Lesson 5:   Islam

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

We are heading into the last homework assignment. This will be about Islamic art, which we will see lots of slides about later. To get you 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.

GROUPS ACTING ON SETS

1 Your color triangles from the last art assignment are example of objects whose spacial symmetries have been broken by the paint. Nonetheless they were painted according to a pattern, or formula, given in the assignment. Is there some sense in which they display a symmetry?

3. More examples might include the geometry which comes from the equations. What is the relationship of these "symmetries" to the spacial symmetries of a regular equilateral triangle? These are both examples of "a group acting on a set." Recall, those equations look like:

 
a = O.   b = O.   c = O.
 
(a+b) + (a+c) = O,  (a+b) + (b+c) = O,  (a+c) + (b+c) = 0,
 
a + b = O,  b + c = O,  a + c = O.
 
Use vectomatic. (With vectomatic, you are really looking at a + b = 0 and c = 0.)

You can think of these equations as formal objects or sets of planes and lines in 3space. Notice that for this to make sense, you have to be willing to give those equations an objective reality, just like a line or a plane.

4. Another example of the same group acting on a set is given by the permutations of three letters, a, b, c. Can you think of others?

5. GROUPWORK 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 reasonable definition.

FUNDAMENTAL REGIONS

1. Here are some slides of repeat patterns.

How is this displaying a group acting on a set? When you have a group acting on a set you can think of the whole set or design as made up of actions by the group on one little piece. This is how the computer works when it uses Terrazzo.

2. Can you find a fundamental region for each of these patterns?

3. What is the fundamental region for the symmetry group of your shibori? When you do a block print, is the block always the fundamental region?

GROUPS AND SUBGROUPS

1. Remember what you learned by coloring mandalas. What is the relationship of a group to a subgroup? The next assignment will require you to know which of the wallpaper groups are subgroups of which others.

2. Groupwork. Divide into groups of 3. Each group has one of the wallpaper groups from the last math homework assignment, either P6mm or P4mm. Find all the subgroups of it that you can. Do this by coloring a pattern with that symmetry. 20 minutes. Use the Dover arabic allover pattern coloring book. You will use this in the homework assignment.

3. Put your results on the board and note the collection of possibilities for reference for this week's homework.

Optional: Proof that the order of a subgroup divides the order of a group.

This assignment is a mathematics assignment and an art assignment and contains a mysterious third component: Netscape!

l. a. Find a friendly nerd who can help you install netscape on your computer. Use the "bookmark" function to make it easy to flip to the Dartmouth Home Page. Everything that is underlined is an interactive function.

b. Click on "Academic Departments, then on "Mathematics Department". After the Math Dept logo appears, click on "Surfing the World Wide Web". When that page comes up, click on "The Geometry Center" which will take you to the University of Minnesota Geometry Center. When their page comes up, click on "Interactive Applications" or something like that. A bunch of logos should appear at the top of the screen after a minute or so. One of them is labelled "Kali".

c. Click on Kali. An explanatory page will come up. Read it, then click on the sentence, "Click here to start." A picture will come up of the simplest drop repeat tesselation. Investigate Kali by reading through these pages. If you desire, you can download some of the software onto your machine, otherwise run it remotely.

2. a. Use the coordinates to change the picture in the box which was used to create the tesselation.

b. Change the labelling of symmetry groups to "Crystallographic". You will then recognize the labels from the previous homework assignment.

2.a Using the groupwork we did in class, pick a string of three groups from those offered by Kali, each of which is a subgroup of the previous group. Extra credit if at least one of these groups has a subgroup of order 3 or 6!!! So we will say these groups are G. H. J. J is a subgroup of H and H is a subgroup of G.

b. Write paragraph or two justifying your choice of groups. How do you know on is a subgroup of the other?

