Enrichment Project #1

ENRICHMENT PROJECT #1

SYMMETRIES OF THE SQUARE

Abstract: This project will help students develop their geometric intuition about the symmetry of certain objects. Students can manipulate a square to explore and find its symmetries. After students have found all possible symmetries of the square they will learn how to compose two or more symmetries to obtain another symmetry. There are also extensions to less symmetric objects.

Format:We start with a description of the project as presented to the students – including instructions, explanations and comments by the teacher. A set of Worksheet Templates is then appended to the end.

Notes for the teacher:

- Worksheet #1. Having the students actually cut out a square to experiment with will be a big help in developing the student’s ability to visualize the symmetries.

- Worksheet #3. Here you may discuss associative property of composition.

- In general, each algebraic statement should be tied to the geometric operation.

- For these three-dimensional symmetry groups in the last few worksheets, you can (and probably should) construct 3D boxes to experiment with. But, reflection symmetries cannot be carried out in 3-space! There are two possibilities. First you could restrict your attention to the group of direct symmetries - those that can actually be carried out in 3-space. In that case, the group sizes are half the numbers listed on the worksheet. Another option is to construct a second box with the mirror image labeling. Also in these complicated examples, the students can easily miss some of the symmetries. They will discover this themselves when they try to build the multiplication table and find that it has some “holes” in it.

What are all the different ways we can manipulate or reorient this square?

To do this experimentally, cut a square out of stiff paper and label it as above. Turn the square over and label the corners on the back. BE SURE THE LABELS ON THE FRONT AND BACK OF EACH CORNER ARE THE SAME.

Worksheet #1. Reorient the square in all possible ways and draw the different possibilities below using the numbers to keep track of your findings.

Worksheet #2. You should have got this list of oriented squares in some order:

Using this labeling of the oriented squares, answer the following questions:

1)Starting the square oriented as in A above rotate it 270 degrees clockwise. What is your ending position?

2)What is the simplest motion (transformation) that will take the square in orientation A into orientation F?

Worksheet #3. Your answer to question 1) should be D. So we will denote the transformation that rotates the square by 270 degrees d. Your answer to question 2) should be the reflection through the vertical line down the middle of the square. We will denote this transformation by f.

For each of the eight possibilities give a geometric description of the simplest motion the transforms A into that position:

Transformation / Geometric Description
b: takes A to B
c: takes A to C
d: takes A to D
e: takes A to E
f: takes A to F
g: takes A to G
h: takes A to H
a: takes A to A

Worksheet #4. You should have got these descriptions:

Transformation / Geometric Description
b: takes A to B / the 90 degrees clockwise rotation
c: takes A to C / the 180 degrees clockwise rotation or half-turn
d: takes A to D / the 270 degrees clockwise rotation
e: takes A to E / the reflection the through the upper left – lower right diagonal
f: takes A to F / the reflection the through the vertical mid-line
g: takes A to G / the reflection the through the upper right – lower left diagonal
h: takes A to H / the reflection the through the horizontal mid-line
a: takes A to A / The identity transformation

With this geometric understanding of these transformations, or symmetriesyou should be able to decide the result of carrying out two of these transformations – one after the other.

For example b (rotating 90 degrees clockwise) and then c (rotating 180 degrees clockwise) results in d(the 270 degrees clockwise rotation).

For each pair of two symmetries, use the table below to record the result of carrying out one symmetry after the other.

Before you start, do you think the order will matter?

a / b / c / d / e / f / g / h
a
b
c
d
e
f
g
h

Are there any patterns that you notice?

Worksheet #5.

You should have filled in the table in one of the following two ways:

a / b / c / d / e / f / g / h
a / a / b / c / d / e / f / g / h
b / b / c / d / a / h / e / f / g
c / c / d / a / b / g / h / e / f
d / d / a / b / c / f / g / h / e
e / e / f / g / h / a / b / c / d
f / f / g / h / e / d / a / b / c
g / g / h / e / f / c / d / a / b
h / h / e / f / g / b / c / d / a
/ a / b / c / d / e / f / g / h
a / a / b / c / d / e / f / g / h
b / b / c / d / a / f / g / h / e
c / c / d / a / b / g / h / e / f
d / d / a / b / c / h / e / f / g
e / e / h / g / f / a / d / c / b
f / f / e / h / g / b / a / d / c
g / g / f / e / h / c / b / a / d
h / h / g / f / e / d / c / b / a

Think of combining symmetries b and e and let’s denote the result by be. It is natural to think of first rotating the square by 90 degrees (b) and then reflecting the through the upper left – lower right diagonal (e). This takes corner 1 to the upper right position and then down to the lower left position; it takes corner 2 to the lower right and leaves it there. So the result is h, as listed in the top table. However, it is also natural to think of these symmetries or transformations as functions on the set of labeled squares: b takes A to B, B to C, F to G, and so on. If we think of the symmetries as functions, we should use function notation: b(A)=B, b(B)=C, b(C)=D, b(D)=A, b(E)=F, b(F)=G, b(G)=H, b(H)=E

But then combining symmetries is composition of functions and b followed by e should be written eb: e(b(X))=h(X) for any configuration X. The second table uses this functional interpretation. Flipping one table over the main diagonal gives the other.

