Logos

Overall Goals:

1) Students are introduced to symmetry functions.2) Students are introduced to composition of functions.3) Students see the connections between the modular arithmetic groups and the symmetry

functions on certain logos.

Activity 1: Symmetry functions

Goals:

1) Students are introduced to symmetry functions on the plane2) Students can identify the “legal moves” on an object.

Introduction:

This introduction is for the instructor and the student.

In this activity we’ll be interested in functions on the plane that map a logo design (a set of point in the plane) to the same logo in the plane (to the same set of points in the plane). This type of function is called a symmetry function. For the time being we’ll call a symmetry function a legal move. Now we must define a legal move?

It should be emphasized that we’re dealing with symmetric functions. These are of closely related to the “symmetries of an object” but they are not the same. We will abuse the language a bit and from this point forward when each time we talk about “symmetries” we’ll be referring to symmetry functions., i.e. machines with inputs and outputs.

Definitions:

MOVE: A move is just a function. We’ll represent a move, when it’s possible, by using the in-out machine model discussed in Activity i of introduction to functions.

LEGAL MOVE: A move is legal if the input object (disregarding any of the surrounding numbers) looks the same as the output , no bigger no smaller, not longer or thinner, etc.

More Formally: A legal move (symmetry function) on the a set of points(your logo) in the plane is function that takes the set of points (logo) onto itself.

Example 1

Below is an example of an illegal move.

A legal move keeps a logo looking the same but the numbers may switch position.

Suggestions for Example 1:

1) Machine Models. We always use the machine model for a legal move (symmetry function) on a logo. This is allows students to get all the information they need for further development of concepts.

2) Proper labeling. You only need as many numbers as will help you distinguish between legal moves. Beware of excessive labeling, it slows down the process. Generally n-sided objects need n or less labels.

3) Keep track of where the numbers go. It will be necessary to distinguish between moves. The numbers are our tool for doing that. It is most important to realize this early on.

4) A legal move maps a logo exactly onto itself. Often students think that an isometry is a legal move. That’s not true. A legal move is an isometry but an isometry isn’t necessarily a legal move.

Now here’s an example of a legal move:

Exercise 1

1) Take the letter V with the numbering below.2) Name two legal move you can perform on the letter V. Make a machine model diagram for

both legal moves. Please make sure to show where the numbers are on the input and output of the machine.

3) Name an illegal move on the letter V. Make a machine model diagram for this move. Please make sure to show where the numbers are on the input and output of the machine.

1 2

VSuggestions for Exercise 1

1) Machine Models. Emphasize that we want to use machine model diagrams for each legal and illegal move.

2) Numbering. Emphasize that proper numbering is a technique that comes with experience.

Too few numbers will not allow a student to differentiate between different symmetry functions. Too many numbers will slow students down.

Typical Student Problems with Exercise 1

1) Machine Models. The students should be asked to redo the exercise if they don’t use the diagrams.

2) Mistaking an isometry for a symmetry function. For example students will make an a 900 turn and call it a legal move. The teacher should use both definitions explain why an illegal move isn’t legal.

3) Students don’t give enough numbers or give too many. Too few numbers will not allow a student to differentiate between different symmetry functions. Too many numbers will slow students down. The best way to get a feel for the amount of labeling to use is by experience and with guidance from an instructor

Exercise 2

This is the first example of naming symmetry functions on works of art.

This a group exercise. The results can be presented by students on the board.

1) Thoughtfully number the peace symbol so that it will make the effects of symmetry function clear. (Notice that numbers are included with the peace symbol below. Please don’t give these numbers to the students, they are the preference of the author.)

2) What are all the legal moves you can perform on the peace symbol given below?

1 2Suggestions for Exercise 2

1) Don’t point out equivalent functions. For example don’t point that a rotation of -3600 is a rotation of 3600. This equivalence will be properly defined in the next activity.

2) Choose a notation convention. A rotation of 720 will be noted as R72. A rotation of n degrees will be noted Rn . A reflection will be notated as M

Typical Errors for Exercise 2

1) Confusing legal and illegal moves. Go through the definitions and examples to remedy this.2) Machine model. They should use the machine model to represent all symmetry functions at

this point. The instructor should check the notes of the students to see that they are using the model in their notebooks.

Exercise 3

Group exercise. Have students put their designs on the board.

1) Create an object with at least three legal moves .2) State what the legal moves on the object are. Use the machine model to represent them.

Suggestions for Exercise 3

1) Creative exercises scare students. It should be emphasized that the goal of the object is not a beautiful design but rather a design with at least three symmetry functions associated to it.

2) Machine Model. Students should represent the functions using the machine model.

