Antthropology for Mathematicians

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Sigma Xi, The Scientific Research Society

Anthropology for MathematiciansSymmetry Comes of Age: The Role of Pattern in Culture by Dorothy K. Washburn; DonaldW. Crowe; Embedded Symmetries, Natural and Cultural by Dorothy K. WashburnReview by: Brian HayesAmerican Scientist, Vol. 93, No. 2 (MARCH-APRIL 2005), pp. 180-182Published by: Sigma Xi, The Scientific Research SocietyStable URL: http://www.jstor.org/stable/27858556 .

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Topical ReviewsMATHEMATICSAnthropology forMathematiciansSymmetryComes of Age: The Role ofPattern inCulture. Edited by Dorothy K.Washburn and Donald W. Crowe, xxx +354 pp. University ofWashington Press,2004. $60.Embedded Symmetries, Natural andCultural.Edited byDorothyK.Washburn.ix + 189 pp. University ofNew MexicoPress, 2004. $69.95.

On a visit to theAlhambra someyears ago, I toted along a copyof Symmetryin Science andArt,aweighty textbyA. V. Shubnikov andV. A. Koptsik, as a field guide to thecarvings and tilings thatdecorate thatextravagant palace overlooking Granada. The two books under review herewould probably serve as better fieldguides?Symmetry Comes of ge even includes a useful flowchart forclassifyingthe symmetrygroups of patterns?butI suspect that the authors and editorswould not entirely approve of thisuseof their ork. The touristwho stalks thehalls of theAlhambra trying o completea checklist of the 17 two-dimensional

symmetrygroups isnot their ideal student of "the role of pattern in culture."When one is looking at an artifactsuchas a tiled floor or a woven fabric or abeadwork ornament, identifying rystallographicgroups isatbest thebeginningof understanding theobject. The classificationmight tellyou something aboutthemeaning of thework in thecontextofWestern mathematics, but it is unlikelyto revealmuch about theobject'smeaningwithin the culture that created it.This point ismade emphatically byBranko Gr?nbaum?a mathematicianwho certainly knows his symmetrygroups?in a previously published article on ancient Peruvian textiles thatis reprinted in SymmetryComes ofAge.Gr?nbaum argues thatgroup theoryoffers ittlehelp inunderstanding thePeruvian patterns because

The concepts of that theoryare entirelyout of tunewith themodes

of thinking of the people whoseproducts are being investigated;moreover, these concepts were totallyabsent even fromthe thinkingofmathematicians during most ofthehistory ofmathematics.He goes on to rail against "the stiflingdictates of the 'symmetry isgroup theory' cult,"writing that

It is aggravating to see sophisticated examples of patterns,madeby cultures forwhich thepatternsheld great importance, consideredas inferioror "mistaken" justbecause theydo not fitsome mathematicians' preconceived notionsof "symmetry."The idea of outsider criticspomtingout the "mistakes" of native patternmakers is turnedupside down inanother essay in thisvolume, by Peter G. Roe,an anthropologist. Roe writes on thegeometric designs of the Shipibo people of theupper Amazon basin. In the1980she presented a selection ofShipibomotifs to a class of art students at the

University of Delaware and, with theassistance of an early computer-graphics system, had the students generatenew patterns inwhat theytook tobe thesame style.The computermade iteasyto apply various symmetry transformations, such as reflectionsand rotations,but it could not enforce other kinds ofdesign rules. Roe took printouts of thestudents' work back to South America,where Shipibo women and men wereasked to evaluate them. The result ofthis ingenious experimentwas not reallya surprise,but it isnonetheless instructive:Patterns that look essentially aliketo theuntrained (or unacculturated) eyecan evoke very differentresponses frominsiderswho understand the art format a deeper level.A patternmight wellhave the right symmetries but still failtowin theapproval of theShipibo judges. In particular, the experiment calledattention to "a nonverbalized rule inShipibo art" requiring a certainkind ofconnectivity inpatterns, allowing "one'sfinger r eye to tracea continuous lineasitmeanders through the lattice."Roe reports that the requirement ofconnectivityhas a clearmeaning in the

