Symmetries of Culture

Mathematicians have teamed up with archaeologists and anthropologists to investigate patterns and designs from around the world. By dividing the patterns into groups, these scientists can study the patterns unique to a culture and gain insight into a group's cultural identity and whether nearby groups influenced each other's work. They have found that people from very early times used highly sophisticated symmetry. Clearly mathematics did not begin in Greece in 500 BC. The Egyptians used patterns with complicated symmetries a thousand years earlier, and people all over the world could recognize the symmetries their own culture accepted, and those they did not.

The study of crystals led to most of the mathematical information we have on the symmetry of repeated patterns. In 1891, E.S. Federov completed a list of the 230 three-dimensional repeated patterns. In 1944, Edith Muller first used the 17 classes of two-dimensional repeated patterns in an analysis of material culture when she studied the Islamic art of the Alhambra in Spain. Through her pioneering work in 1944, she identified 11 of the 17 classes, and it was not until 1987 that mathematicians were able to document all 17 in the incredibly beautiful artistry achieved by the builders of the Alhambra. In 1948, Ann O. Shepard used symmetry analysis in the study of designs from the American Southwest, the Anasazi, Mimbres, and Rio Grande Pueblos. This important work is only now being fully appreciated.

Mathematicians, cognitive scientists, and anthropologists are working together to unravel the mysteries of the patterns. This study of symmetries from around the world may be able to provide us with a deeper understanding and appreciation of our human heritage.

- Kay Gilliland, EQUALS

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Mathematics found in Seminole Patchwork

There are four rigid motions of the plane, a rigid motion is a motion that does not alter the size or shape of the object being moved. The four rigid motions are reflection, rotation, translation, and glide reflection. These four motions, or symmetries, are used to describe the geometry of repetition found in repeated decorative patterns.

Seminole patchwork designs are strip patterns, also called border or freize patterns. The repeated design of strip patterns only occurs in one linear direction; this distinguishes them from other repeated patterns where the recurring pattern appears in more than one linear direction or may just rotate or reflect.

There are only 7 combinations of symmetries that can exist in strip patterns. Every strip pattern has a translation of the design since it occurs in a linear direction. Reflections in strip patterns can exist in two ways, in the direction of the pattern, parallel to the pattern, or perpendicular to the direction of the pattern. Rotations are limited to 180° (half-turn) since the pattern must retain its original orientation. Glide reflections can only occur in the direction of the pattern. The seven combinations of symmetries, pattern types, are listed below with an example of each.

Translation only
Perpendicular Reflection
Parallel Reflection
Parallel & Perpendicular Reflections
Glide Reflection
Half Turn Rotation
Perpendicular Reflection & Half Turn Rotation

Strip patterns that have the same rigid motions, the same combinations of symmetries, are classified as one pattern type. Two designs can look vastly different but be of the same pattern type. A classification system was established to distinguish between pattern types without having to write down all the symmetries.

This classification system was first introduced by crystallographers. The notation has a symbol in each of four positions, each position describing characteristics of the pattern. The symbol in the first position describes the basic unit found in the pattern. Strip patterns have several basic units that might occur, but all are considered "primitive", so the first symbol in all strip pattern classifications is a p.

The second position indicates whether the pattern has a reflection perpendicular to the direction of the pattern. If there is a perpendicular reflection the symbol m, for mirror, is used; if not, a 1 is used. The third position in the notation denotes whether there is a reflection parallel to the direction of the pattern or a glide reflection. These two symmetries will not occur together so the third position is either an m or a g, for glide reflection. The fourth position indicates whether there is a half-turn rotation; 2 if there is, 1 if not.

The classification of each of the seven strip patterns, including a list of symmetries that exist, is given in the table below. A "distinct" symmetry is one that occurs in a specific location of the design. Strip patterns can have one symmetry occuring in more than one location of the design.