Typical Problems with Exercise 3

1) Students don’t work. Students tend to take creative exercises personally. They tend to not want to show their work for reasons of shyness. Please be encouraging.

2) Number appropriately. This is an ongoing issue that needs to be corrected.

Exercise 4

1) Name the legal moves on the Mercedes Benz logo. (insert the numbers 1, 2, and 3 counter clockwise at the points on the Mecedes Benz.)

Suggestions for Exercise 4

1) The legal moves are rotations and reflections. The reflections should be notated by M1, M2, M3.

2) Lines of reflection. There are three lines of reflection. M1, M2, M3. These lines do not correspond to any numbering system on the symbol. They correspond to a reflection over a vertical line, a reflection over a line with a negative slope, and a line over a positive slope.

3) Equivalence of symmetry function. In the next activity we’ll precisely define what it means for two functions to be equivalent. For now, the instructor may point out that the number configurations of for example R-360 and R360 are the same.

Homework

1) Appropriately label the following letters with numbers. 2) Name at least 2 legal moves associated to the letters. Use the machine model to show where

your numbers go with each move.

B A M3) Create 2 illegal moves for each letter given above.4) Extend the definition of a move in 2 dimensions to a move in 3 dimensions. Name at least 3

legal moves on a cube.

Activity 2: Equivalence of legal moves

Goals:

1) Students can determine when moves are equivalent2) Students can name the legal moves up to this concept of equivalence.

This is the single most important definition for the “Logos” module. It allows us to put a group structure on symmetry functions.

Definition: Two legal moves are considered equivalent if both outputs have the same number configuration.

Example1

Using the definition of equivalence on the symmetry functions for the Mercedes Benz symbol we see Ro is equivalent to R360 . Notice also that R360 is equivalent to R720.

Exercise 1

Part 1

1) Find two legal moves on the Mercedes Benz symbol that are equivalent to R120.2) Find two legal move that are equivalent to R-120.3) Can you find a legal move equivalent to M1?

Sample Answers:

1) R120=R480 or R120=R-240

2) R-120=R240 or R-120=R600

3) Not yet, in activity 4 we will be able to find the answers.

Part 2

1 1

32 32

R0

1 1

32 32

R360

1) How can you classify rotations that are equivalent to R0?2) How can you classify rotations that are equivalent to R120?3) How can you classify rotations that are equivalent to R-120?

Sample Answer:

2) A final answer may look like R120=R(120+n360)

Part 3

1) Make a list of all the legal moves for the Mercedez Benz logo. If two moves are equivalent only include one of the moves in your list.

Answer: For Part 3 number 1, the list below represents one solution. Note that students might have chosen to list R360 instead of R0 . It will be the convention of the class to use R0 and the other representatives below.

R0 R120 R240 M1 M2 M3.

Exercise 2

1) List all the legal moves for the peace symbol . If you have two moves which are equivalent in your list only list one of them.

Answer:

1) R0 , M where M is a reflection over a vertical line.

Important Remark:

To the students and the instructors

You might notice have noticed that:

all logos have the symmetry function R0 defined on it

Even if an object has no symmetries it will have R0 as a symmetry function. This function will become important as we start to find group structures on the sets of symmetry functions on a logo.

Homework

1) List the legal moves on the Mac Donalds Archs Logo. If two moves are equivalent only include one in your list (disregard the writing and the yellow lines).

2) Extend the definition of legal moves from 2 to 3 dimensions. What are the legal moves on a donut?

Activity 3: Introduction to composition of functions.

Goals:

1) Students are introduced to the idea of composition of functions.2) Students are comfortable with the standard notation for composition of function3) Students can compose two symmetry and real valued functions .

Introduction

(This introduction is for the instructor and the student)

By this time we’ve named all of the symmetry functions associated with the peace symbol and the Mercedes Benz logo. We should feel comfortable moving the numbers around as we perform the functions.

The act of composing two functions in this case is simply inputting your logo into the one function and taking the output to input into the second functions . The value of the composed functions is the final output after going through the second function. See the Example 1.

Example 1

The teacher should note that the order the functions are used is read from right to left. Exercise 1 deals with composition of function notation.

Exercise 1 Notation practice

Write out the notation for the following composition instructions.

1) I put the Mercedes Benz in M1 then take the output and put it in R120

2) I put the Mecedes Benz symbol in R120 and then put the output into M1

3) I put the Mercedes Benz symbol in R120 and then put the output in M1 and finally take that output and input it into M3

Typical Student Problems with Exercise 1

1) The ordering is wrong. For example, the answer to 1) is M1 ◦ R120 , often students write R120 ◦ M1 instead.

