BOOK REVIEW
Anthropology for Mathematicians
Brian Hayes
Symmetry Comes of Age: The Role of Pattern in Culture.
Edited by Dorothy K. Washburn and Donald W. Crowe. xxx + 354 pp.
University of Washington Press, 2004. $60.
Embedded Symmetries, Natural and Cultural. Edited by
Dorothy K. Washburn. ix + 189 pp. University of New Mexico Press,
2004. $69.95.
On a visit to the Alhambra some years ago, I toted along a copy of
Symmetry in Science and Art, a weighty text by A. V.
Shubnikov and V. A. Koptsik, as a field guide to the carvings and
tilings that decorate that extravagant palace overlooking Granada.
The two books under review here would probably serve as better field
guides—Symmetry Comes of Age even includes a
useful flowchart for classifying the symmetry groups of
patterns—but I suspect that the authors and editors would not
entirely approve of this use of their work. The tourist who stalks
the halls of the Alhambra trying to complete a checklist of the 17
two-dimensional symmetry groups is not their ideal student of
"the role of pattern in culture." When one is looking at
an artifact such as a tiled floor or a woven fabric or a beadwork
ornament, identifying crystallographic groups is at best the
beginning of understanding the object. The classification might tell
you something about the meaning of the work in the context of
Western mathematics, but it is unlikely to reveal much about the
object's meaning within the culture that created it.
This point is made emphatically by Branko Grünbaum—a
mathematician who certainly knows his symmetry groups—in a
previously published article on ancient Peruvian textiles that is
reprinted in Symmetry Comes of Age. Grünbaum argues
that group theory offers little help in understanding the Peruvian
patterns because
The concepts of that theory are entirely out of tune with
the modes of thinking of the people whose products are being
investigated; moreover, these concepts were totally absent even from
the thinking of mathematicians during most of the history of
mathematics.
He goes on to rail against "the stifling dictates of the
'symmetry is group theory' cult," writing that
It is aggravating to see sophisticated examples of
patterns, made by cultures for which the patterns held great
importance, considered as inferior or "mistaken" just
because they do not fit some mathematicians' preconceived notions of
"symmetry."
The idea of outsider critics pointing out the "mistakes"
of native pattern makers is turned upside down in another essay in
this volume, by Peter G. Roe, an anthropologist. Roe writes on the
geometric designs of the Shipibo people of the upper Amazon basin.
In the 1980s he presented a selection of Shipibo motifs to a class
of art students at the University of Delaware and, with the
assistance of an early computer-graphics system, had the students
generate new patterns in what they took to be the same style. The
computer made it easy to apply various symmetry transformations,
such as reflections and rotations, but it could not enforce other
kinds of design rules. Roe took printouts of the students' work back
to South America, where Shipibo women and men were asked to evaluate
them. The result of this ingenious experiment was not really a
surprise, but it is nonetheless instructive: Patterns that look
essentially alike to the untrained (or unacculturated) eye can evoke
very different responses from insiders who understand the art form
at a deeper level. A pattern might well have the right symmetries
but still fail to win the approval of the Shipibo judges. In
particular, the experiment called attention to "a nonverbalized
rule in Shipibo art" requiring a certain kind of connectivity
in patterns, allowing "one's finger or eye to trace a
continuous line as it meanders through the lattice."
Roe reports that the requirement of connectivity has a clear meaning
in the lives of the people who decorate their homes and belongings
with these patterns. "The continuity of the formline ... does
not let contagion intrude," he writes; "designs are
supernatural armor, a Shipibo's most important protection from
bewitchment and death." An explanation in such terms—at
odds not just with mathematics but with the whole tradition of
Western rationality—tends to emphasize our remoteness from the
Amazonian way of life. But keep in mind that less exotic cultures
make equally dogmatic and arbitrary aesthetic judgments about
decorative patterns, and the explanations offered are no better. In
our own culture, for example, a quite narrow spectrum of patterns
and color combinations is deemed acceptable for men's neckties. A
Wall Street banker could doubtless classify any given tie as
wearable or not but probably couldn't articulate the reasons any
more sensibly than could Roe's Shipibo informants. (And, as
Grünbaum would point out, neither the banker nor the Shipibo
would explain their preferences in the vocabulary of mathematical
group theory or crystallography.)
In another chapter of Symmetry Comes of Age, Frank Jolles
of the University of Natal in South Africa examines the colors and
patterns of Zulu beadwork, again discovering hidden design rules.
(In some cases, the rules are apparently hidden even from the
designers.) Jolles focuses extended attention on a particular
beadwork belt (see illustration on facing page) of a type worn by
women after the birth of their first child. As beaded garments go,
it is not particularly intricate or impressive, but understanding
the decisions that went into its construction makes a good puzzle.
At first glance, the belt pattern seems to be composed of 15
adjacent rectangles, split along zigzag diagonals so that each
rectangle breaks into two triangles of contrasting color. Closer
examination, however, shows that the diagonals don't actually reach
the corners of the rectangles, and so the triangles are imperfect;
they are really wedge-shaped trapezoids. Here is a case where the
siren song of symmetry might tempt one to see a "mistake,"
but Jolles makes clear that the reduced symmetry of trapezoids
rather than triangles is surely not an accident or a defect; it was
part of the plan.
