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).
a * ( b * c ) = ( a * b ) * c
Reading: Davis and Hersch: "Group theory and the Classification of Finite Simple Groups"
More sona pictures
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.
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