Exercise 2

Since there are only two functions to work with, M and R0 , it’s easy to work out all the possible compositions of two functions. Let’s set R=a rotation of 0 degrees and M=a vertical reflection.

1) Using the machine model for composition of functions given above, make charts for all of the compositions listed below.

1 2 3

2 13 21

M3 R120

The notation for the composition of the reflection and then rotation is R120 ◦M3

a) R ◦Rb) R ◦Mc) M◦Rd) M◦M

Typical Student Problems with Exercise 2

1) Machine Model. The machine model will tell the instructor whether the student is composing in the proper order or not. Also, facility with the machine model will help in later activities when a group structure on a set of symmetry functions is discussed.

Composition of Functions Using Standard Precalculus Functions

The concept of composition of function has been introduced to students in the context of symmetry functions on Logos. The goal now is to transfer that concept to a precalculus setting.

To make this transfer most effective the instructor should use the machine model and the same notation.

Example 2

We first introduce F◦G(x) as follows by the exampleLet F(x)=2x+3 and G(x)=5x-4

1 2(1)+3 5(2(1)+3)-4 X 2(x)+3 5(2(x)+3)-4

FG

G F

Exercise 3 and beyond

The instructor should proceed to do composition of functions exercises using standard precalculus functions (SPF’s). It’s important to transfer what was learned on the symmetry functions to standard precalculus functions. For that reason, the author prefers you to use the machine model at first. When the transfer has occurred further notation can be employed.

Homework

1) There are 36 different ways to compose two symmetry function on the Mercedes Benz symbol. Split the class into 9 groups each group will be responsible for 4 compositions.

2) At this point the instructor can start to assign problems from the textbook that deal with composition of functions.

Activity 3.25, 3.5, and 3.75

The instructor can develop composition of functions on standard precalculus functions (SPF’s). The course will allow for at least 2 days of standard development. It’s important to cover domain issues.

Activity 4 Equivalence of functions and composition

Goals:

1)Students compose symmetry functions and give the name of the resulting function using our definition of equivalence.

Introduction

(this introduction is for the instructor and the student)

This activity takes off from where the previous exercise left us. Two functions that are composed make a “new” function. What is that new function equivalent to? When we compose two symmetry functions on a logo it makes a symmetry function. We’ll see that the function is equivalent to one we already know.

Exercise 1

Have groups of students put the results of the Mercedes Benz homework assignment from Activity 3 on the board. Give students a chance to inspect the work. The instructor should comment on the work and make sure it’s correct.

Common Errors with Exercise 1

1) Reflections. Students think M3 is a reflection determined by where the number 3 is on the symbol. It should be made clear that M3 is a reflection over a specific line in the plane.

2) Composition. When inputting into the second function during composition, students often don’t use the number configuration of the output from the first function. Instead students will mistakenly use the initial number configuration.

Before doing the next example students should have the following chart in their notebooks or in a handout.

Example 1

Example 1 will show two composed symmetry functions on an object gives a symmetry function when we use are definition of equivalence. In fact the resulting symmetry functions will be one of the functions from the chart given above.

Now make the following observation :

R120◦M3 is equivalent to M2.

In this case we write

R120◦M3 =M2

Common Problems with Example 1

1) The value of the composition. Students often don’t realize that the value of the new function made from the composition process is final output. To facilitate this the teacher should emphasize the input and final output as the pair that determines the new function.

Exercise 2

Split the class into the same groups as in exercise 1.

Using the symmetry functions on the Merceds Benz, groups should state what each composition they were responsible for homework is equivalent to.

1 2 3

32 31 12

M3 R120

The notation for the composition of the reflection and then rotation is R120 ◦M3

The class should review the results. The material should be compile and put into a cayley table.

Introduction to Cayley Tables

This introduction is for the students and the instructor

Cayley tables are wonderful in that they give us lots of information about a particular group of functions very quickly. The cayley table looks just like a multiplication table.

Our convention for using Cayley tables:

we input the logo into a functions on the top and then the output goes into a function on the side. We write what the composition is equivalent to in the box where the column and row intersect.

Below is the table for the Mercedes Benz Logo( It should be used in a handout)

Homework

1) Construct a cayley table for the symmetry functions on the peace symbol

Activity 4.5

Continuing Composition of functions and Cayley Tables

Exercise 1

Students should be split into 8 groups. Each group will be responsible for filling in 8 boxes of the cayley table.

1) Find the symmetry functions for the Red Cross logo given below. 2) Create a cayley table for symmetry functions.

Activity 5 (this activity can be given after the groups are introduced in the Groups module)

Commutative Functions

Goals:

1) Show the relationship between groups ( integers with addition) and groups of symmetry functions on logos.

