experienced agency by using Blockly to define our coding environment to suit our needs. We vicariously experienced the low floor, high ceiling of the coding environments we created through the eyes of children and teachers. We experienced surprise when what-might-be drafts of coding environments gave us new mathematical and pedagogical insights into group theory, such as the Bumper Symmetries coding environment and its playful look at symmetry composition. And we did all this for an audience of teachers and children (and their parents) that invited us into their classrooms.

Gadanidis, G, Clements, E. & Yiu, C. (2018). Group theory, computational thinking and young mathematicians. Mathematical Thinking and Learning 20(1), 32-53.