Lesson 1:
Ritual Geometry, the Mandala, and its Symmetries

Goals:

To understand that symmetry is a human construction laid upon nature, to distinguish between the symmetric object and the symmetry itself, to understand that symmetries can be labeled, listed, and composed with each other, to construct examples of multiplication tables arising from this activity, and to see some cultural context for the math.

In India around 400 B.C., altars in shapes such as this were used for religious purposes. The belief was that, in times of plague, the altar needed to be doubled in size to placate the gods. Consequently, priests were expected to be able to figure out how to double the area of a relatively complex geometric figure. (source: Pi in the Sky) Given that the much simpler problem of doubling the area of a square presupposes knowledge of the Pythagorean theorem (written down by Pythagoras around 550 B.C.E. but in a vastly different place), we can see that priests were also expected to be competent geometers.

Another example of math and religion interacting is the mandala -- a design used for meditation. This one is from at least 600 C.E., but there are examples dating to the 12th century B.C.E.

Here symmetry is the crucial mathematical element. The symmetries increase as we move from the center, which has only bilateral symmetry, toward the outer circle, which has infinite symmetry. Correspondingly, the center of the mandala represents order and the realm of the gods, while the outside represents the chaos of the material world. We don't know if the mandalas were created using overt, systematic geometric knowledge or if this is an example of "geometric intuition" on the part of the artist, preceding the formalization of the math.

The symmetries of an object such as a mandala can be thought of as objects in themselves. Consider an equilateral triangle and notice that it has six symmetries. If we label them (for example as rotations and flips of different degrees), we can compose them and make a multiplication table, giving them a mathematical structure of their own.

• 1. Can you see how to double the area of the altar? How about of a simple square?

• 2. Why do we count the "do-nothing" symmetry?

• 3. What other kinds of composition of different things are there?

Dream-Catcher Mandalas
Mathematical Art by Louis A. Talman

Mandalas pictures from Dover book.

Each group has two mandalas, including a transparency of each. For each of these do the following exercise, write it up and hand it in with your names on it at the end of class. Include a copy of the mandalas and the transparency. Keep a copy of your work for yourself.

• 1. Count the symmetries of the mandala.

• 2. Label each symmetry in a way that helps you remember what it does.

• 3. Fill in a multiplication table for these symmetries. Check it carefully!

• 4. Do the same for the second mandala.

• 5. Make copies of these to take home for yourselves. You will need them for the homework assignment.

• 6. If you have time, finish the triangle exercise stared in lecture.

  Turn in the assignment, pick up the handout for the homework assignment and you are free to go.

This homework is due to at the start of class on July 1, in rough form. You cannot participate in the class activities without it. The final version will be due at the start of class on July 8!

• (1) Label symmetries and construct multiplication tables for the two mandalas you drew, just as you did for the Japanese optical designs in class.

  You now have five examples of symmetry sets for five different figures (two optical designs from class, the equilateral triangle from class, your two mandalas). Based on an investigation of the tables for each of them, discuss the following points:

• (2) For which of your mandalas is composition of symmetries a commutative operation? That is, if you compose two symmetries in reverse order, when do you get the same thing? How is commutativity reflected in the structure of the multiplication table?

• (3) Can you tell from the picture itself when the symmetries will commute? How? It might be good to look at more examples, such as those done by others in class, to help you figure this one out.

• (4) What properties do you think all multiplication tables of this sort ought to have? Here you are making conjectures as to the nature of these symmetries, and you should try to justify your conjectures if possible.

• (5) Along with the exercises above please hand in a Xerox copy of the five mandalas you are looking at along with their multiplication tables. Keep originals for your future use.