2) Give examples of logos with and without a set of commutative functions .

Introduction

(For the instructor and the student)

We’ve known for a long time that for any two integers A and B , A+ B=B+A, i.e. that 2+3=3+2. We say that integers, with the operation of addition, form a commutative group. The set of polynomials , with addition, forms a commutative group for example (2x+3) + (x2+6)=(x2+6) + (2x+3) . Shortly we’ll investigate whether the symmetry functions on a logo, using the operation of composition, form a commutative group.

Definitions

Definition 1: Two elements A and B of a group commute if A#B=B#A where # is the operation.

Definition 2: If all pairs of elements of a group commute, we say the group is a commutative group.

Example 1:

Let’s look at the Mercedes Benz example. Look at its cayley table

Notice that R240◦R120=R120◦R240 ie they are both equal to R0. These two functions commute.

Now exam the following R120◦M3 and M3◦R120 . They’re not equal, R120◦M3=M1 and M3◦R120=M2

Exercise 1

1) Using the cayley table, find an example of two other functions on the Mercedes Benz that don’t commute.

2) Find a function that commutes with every other function. A function F commutes with every other function means that given any other function B , F◦B=B◦F.

Exercise 2

1) Generate an object with 3 rotations and 0 reflections.2) Name the symmetry functions for this object.3) Create a cayley table for this object.4) Can you find an example of two functions that don’t commute. (insert composition table).

Exercise 3

1) Examine the cayley table for the peace symbol. Can you find an example of two functions that don’t commute.

Activity 5

Connections between Groups on Logos and Modular Arithemetic Groups

Goals :

1) Students define modular arithemetic, i.e they understand mod 3 mod 4 mod5 … arithemetic.

2) Students find connections between logos and modular arithemetic.

3) Students can create maps between symmetry functions and

We’ve created the cayley tables for the peace symbol , Mercedes benz logo and objects of our design with n- rotations and 0- reflections. The operation of composition has been compared to standard operations of multiplication and addition. In the present activity we’ll explore the connection between composition of symmetry functions on logos and some very famous groups seen in group theory, the modular arithmetic groups .

Example 1

Below is logo we created with 3-rotations and 0-reflections.

(Insert image)

The symmetry functions are R0 ,R120, and R240. Rename them 0, 1, and 2 respetively

Below is the cayley table for this logo.

R0 R120 R240

R R0 R120 R240

R R120 R240 R0

R R240 R0 R120

Here’s a table where R0 ,R120, and R240 is replace by 0, 1, 2 and the composition is replaced by +.

Here’s the altered cayley table

0 1 20 0+0=0 0+1=1 0+2=21 1+0=1 1+1=2 1+2=02 2+0=2 2+1=0 2+2=1

Exercise 1

Groups of 3-4 students are assigned one of the following examples:

1) Insert image (4 rotation 0reflections)2) Insert image (5 rotation 0reflections)3) Insert image (6 rotation 0reflections)4) etc.

Part1

i) Name the symmetry functions for your logo and make the cayley table for it.ii) Rename your symmetry functions 0, 1, 2,…., n- 1 where n is the number of rotations your

object has.

iii) Create a new cayley table using the numbers and the operation of addition +.

Groups should put their cayley tables on the board for review.

Part 2

i) The following instructions should be given to the students. Each table of numbers works like normal addition in some ways, and doesn’t work like normal addition in others. Can you point out what looks “ normal” and what looks “ abnormal”. (have students fill out a copy of the chart below for each cayley table)

Normal arithmetic Abnormal Arithmetic

ii) (give the following instructions to the class) Pretend the abnormal aspects of a Group’s arithmetic are normal. Fill in the following chart for your new arithmetic.

Actual Value Value in the new arithmetic11+11+1+11+1+1+11+1+1+1+11+1+1+1+1+11+1+1+1+1+1+1

iii) (Each group should put their table on the board.) Make a clear statement of how you found the one can find the value in the new arithmetic

iv) (to each group) What are the following numbers equivalent to in your funny arithmetic12, 17,19.

The instructor should now name each modular arithmetic group ,Z4 , Z5 , Z6 ,etc.

Exercise 2

1) What numbers are in the set Zn.

2) What is are the numbers 12, 13, 59,63, and 80 equivalent to in Zn

Exercise 3

i) After seeing the results of the past exercises can you say describe the group you would associate to a logo with 10 rotations and 0 reflections?

ii) Can you make a statement about the connection between logos with n-rotations and 0-reflection and our the modular arithmetic groups.

Homework

Answer the following questions

1) What parts of the cayley tables for the Mercedes Benz symbol correspond to an arithemetic we’ve seen today?