lives of thepeople who decorate theirhomes and belongings with these patterns. "The continuity of the formline. . .does not let contagion intrude," hewrites; "designs are supernatural armor, a Shipibo's most important protection frombewitchment and death."An explanation in such terms?at oddsnot justwith mathematics butwith thewhole tradition ofWestern rationality?tends to emphasize our remotenessfrom the Amazonian way of life. Butkeep inmind that less exotic culturesmake equally dogmatic and arbitraryaesthetic judgments about decorativepatterns, and the explanations offeredare no better. In our own culture, for example, a quite narrow spectrum ofpatternsand color combinations isdeemedacceptable formen's neckties. A WallStreet banker could doubtless classifyany given tie as wearable or not butprobably couldn't articulate the reasonsany more sensibly than could Roe'sShipibo informants. And, asGr?nbaumwould point out, neither thebanker northeShipibo would explain theirpreferences in thevocabulary ofmathematicalgroup theoryor crystallography.)In another chapter of Symmetry omesofAge, Frank Jolies of theUniversityofNatal in South Africa examines thecolors and patterns of Zulu beadwork,again discovering hidden design rules.(In some cases, the rules are apparently hidden even from thedesigners.)Jolies focuses extended attention on aparticular beadwork belt (see illustration on facing page) of a typewornby women after the birth of their firstchild. As beaded garments go, it is notparticularly intricate or impressive,but understanding the decisions thatwent into itsconstructionmakes a goodpuzzle. At firstglance, the belt patternseems to be composed of 15 adjacentrectangles, split along zigzag diagonalsso that each rectangle breaks into twotrianglesof contrasting color.Closer examination, however, shows that thediagonals don't actually reach the cornersof the rectangles, and so the trianglesare imperfect; they are really wedgeshaped trapezoids. Here is a casewherethesiren song of symmetrymight temptone to see a "mistake," but Joliesmakesclear that thereduced symmetryof trap

180 American Scientist, Volume 93

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This braided grass belt covered with cotton fabric and glass beads measures 65 x 7 centimetersand is from the Hlabisa District in South Africa, where such belts are worn by women afterthe birth of their first child. The women who made the belt have no explanation for its traditional pattern, but one scholar has proposed that color placement may have been guided bya rule that red and black, which in Zulu culture represent female fertility, should never sharean edge, and neither should colors from thewhite-yellow-green family, which are associatedwith courtship, love and youth. From Symmetry Comes of Age.

ezoids rather than triangles is surelynotan accident or a defect; itwas part oftheplan.The arrangement of colors in thebeltalso suggests a not-quite-perfect symmetry. The trapezoids come infive colors, assigned in such away thatvariousrotations,reflections nd translationsarealmost syrnmetriesof thepattern, but ineach case the symmetry breaks downwhen you lookmore closely. For example, the red trapezoids have a center oftwofold rotational symmetry?a pointwhere twirling thebelt by 180 degreesreturns red trapezoids to all the samepositions?but that rotation scramblestheother colors. The green and thewhitetrapezoids share a symmetryoperationcalled a glide reflection,but again thepartial symmetry fails topreserve other colors.And so thequestion arises: Ifthese symmetries,which seem like theobvious ones toWestern eyes, did notdetennine theplacement of the colors,thenwhat is the rule thatguided the fabrication of the belt? Joliesproposes thattheunderlying principle is the divisionof thefive-color spectrum intotwo families: red and black on theone hand andwhite, yellow and green on the other.The paramount rule is to assign the colors so thatno two trapezoids thatsharean edge have colors fromthe same family. he color families have an interpretation in ulu culture thatseems appropriate to thebelt's function: The red-blackfamilyrepresents female fertilitynd thewhite-yellow-green family is associatedwith courtship, love and youth; thus therepeated juxtapositions of likewith unlike seem to celebrate sexual union andchildbearing. But it hould be noted thatthis ismerely Jolles's inference bout themeaning of thepattern.When he interviewed themaker of thebelt, he reports,"Neither she, nor her friends,nor anyof theolder women who were present,were able togive any information aboutthepiece."A fewminutes ofplayingwith the coloring of the belt pattern suggests theremust be stillmore unstated rules,beyondtherequirementof color-family xogamy.The rule forbidding like-with-likeadjacencies could be satisfied ina trivialwayby simply discarding one color from thewhite-yellow-green family and repeating theother four colors ina regular sequence. Presumably thisscheme wouldnot earn approval from themakers ofthebelt. Perhaps there s also an unstatedrule that ll five colorsmust be used andthattheyshould be distributed as evenly

as possible. In that ase, theobserved beltpattern is a plausible solution, althoughcertainlynot the only one, or themostsymmetrical one. As I continue to lookat thisobject, Ihave a persistent urge to"improve" it, hich probably indicatesthat I still don't understand what goalwas in themaker's mind.