Using the second (function notation) table, find the following compositions.

1)(bd)e =

2)(ce)d =

3)b3 = bbb =

4)d2be2 =

5)Find the symmetry x so that bx = g

6)Find the symmetry x so that xb = g.

You should verify each of your answers experimentally. For example, fbg=c. To verify this equality experimentally, start with your square in position A. Apply transformation g to it to get orientation G. Next, apply transformation bto G to get orientation H. Finally, apply transformation fto H to get orientation C. Since the transformation fbgapplied to A gives C, it must be the transformation c.

We may think of this table as the multiplication table for a kind of abstract number system called a group. In a group, just like in our number system, we can do algebra: let x denote and unknown and then solve an equation for x. The key to “doing algebra” in our number system is the special number 1 (the identity) so that 1x= x1 = x and multiplicative inverses. To solve 2x = 4, we multiply both sides by ½ (the multiplicative inverse of 2). In the group of symmetries of the square, we have the identity a: ax = xa = x, for all symmetries, and we have inverses for all symmetries. For example, the inverse of b is d: bd=db=a. Using this approach, we may solve 5) and 6) on the previous worksheet algebraically:

Given bx = g

Multiply both sides on left by d: dbx = dg

Which simplifies to: x=f

Similarly, x = xa =xbd = gd = h.

The main difference between the multiplication of numbers and the multiplication of symmetries is that the multiplication of numbers is commutative (pq=qp) but the multiplication of symmetries is usually not commutative. For example: dg = f but gd = h.

We can still do algebra in a non-commutative group; we simply have to be careful

Worksheet #6. Find all solutions to each of the following equations involving the unknown symmetry x.

1)bxc=d,

2)bxb = g,

3)x2= c,

4)x2 = g,

5)x2= a,

6)x3 = e.

Worksheets #7 to #11: Extensions!

For each of the following geometric objects list all symmetries and then fill out the multiplication table.

7) The symmetries of the rectangle:

8) The symmetries of the pentagon:

9) The symmetries of a rectangular box. Assume that the height, width and length are all different. You should find

8 symmetries – one of them is hard to visualize.

10)Harder: Assume height = width length. There are 16 symmetries for the box has a square end.

11) Much Harder: Assume height = width = length. The cube has 48 symmetries!

Worksheet #1.

List all possible orientations of the labeled square.

There are usually two problem in making such lists: making sure that you find all possibilities and making sure that you have no duplicates in your list. We have helped you out by indicating that you should end up with exactly 8 distinct orientations.
Worksheet #2

You should have got this list of oriented squares in some order:

Using this labeling of the oriented squares, answer the following questions:

1)Starting the square oriented as in A above rotate it 270 degrees clockwise. What is your ending position?

2)What is the simplest motion (transformation) that will take the square in orientation A into orientation F?

Worksheet #3.

Your answer to question 1) should be D. So we will denote the transformation that rotates the square by 270 degrees d. Your answer to question 2) should be the reflection through the vertical line down the middle of the square. We will denote this transformation by f.

For each of the eight possibilities give a geometric description of the simplest motion the transforms A into that position:

Transformation / Geometric Description
b: takes A to B
c: takes A to C
d: takes A to D
e: takes A to E
f: takes A to F
g: takes A to G
h: takes A to H
a: takes A to A

Worksheet #4.

You should have got these descriptions:

For each pair of two symmetries, use the table below to record the result of carrying-out one symmetry after the other. Do you think the order will matter?

a / b / c / d / e / f / g / h
a
b
c
d
e
f
g
h

Are there any patterns that you notice?

Worksheet #5.

a / b / c / d / e / f / g / h
a / a / b / c / d / e / f / g / h
b / b / c / d / a / f / g / h / e
c / c / d / a / b / g / h / e / f
d / d / a / b / c / h / e / f / g
e / e / h / g / f / a / d / c / b
f / f / e / h / g / b / a / d / c
g / g / f / e / h / c / b / a / d
h / h / g / f / e / d / c / b / a

Using the function notation table above, find the following compositions.

1)(bd)e =

2)(ce)d =

3)b3 = bbb =

4)d2be2 =

5)Find the symmetry x so that bx = g

6)Find the symmetry x so that xb = g.

Verify each of your answers experimentally

Worksheet #6.

Find all solutions to each of the following equations involving the unknown symmetry x.

1)bxc=d,

2)bxb = g,

3)x2= c,

4)x2 = g,

5)x2= a,

6)x3 = e.

Worksheets #7

List all symmetries of the rectangle and then fill out the multiplication table.

Worksheets #8

List all symmetries of the pentagon and then fill out the multiplication table.

Worksheets #9

List all symmetries of the box with height, length and width all different. Then set up and fill out the multiplication table.

Worksheets #10

List all symmetries of a long box with square ends. Then set up and fill out the multiplication table.

Worksheets #11

List all symmetries of a cube. Then set up and fill out the multiplication table.