Lesson 2: Mandala Symmetries, Group Elements, Lusona

To investigate properties common to all sets of mandala symmetries, go up one level of abstraction, and define group elements as objects.

"These rules, the sign language and grammar of the Game, constitute a kind of highly developed secret language drawing upon several sciences and arts, but especially mathematics and music (and musicology), and capable of expressing and establishing interrelationships between the content and conclusions of nearly all scholarly disciplines. The Glass Bead Game is thus a mode of playing with the total contents and values of our culture; it plays with them as, say, in the great age of the arts a painter might have played with the colors on his palette."

-- Hesse, Magister Ludi

Chicken coop and nest
The Lusona called tshikalanga tsha tusumbi (see Figure 159) illustrates a coop for transporting chickens (Fontinha, p.273). The drawing has a 90 degree rotational symmetry. Without counting the small circle arounding the center, the pattern is 2-linear: a square and a ornamental line of double loops. The ornamental line can easily be generalized.

The pattern of Figure 159 can be transformed into a monolinear pattern, introducing a cut between the second and third point of the second row (counting from above). Figure 161 presents the result. With this transformation, the drawing lost its rotational symmetry.

Lusona are line drawings made by the Lunda-Tchokwe group of the Bantu people in eastern Angola, northeast Zambia, and parts of Zaire. Like the game Hesse describes, sona making integrates mathematics, art, and culture -- the lusona follow certain aesthetic and mathematical guidelines, and each is a pneumonic for a story or lesson important to the culture. Traditionally, lusona were passed from generation to generation during a 6 to 8 month period of male initiation rites.

Notice that the two lusona pictured are very similar. In fact, the second is obtained from the first by symmetry breaking, resulting in an asymmetical but monolineal sona. In 1935, an ethnographer named Baumann collected hundreds of lusona in an attempt to keep them from dying out with the Tchokwe culture. Of those collected, 54% were monolineal and 77% displayed some sort of symmetry, indicating that these were desirable features in sona drawing. Furthermore, of the lusona with reflectional symmetry, all had either one or two mirror lines, and of those with rotational symmetry, all had either 90 or 180 degree rotational symmetry. (source: Paulus Gerdes, Sona Geometry )

In the homework, we wrote down multiplication tables for the symmetries of different mandalas. Notice that there are certain properties common to all the tables -- they have an identity (the "do-nothing" symmetry), they have inverses (for each symmetry there is another symmetry such that when the two are composed the result is the identity), and they are associative (this can be spot-checked). In this sense, the collection of symmetries of any of the mandalas is a recognizable mathematical object (a group).

• 1. In your group no two people have the same mandalas. Spend a few minutes going over the symmetries of the four examples you have worked out for the others in your group. You may decide to make extra copies of your examples to trade for those done by others n your group.

• 2. Based on these examples and on your research done in the homework due today, make conjectures as to properties that all sets of symmetries ought to have. In other words, discuss your various responses to question 4 on the homework. Do you all agree on a list of properties?

• 3. For many systems of multiplication, such as multiplication of actual numbers, there is a property called the associative law which states that for any three elements, a, b, c, of the set in question and its operation, *, (e. g. actual numbers with actual multiplication), it is true that:

  a * ( b * c ) = ( a * b ) * c
  
  

  What does this mean in the context of the symmetry tables you have constructed? Check some triples for each table and guess whether this law ought to hold.

• 4. What happens to the number of symmetries left when you use color to break a symmetry of a black and white mandala? Do you have any conjectures as to which numbers are or are not possible?

• 5. Use what you learned today to help you with the homework.

Reading: Davis and Hersch: "Group theory and the Classification of Finite Simple Groups"

• 1. Each of the attached sheets has a pair of figures. Your job in each case is to decide when the two figures have the "same" symmetries or not. You should consider the following ideas when making your arguments:

  A. The geometric description of the symmetries themselves.
  B. The group of symmetries as described via a multiplication table that you construct.
  C. Insight provided by the readings.
  

• 2. After discussing each example, propose a general criterion for deciding whether two symmetry groups presented via their multiplication tables should really be considered the same or not. Your job is to convince the reader (your peer) that your criteria are the best possible for making this decision.

• 3. Prepare your responses to questions 1 and 2 as an essay of no more than 10 pages, including diagrams, tables, figures, etc. Please bring two copies of your essay to class on the 13th.

• 1. Has anyone ever heard of a game like the one described by Hesse? What sort of game would this be? What sort of culture would you expect to find it in?
  
  
• 2. What do you notice about the stories and diagrams of the sona? What sorts of artistic and mathematical values are displayed?
  
  
• 3. Which mandalas look like they should have the same symmetries? Why?
  
  
• 4. How would you go about convincing someone that two different mandalas have the same symmetries?
  
  
• 5. Is there a systematic way to decide if two mandalas have the same symmetries?
  
  
• 6. What happens when symmetry is broken? Consider the following mandala.

It has 3-fold rotational symmetry and 3 mirror lines. However, if we color one triangle, we are left with only one rotational symmetry -- the identity -- and one mirror line.

By symmetry-breaking, we get a new symmetry group of size two from the old one of size six. What can you say in general about what happens when symmetry is broken? What relationship does the new group have to the old one, if any?

Davis and Hersch: "The Mathematical Experience", page 202-209