3. a. You are going to use G. H. and J as the symmetry groups for your block print. Make a sketch of a block (or series of blocks.)

b. Use the little box in Kali that contains.just one design unit to draw an approximation to your block. The coordinate function is somewhat limited, you only get five line segments to approximate your design. Coordinates range from 0 to 1, with decimal points. It took me an hour to figure this out. After a few tries with the group pl, you should be able to figure out which coordinates are a best approximation.

c. Use the groups you chose in part 2a, G. H and J. to tesselate the plane with your fundamental region. Use Kali's print function to print out the resulting pictures. Some people have trouble with the print function. If you get desparate, there is a way to save a picture of your computer screen (a screen dump) and print that. Your friendly nerd can help with this too. (Instructor's note: We are not nerds. As far as we can tell, computers smell fear just like dogs. You need your own nerd! But trust us, you are surrounded by them.)

4. a. Line groups.
See sample a. Look at the attached handout and classification of the so-called "line groups" or "frieze patterns". These are the basis for border designs and you will choose one of them as a basis for a border print for one of your block designs.

Line Groups for Borders (also shown as frieze groups)

Mathematically, it would be interesting to know the relationship between the line group you choose and the three subgroups you have chosen.

5. a. Creating the design.
Use G. the most complex group, and use one or more blocks, to make art out of the design Kali has generated. Enlarge this design by hand using a pencil compass, protractor, rulers etc. and graph paper. Please refer to the attached diagrams for ideas and help. Create two repeats. This drawing will be turned in as part of your homework. Use it to help design your block. Your block will be 3 to 4 inches across and down, so make your drawing with that in mind. You will be using three full sheets of Rives printing paper, one for each proof. These are 19 by 25 inches. We will be looking for an interesting assortment of polygons in your overall basic design.

b. Your first print will be based on the group G and your sketches. Using the techniques you tried on Tuesday, aim for a richly colored pattern with at least 3 or 4 colors in it. But do not break the symmetry of the group G! To help with accuracy, make use of your drawing from part 4a and make use of the Kali printout and Xerox enlargement.

c. For your second print you will start with a new print of the design with symmetry group G. Study the symmetry group H using the Kali picture. You are going to alter this new print so that it has the same symmetry group H. but it will not necessarily look like the computer printout! Using another block or another color or any other technique that you have learned, break the symmetry of your G block print so as to create a print with symmetry H. In this exercise the object is to move to a simpler type of geometry. Visually there may be fewer shapes. See sample c. The color can break the symmetry of G but must support the symmetry of H. We want a minimum of three colors.

d. For the third print you will create the symmetry group J by using layering, coloring and introducing motifs into the spaces of the design. Again, J is a subgroup of G and so your new print will be derived visually from the original pattern by these techniques. For this exercise, the J symmetry can be an implied symmetry. That is, the eye and brain should be able to pick it out of the design even if you have gotten so creative that there is no strict symmetry left. Think of an oriental carpet and look at the attached examples. Motifs may be introduced by painting as well as stamping.

e. Look at your field pattern and develop an appropriate border using the group you chose in part 4a. Do this for at least one of your three prints. The border may completely encircle the design or not, as is appropriate. Printing can be done with stamps, sponges or small cut blocks. Coordinate the colors and think in terms of reversing figure ground and repeating some of the rhythms and shapes in the field pattern.

6. Assume that your series of prints will go on display in a public location where the mathematical significance of them will not be understood immediately by the viewer. Write a few excellent paragraphs explaining the mathematical principles illustrated by your art. Write for an audience of people like yourself, but who haven't taken our course. You may assume that Kali's printouts will be displayed along with your art and you may refer to them or even explain them in your paper. This is a hard assignment because it is a short one. Your paper must fit on one page.

7. Write a few excellent paragraphs explaining the printing techniques you used. This must also fit on one page. Do not assume any knowledge of printing from your audience.

8. You will need to turn in: Your prints, your drawn enlargements, your two stupendous short papers and your Kali printouts.

9. DATES Printouts from Kali, Xerox of your drawn sketches, print of one of your blocks due on Thursday, Aug 1st. Final version with all papers and so on included, due Aug 5th.