Pattern	Classification	Translation	Perpendicular Reflection	Parallel Reflection	Glide Reflection	Half-Turn Rotation
Translation only	p111	Yes
Perpendicular Reflection	pm11	Yes	Yes, 2 distinct lines of reflection
Parallel Reflection	p1m1	Yes		Yes, 1 distinct line of reflection
Parallel & Perpendicular Reflections	pmm2	Yes	Yes, 2 distinct lines of reflection	Yes, 1 distinct line of reflection		Yes, 2 distinct centers of rotation (where the reflection lines cross)
Glide Reflection	p1g1	Yes			Yes
Half-Turn Rotation	p112	Yes				Yes, 2 distinct centers of rotation
Perpendicular Reflection & Half-Turn Rotation	pmg2	Yes	Yes, 1 distinct line of reflection		Yes	Yes, 1 distinct center of rotation

It is an interesting exercise to verify why certain combinations of symmetries exist together; for example, if both parallel and perpendicular reflections occur, rotation must also occur.

Looking for symmetries in design is an engaging activity that sometimes yields surprising results!

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The Tradition of Storytelling

Many cultures throughout the world maintain their history through oral traditions of storytelling. In some cultures, certain members of the society have the responsibility to maintain the oral history.

Storytelling is an important aspect of community gatherings for many American Indian tribes. Everyone sits in a circle listening to the stories which are told by a variety of people. Many of the stories explain natural events or the presence of living beings. For example, there are stories to explain why leaves change colors in the fall and the existence of the woodpecker. Other stories are lessons; stories which teach how to live within the culture. These stories reflect the society's values, explaining that members need to help each other and get along.

The Seminole tribe has a Creek ancestry. The Creek Nation was composed of many tribes. The Creek Nation, in turn, has elements of its culture that can be traced to the Mayan Indians of South America. This ancestral tie is evidenced both in the traditional stories of the Creek Nation and the similarity of designs found in both cultures.

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The Origins of Seminole Clans

In the beginning, the Muscogee people were born out of the earth itself. They crawled up out of the ground through a hole like ants. In these days, they lived in a far western land beside the tall mountains that reached the sky. They called the mountains the backbone of the earth.

Then a thick fog descended upon the earth, sent by the Master of Breath, Esakitaummesee. The Muscogee people could not see. They wandered around blindly, calling out to one another in fear. They drifted apart and became lost. The whole people were separated into small groups, and these groups stayed close to one another in fear of being entirely alone.

Finally, the Master had mercy on them. From the eastern edge of the world, where the sun rises, he began to blow away the fog. He blew and blew until the fog was completely gone. The people were joyful and sang a hymn of thanksgiving to the Master of Breath. And in each of the groups, the people turned to one another and swore eternal brotherhood. They said that from then on these groups would be like large families. The members of each group would be as close to each other as brother and sister, father and son.

The group that was farthest east and first to see the sum, praised the wind that had blown the fog away. They called themselves the Wind Family, or Wind Clan. As the fog moved away from the other groups, they, too, gave themselves names. Each group chose the name of the first animal it saw. So they became the Bear, Deer, Alligator, Raccoon, and Bird Clans. But the Wind Clan remained the most important clan of all.

The Seminole Clans are Alligator, Bear, Beaver, Bird, Deer, Otter, Tiger, Raccoon, Snake, Sweet Potato, Wolf, and Wind. Clans are named for animals and manifestations of nature. Descendancy and inheritance comes through the mother's side of the family.

- Cultural Curriculum for Communities

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Bibliography

Crowe, Donald. 1986. Symmetry, Rigid Motions, and Patterns. COMAP: Arlington, MA.

Cultural Curriculum for Communities.

Gilliland, Kay. 1994. Rafter Patterns of New Zealand Maori: An Interdisciplinary Approach to Symmetry. A presentation at the Southern Regional Conference of the National Council of Teachers of Mathematics, Tulsa, Oklahoma, 13-15 October 1994.

Grünbaum, Branko and G.C. Shepard. 1989. Tilings and Patterns: An Introduction. Freeman: New York.

Schattschneider, Doris. 1978. The Plane Symmetry Groups: Their Recognition and Notation. American Mathematical Monthly, 85(6) 439-450.

Tannenbaum, Peter and Robert Arnold. 1992. Excursions in Modern Mathematics. Prentice Hall: Englewood Cliffs, NJ.

©1998 Vera Preston & Mary Hannigan
Austin Community College