Lesson 3:
Groups and Groups Acting on Sets

To understand groups acting on sets, fundamental region and subgroups.

• 1. We are heading into the last homework assignment. This will be about Islamic art, which we will see lots of slides about on Monday. To get ready for this, it helps to know more about subgroups of a group, and also about something called the fundamental region of a group acting on a set.
  
  
• 2. Your color triangles. Is there a sense in which they display a symmetry? Class discussion. Cut up mine and place on template. How many ways? What is happening? Pure color permuted. How about the equations that give the color triangles?
  
  
• 3. More examples, such as geometry which comes from the equations. What is the relationship of these "symmetries" to the special symmetries of a regular equilateral triangle? These are both examples of "a group acting on a set."
  
  
• 4. Here are some more examples:
  • a. permutations of three letters, a, b, c. Use three volunteers.
    
  • b. Set of planes in 3-space given by:

    a = 0,  b = 0,  c = 0,  
    (a + b) + (a + c) = 0, 
    (a + b) + (b + c) = 0, 
    (a + c) + (b + c) = 0, 
    a + b = 0,  b + c = 0,  a + c = 0.
    

    Use vectomatic. (With vectomatic, you are really looking at a + b = 0 and c = 0.)
    
  • c. Set of equations above. Notice that for this to make sense, you have to be willing to give those equations as objective reality, just like a line or a plane. Can you think of others? yes, permutations of triangle, resorting of color triangles.
  
  
• 5. Frame a formal definition of a group acting on a set. (That starts by assuming the formal definition of group given in the reading.) Groups of 2 or 3, 10 minutes. Put ideas on board, settle on resonable definition.

Consider the following multiplication tables.

*  a  b  c  d
a  a  a  a  a
b  a  b  c  d
c  a  c  a  c
d  a  d  c  b

*  a  b  c  d
a  a  b  c  d
b  b  a  d  c
c  c  d  b  a
d  d  c  a  b

We want to know what kinds of tables can come from mandalas. Try to imagine what a mandala with the first multiplication table would look like: any motion composed with the motion a would have to result in a again. But there is only one motion which does this -- the "do-nothing" -- and the multiplication table claims there are three distinct such motions. Clearly there is no mandala with symmetries like that, so the first table doesn't come from a mandala.

The second table, on the other hand, probably looks familiar. Can you draw a picture of a mandala with this table?

Recall the definition of group and notice that the set with the first table is not a group (why?) while the second is.

So far we've only looked at patterns with finite sets of symmetries. However, we can take any one of those patterns and repeat it throughout the infinite plane. Depending on how we do this, we may maintain some or all of the original symmetries and we may create some new ones.

Notice that whenever a pattern is repeated, a new kind of symmetry is created -- translational symmetry. When we repeat a pattern infinitely, we get infinite translational symmetry. Hence we now have an infinite set of symmetries. In fact, this new infinite set is a group (how can we see this?). Recall that, in the mandala case, we could break or ignore some of the symmetries and be left with a smaller set, a subset of the complete set. Such a smaller set is called a subgroup of the original group. In the case of this new infinite group, we have infinite subgroups (find some).

Another new kind of symmetry, one that is a little less obvious, is displayed in this pattern:

From J. Bourgoin: "ARABIC ALLOVER PATTERNS"
Dover Publications, Inc., 1985

In this case, the symmetry is created by a combination of translation and mirror flip. Such a symmetry is called a glide reflection. Neither a mirror flip nor a translation alone will allow the red design to lie down on the blue design -- we need to do both.

We define repeat patterns in the plane to be patterns with infinite translational symmetry in two directions. Such patterns can be used to make strong artistic statements; for example, they were used in Islamic art to suggest the infinite.

• 1. What prevents a multiplication table from coming from a mandala? How can you tell if a given table does or not?
  
  
• 2. Consider the following two mandalas:
  

  Do they have the same symmetry? Why or why not? Their multiplication tables are the same, yet they are different geometric objects. By removing the symmetries from the mandalas, we go up a level of abstraction and say the groups look the same.
  

Find the symmatries of each of the following patterns.

From Istvan Hargittai and Magdolna Hargittai: "SYMMETRY A Unifying Concept"
Shelter publications, Inc. p.187

• 1. How would you describe the symmetries? Is there a good geometric description? a good verbal description? a good algebraic description?

• 2. Do the types of symmetry you see fall into natural classes geometrically? If so, how would you describe the classes?

Attached to this sheet are three descriptions of patterns with different symmetry groups taken from three different sources. Feel free to work together with one or two friends on this assignment. If two people work together it is not necessary to hand in more than one copy. (With all your names on it.) Please also hand in your Group Mtork from the July 11th class as part of this assignment also.

• 1. Correlate the patterns from the various sources. Which pattern from the second source exhibits the same symmetries as the first pattern from the first source? And so forth. Give me a correlated list.

• 2. Do they match up perfectly? Are there any patterns from one of the sources that don't match up with any from the other two? What do you conjecture based on this? Give me a well written paragraph, please.

• 3. Can you find or make a pattern that is not represented by any of the ones in the handout? For example, can you make a repeat pattern with 5 fold symmetry? If you can't construct a new one, write a paragraph or two describing your efforts to do so and why or how they failed.

• 4. One of the lists of patterns offers standardized notation for the symmetry groups. Can you decode it? What might pmg mean? Justify your answer.