Symmetry: Culture and Science
Vol. 12, Nos. 1-2, 159-166, 2001
RETHINKING SYMMETRY IN
ETHNOMATHEMATICS
Ron Eglash
Address: Dept of Science and Technology Studies, Rensselaer Polytechnic Institute, Troy NY 12180. E-mail:
eglash@rpi.edu.
Abstract: The use of crystallography classifications for symmetries, in particular of the
seven possible symmetry classes for repeating strip/frieze patterns using rigid motions
of the plane (reflection, rotation, and translation) has been a persistent element in
ethnomathematics. Popularized in Crowe’s chapter in Zaslavsky’s (1971) African
Counts, and later in Crowe and Washburn’s Symmetries of Culture, this has become a
standard activity in ethnomathematics. But it bears a curious relationship to the
fundamental concept of ethnomathematics as a discipline. Here I will briefly discuss the
nature of this relationship, and some directions that might lead to alternative
frameworks.
1. INTENTIONALITY IN ETHNOMATHEMATICS
The fundamental concept of ethnomathematics is perhaps best illustrated in a
comparison with mathematical anthropology. Mathematical anthropology uses
mathematical modeling in historic, ethnographic, and material culture studies to
describe material and cognitive patterns, typically without attributing conscious intent to
the population under study. The patterns are instead seen as the structural basis of
underlying social forces, or as epiphenomena resulting unintentionally from the nature
of the activity itself. In part this is due to a reasonable supposition that much of the
underpinnings of society would be forces unnoticed by its members (not only because
such forces operated at levels beyond individual awareness, but also because regulatory
mechanisms would have to be covert, obscured, or otherwise protected from
manipulation and conscious reflection). But it also arose from imitation of the
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researcher-object relation in the natural sciences: if anthropologists were simply
reporting indigenous discourse, then they would not count as scientists (as was indeed
the case for non-western mathematics, traditionally only a subject for historians).
Classificatory systems for kinship (e.g., Morgan 1871) were the first of these models.
Later refinements of mathematical anthropology (e.g., Kay 1971) expanded this analysis
to a variety of social phenomena, and increasingly complex mathematical tools.
Ethnomathematics, in contrast, stresses conscious intent in the opposite direction. I say
“stresses” because neither ethnomathematics nor mathematical anthropology is absolute
in that regard. Crowe is a case in point; his work has emphasized the possibility of
conscious intent or knowledge, simply by virtue of fact that he is working with
intentional designs. How might alternative approaches to symmetry further this
exploration at the intersections of external modeling and indigenous intentionality?
2. CULTURALLY SITUATED DESIGN TOOLS: AN ALGORITHMIC
APPROACH TO SYMMETRY IN ETHNOMATHEMATICS
The Virtual Bead Loom, developed in collaboration with teachers and students at the
Shoshone-Bannock reservation school in Idaho, is available online at
http://www.rpi.edu/~eglash/csdt/na/loom/overvw.htm. The web page begins by showing
the prevalence of fourfold symmetry in many Native American designs (textiles,
sandpaintings, pottery, etc.), where the “four winds” or “four directions” provide an
indigenous analog to the Cartesian coordinate system with its x and y axes.
Division into four equal parts, or “four-fold symmetry”, is common in many Native
American designs.
Shoshoni beadwork
Embroidery - Plains Indians
RETHINKING SYMMETRY IN ETHNOMATHEMATICS
Sand painting – Navajo
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Pawnee buffalo hide drum
The best case for this analogy (which from the ethnomathematics point of view is a
“translation” between two different mathematics traditions) is probably Navajo sand
painting. In the sandpainting on our website, we see dark figures on the X axis, and light
figures on the Y axis; with clear faces where the Cartesian graph has positive values and
white faces for negative values. Thus dark body clear face for X positive, light body
clear face for Y positive, and so on.
Virtual bead loom
The traditional bead loom also uses a Cartesian-like system (rows and columns of
beads), so it was a simple matter to create a virtual loom. The virtual loom enables
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students to enter x and y coordinates for bead positions of various colors to create
patterns similar to those on a real loom. Selecting one bead at a time was too tedious, so
we have been adding “design tools.” For example, you can create a filled rectangle of
beads by specifying the x,y coordinates of the corners. We found that native
beadworkers used iterative algorithms (e.g., if the first row of red beads is N0, and you
subtract one bead each time you go up one row, then your general rule is Ni+1 = Ni-1) so
our latest version of the bead loom includes design tools that use these iterative rules as
well.
Kristine Hansen, a math teacher at the school, had been interested in the symmetries in
traditional Shoshone-Bannock beadwork, and found that the Cartesian system made
teaching reflection symmetry quite easy, since the design tool allowed speedy
replication with reflection to any of four quadrants, simply by changing the sign of the
figures. Hansen reports an assignment for creating a “Christmas Tree” pattern by
replicating triangles; her students reduced the amount of translation between triangles
and turned the Christmas tree into a Shoshone feather pattern. In other words, rather
than the static modeling approach, in which external analysis implies that there is only
one correct symmetry analysis, the design tool approach allows us to move dynamically
between indigenous math and western math, creating new hybrids as well as shedding
light on indigenous math that might otherwise be overlooked.
RETHINKING SYMMETRY IN ETHNOMATHEMATICS
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Shoshone-Bannock beadwork examples
In addition to Hansen’s work at the Shoshone-Bannock school, the loom is currently
used in middle school classrooms with Mimi Thomas at a school serving students from
the Ute reservation in Northern Utah, and with Joyce Lewis at a school serving students
from the Onondaga Nation in upstate New York.
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The bead loom has columns of fine thread. As the beads fill in vertically, they are aligned in rows.
Using a simple Cartesian grid, Shoshoni beadwork provides an
astonishing array of geometric forms (Candy Titus, April 1999).
RETHINKING SYMMETRY IN ETHNOMATHEMATICS
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3. CONCLUSION
Other Culturally Situated Design Tools have been developed for examining scaling
symmetries
in
African
and
African
American
patterns
(see
http://www.rpi.edu/~eglash/csdt.html).
Scaling design from Africa is this Mangbetu ivory sculpture: Note that this sculpture not only shows scaling;
it is also shows an underlying structure making use of right angles. The Mangbetu live in the Democratic
Republic of Congo (formally Zaire). Their beautiful art makes striking use of geometric principles.
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Fractal shapes abound in traditional African designs. One African fractal design that is
also part of African American innovation in the U.S. is the scaling patterns of cornrows.
Adding scaling as a symmetry transformation opens a great deal of indigenous patterns
to mathematical analysis. Examination of aperiodic patterns (cf. Eglash 1999, Figure
10.13, From order to disorder in a Bakuba cloth) can also bring indigenous designs and
concepts into new mathematical appreciation. While the crystallography classifications
for symmetries should remain an important tool in ethnomathematics, the development
of alternative frameworks have much to offer.
REFERENCES
Crowe, D. W. (1971) The geometry of African art I., Bakuba art, J. Geometry 1, 169-182.
Eglash, R. (1999) African Fractals: modern computing and indigenous design, New Brunswick: Rutgers
University Press.
Washburn, D. and Crowe, D. (1988) Symmetries of Culture. Theory and Practice of Plane Pattern Analysis,
Seattle: University of Washington Press.
Zaslavsky, C. (1973) Africa Counts, Boston: Prindle, Weber & Schmidt Inc.