The arrangement of colors in the belt also suggests a
not-quite-perfect symmetry. The trapezoids come in five colors,
assigned in such a way that various rotations, reflections and
translations are almost symmetries of the pattern, but in
each case the symmetry breaks down when you look more closely. For
example, the red trapezoids have a center of twofold rotational
symmetry—a point where twirling the belt by 180 degrees
returns red trapezoids to all the same positions—but that
rotation scrambles the other colors. The green and the white
trapezoids share a symmetry operation called a glide reflection, but
again the partial symmetry fails to preserve other colors. And so
the question arises: If these symmetries, which seem like the
obvious ones to Western eyes, did not determine the placement of the
colors, then what is the rule that guided the fabrication
of the belt? Jolles proposes that the underlying principle is the
division of the five-color spectrum into two families: red and black
on the one hand and white, yellow and green on the other. The
paramount rule is to assign the colors so that no two trapezoids
that share an edge have colors from the same family. The color
families have an interpretation in Zulu culture that seems
appropriate to the belt's function: The red-black family represents
female fertility and the white-yellow-green family is associated
with courtship, love and youth; thus the repeated juxtapositions of
like with unlike seem to celebrate sexual union and childbearing.
But it should be noted that this is merely Jolles's inference about
the meaning of the pattern. When he interviewed the maker of the
belt, he reports, "Neither she, nor her friends, nor any of the
older women who were present, were able to give any information
about the piece."
A few minutes of playing with the coloring of the belt pattern
suggests there must be still more unstated rules, beyond the
requirement of color-family exogamy. The rule forbidding
like-with-like adjacencies could be satisfied in a trivial way by
simply discarding one color from the white-yellow-green family and
repeating the other four colors in a regular sequence. Presumably
this scheme would not earn approval from the makers of the belt.
Perhaps there is also an unstated rule that all five colors must be
used and that they should be distributed as evenly as possible. In
that case, the observed belt pattern is a plausible solution,
although certainly not the only one, or the most symmetrical one. As
I continue to look at this object, I have a persistent urge to
"improve" it, which probably indicates that I still don't
understand what goal was in the maker's mind.
Symmetry Comes of Age is presented as a sequel to an
earlier volume by the same pair of editors, Symmetries of
Culture (University of Washington Press, 1988), but in some
respects the new book seems more like a dialectical response. The
1988 volume emphasized the identification and analysis of symmetry
patterns in cultural artifacts; it was mathematics for
anthropologists. This new work is more anthropology for
mathematicians. Of the 10 chapters, only the introductory one by
coeditor Donald W. Crowe puts mathematics out front; it is a guide
to the two-dimensional symmetry groups (including the handy
flowchart for field identification). The volume grew out of a 1999
workshop on "symmetries of patterned textiles" held at the
University of Wisconsin, and so it is no surprise that cloth is the
most common medium of expression discussed in these chapters. Carrie
Brezine, who is both a mathematician and a weaver, writes a
complement to Crowe's mathematical tutorial: Her chapter is a guide
to the capabilities of the loom as an instrument for generating
symmetrical patterns. In subsequent chapters, Paulus Gerdes writes
on Yombe woven mats, Mary Frame on Nasca embroidery, E. M.
Franquemont and C. R. Franquemont on weaving in the Andean village
of Chinchero, and Patricia Daugherty on Turkish-Yörük
weavers. Outside the world of cloth, coeditor Dorothy K. Washburn
discusses patterns on Inca-era ceramics from Peru. Although
Symmetry Comes of Age began with a symposium, it is
more cohesive and coherent than a typical proceedings volume.
Embedded Symmetries, Natural and Cultural also derives from
a symposium, held in 2000 at the Amerind Foundation in Arizona. Some
of the same themes and authors reappear: Peter G. Roe and E. M.
Franquemont contribute chapters, and Dorothy K. Washburn is both an
author and the editor. Some of the same ancient Peruvian fabrics
discussed by Grünbaum are analyzed here by Anne Paul. But three
introductory articles offer a point of view quite different from
that found in Symmetry Comes of Age, a point of view
oriented at right angles to both the mathematical and the
anthropological approaches to symmetry. Diane Humphrey describes
studies of the developmental biology of symmetry. Infants as young
as four months seem to show a preference for patterns with certain
kinds of symmetry. Michael Kubovy and Lars Strother examine the
perceptual psychology of symmetrical patterns; for example, they
point out that when we see a frieze pattern that has both a twofold
rotational symmetry and reflection symmetries (it can be brought
into coincidence with itself either by a 180-degree turn or by a
mirror reflection), the mirror symmetry seems to take precedence.
Finally, Thomas Wynn looks at the cognitive development of symmetry
over the course of human evolution, citing evidence in stone tools
and cave paintings.
Both of these books are eye-openers. I am inclined to describe
Symmetry Comes of Age as a basic text, and Embedded
Symmetries as important supplementary reading. If you must
choose between them, it's only fair to note that Symmetry Comes
of Age offers brighter paper, better illustrations
(including some in color) and twice as many pages for $10
less.—Brian Hayes