Indigenous mathematics in Australia and New Zealand
Indigenous mathematics in Australia and New Zealand reflects the rich cultural heritage and sophisticated mathematical concepts developed by Aboriginal and Maori peoples. In Australia, the kinship systems of Aboriginal societies can be mathematically modeled using group theory, illustrating complex relationships that govern social structures and obligations. The Warlpiri community's kinship system, for instance, demonstrates cyclic relationships with an algebraic structure that resonates with abstract mathematical concepts, despite not being formally recognized in their culture.
In New Zealand, Maori mathematics is prominently expressed through geometric art found in traditional practices such as wood carving, tattooing, and weaving. These art forms utilize symmetry and geometric principles, serving both aesthetic and cultural functions. Maori carvings and tattoos often embody tribal history and individual identity, reflecting a worldview that balances unity and separation. The study of these indigenous mathematical expressions not only enhances our understanding of their cultural significance but also contributes to broader fields like anthropology, art, and mathematics education. This intersection of culture and mathematics reveals the depth of indigenous knowledge systems and their relevance in contemporary discussions on mathematics and identity.
Indigenous mathematics in Australia and New Zealand
Summary: The indigenous cultures of Oceania are mathematically interesting.
The United Nations classification for Oceania includes Australia and New Zealand as well as the hundreds of Pacific Islands groups under the headings Melanesia, Micronesia, and Polynesia. The Australian Mathematical Society was founded in 1956 and promotes mathematics and its applications. The New Zealand Mathematical Society was found in 1974 and promotes research and the dissemination of mathematics. Mathematicians born in Australia and New Zealand include Field’s Medal winners Terence Tao (2006) from Australia and Vaughan Jones (1990) from New Zealand. High school students participate in the International Mathematical Olympiad. Australia began its participation in 1981 and hosted the contest in 1988, while New Zealand first participated in 1988. Mathematics historians and ethnomathematicians have researched the mathematics of the indigenous inhabitants of Australia and New Zealand. For example, the structures of Australian Aboriginal kinship systems can be modeled by the algebraic theory of groups, while the wood carving and tattooing done by the Maori of New Zealand embody geometrical principles of symmetry. These cultural achievements interest mathematicians and teachers of mathematics and also have influenced the humanities, the social sciences, and popular culture.
Australia
Studying Aboriginal kinship systems has greatly influenced anthropology and can be mathematically modeled. To give just one important example, Claude Lévi-Strauss, in support of his ideas on structural anthropology, cited what he called “the Australian facts” to help argue that a system of exchange (as illustrated by marriage partners reciprocally chosen from paired sections) underlies the origin of marriage rules.
Many Aboriginal societies are divided into two halves, with four sections in each half, for the purpose of determining kinship. The example best known to mathematicians, because of the classic work of Marcia Ascher, is that of the Warlpiri of Australia’s Northern Territory. A schematic diagram is shown below.
The equal sign designates allowed marriages. That is, for members of a section in either half, there is one section in the other half from which marriage partners come. For instance, women in section A marry men from section W, and men from section A marry women from section W. Children’s sections are determined by their mother’s; the directed arrows show how. For instance, if a mother is in section A, her children are in section C; mothers in C have children in B; mothers in B have children in D; and mothers in D have children in A—completing a cycle. Similarly, mothers in W have children in Z, and so on. Thus, the matrilineal cycle has a length of 4. For fathers, if a man is in A, for instance, following the arrow backward shows that his mother is in D, so his father is in Z. Then his father’s mother is in W, so his father’s father is in A again. Thus the complete patrilineal cycle has a length of 2.
If one writes I for one’s own section, m for one’s mother’s section, m2 for one’s mother’s mother’s section, f for one’s father’s section, and so on, the cyclic relationships can be expressed by m4=I and f2=I.. Other algebraic relationships, like (mf)(mf)=I, can be verified from the diagram. The resulting algebraic structure is that of the dihedral group of order 8. The Warlpiri, of course, do not have the concept of group, but those learning the system are asked to solve word problems like, “If someone’s mother is in a particular section, then in what section is such-and-such a relative?” The Warlpiri abstract from the personal relationships to conceptualize the system itself. General terms of address reflect the individual’s place in the structure. Kin relationships determine a person’s behavior, obligations, place to live, and relationships to plants, animals, and landscape; they also link past, present, and future generations.
The Aboriginal view of the origin of their kinship system in the journeys of their ancestors during the ancestral past (known as the “dreamtime”) is reflected in Aboriginal paintings. Such paintings are noted for their symmetry, and particular geometric elements indicate individual places, ancestral beings, or clans. The current interest in Aboriginal art has brought these geometric forms to a worldwide audience.
New Zealand
Geometric art pervades Maori culture in dance, song, music, weaving, painting, latticework, carving, and tattooing. Wood carving is the most prominent, though facial and body tattoos also continue to be symbols of Maori identity. Traditional Maori carving uses a small number of design forms and motifs, combined according to well-established rules. Rafters and ridgepoles of the Maori meetinghouse are decorated with carvings that embody tribal history. These carvings employ all seven of the symmetry groups that characterize strip patterns. They are often colored in ways that complement, rather than echo, the symmetries. Maori art also uses bilateral symmetry, but the symmetry is often broken by the nonsymmetrical use of colors or by the addition of small figures that vary. Maori tattoos use many of the same themes and motifs as does carving. Also, individuals’ tattoos serve to identify family, tribe, community, birthplace, and inherited or achieved authority.
Maori symmetric forms are united by their near identity while differing in their asymmetries. This aspect reflects the way the Maori characterize reality by pairs of things existing in a tension between union and separation. Understanding the formal geometric patterns thus gives insight into Maori culture. Maori geometric art has become part of global culture. For example, Maori carved wooden bowls appear in Paul Gaugin’s paintings. In Herman Melville’s Moby Dick, the tattooed harpooner Queequeg possesses—and sells—Maori tattooed ancestral heads. Enlightenment philosopher Immanuel Kant felt that he had to discuss Maori tattoos in examining the nature of beauty, though he concluded that Maori tattoo designs could be beautiful only if they were not on a human face. Additionally, Maori tattooing plays a key role in the acclaimed 1994 film Once Were Warriors.
Bibliography
Blakers, A. L. “The Australian Mathematical Society: Foundation and Early Years. I: Events Leading Up to the Foundation of the Society.” Australian Mathematical Society Gazette 3, no. 2 (1976).
Greer, Brian, et al. Culturally Responsive Mathematics Education. New York: Routledge, 2009.
Kaeppler, Adrienne Lois. The Pacific Arts of Polynesia and Micronesia. New York: Oxford University Press, 2008.
Munn, Nancy D. Walbiri Iconography. Ithaca, NY: Cornell University Press, 1973.
Starzecka, D. C., ed. Maori Art and Culture. London: British Museum, 1996.
Tee, G. J. “The First 25 Years of the New Zealand Mathematical Society.” New Zealand Mathematical Society Newsletter 76 (1999).
Washburn, Dorothy, and Donald Crowe. Symmetries of Culture: Theory and Practice of Plane Pattern Analysis. Seattle: University of Washington Press, 1998.