SymmetryComes ofAge ispresentedas a sequel to an earlier volume by thesame pair of editors, Symmet?es ofCulture (University ofWashington Press,1988),but insome respects thenew bookseems more like a dialectical response.The 1988 volume emphasized the identification nd analysis of symmetry patterns in cultural artifacts; itwas mathematics foranthropologists. This newwork ismore anthropology formathematicians. Of the 10 chapters, only theintroductory ne by coeditor Donald W.Crowe putsmathematics out front; t saguide to thetwo-dimensional symmetrygroups (including thehandy flowchartfor field identification). The volumegrew out of a 1999workshop on "symmetries of patterned textiles" held attheUniversity ofWisconsin, and so it isno surprise thatcloth is themost com

mon medium of expression discussedin these chapters. Carrie Brezine, whoisboth amathematician and aweaver,writes a complement toCrowe's mathematical tutorial:Her chapter is a guideto the capabilities of the loom as an instrument forgenerating symmetricalpatterns. In subsequent chapters,PaulusGerdes writes on Yombe woven mats,Mary Frame on Nasca embroidery, E.M. Franquemont and C. R. Franquemont onweaving in theAndean villageof Chinchero, and Patricia Daugherty

on Turkish-Y?r?k weavers. Outsidetheworld of cloth, coeditor Dorothy K.Washburn discusses patterns on Incaera ceramics fromPeru. Although SymmetryComes of ge began with a symposium, it ismore cohesive and coherentthan a typicalproceedings volume.EmbeddedSymmetries,atural andCultural also derives from a symposium,held in2000 at theAmerind FoundationinArizona. Some of the same themesand authors reappear: Peter G. Roe andE. M. Franquemont contribute chapters,and Dorothy K. Washburn is both anauthor and the editor. Some of the sameancient Peruvian fabrics discussed byGr?nbaum are analyzed here by AnnePaul. But three introductoryarticles offera point of view quite different fromthat found inSymmetry omes ofAge, apoint of view oriented at right anglesto both themathematical and the anthropological approaches to symmetry.Diane Humphrey describes studies ofthedevelopmental biology of symmetry. Infants as young as fourmonthsseem to show a preference forpatternswith certainkinds of symmetry. ichaelKubovy and Lars Strother examine theperceptual psychology of symmetricalpatterns; for example, they point outthatwhen we see a friezepattern thathas both a twofold rotational symmetry nd reflection symmetries (itcan bebrought into coincidence with itselfeitherby a 180-degree turnor by amirrorreflection), themirror symmetry seemsto take precedence. Finally, ThomasWynn looks at the cognitive development of symmetry over the course ofhuman evolution, citing evidence instone tools and cave paintings.

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Both of thesebooks are eye-openers. Iam inclined todescribe Symmetry omesofAge as a basic text, and EmbeddedSymmetriesas important supplementaryreading. Ifyoumust choose betweenthem, it'sonly fairtonote thatSymmetryComes ofAge offersbrighter paper, better illustrations (including some in color) and twice as many pages for $10less.?Brian HayesServing a SiliconMasterMathematics by Experiment: PlausibleReasoning in the 21st Century. Jonathan Borwein and David Bailey, + 288pp. A Peters, 2004. $45.Experimentation inMathematics: Computational Paths to Discovery. JonathanBorwein, David Bailey and RolandGirgensohn. + 357 pp. A Peters, 2004.$49.

Once upon a time, in ancientGreece, science was platonicand a priori.The Sun revolvedaround theEarth in a perfect circle,because the circle is such a perfect figure;there ere fourelements,because four issuch a nice number, and so forth. henalong came Bacon, Boyle, Galileo, Kepler, Lavoisier, Newton and theirbuddies, and revolutionized science,makingitexperimental and empirical.Butmath remains a prioriand platonicto this day. Kant even went to excruciating lengths to "show" that geometry, lthough synthetic, isneverthelessa priori.Sure, allmathematicians, greatand small, conducted experiments (untilrecently, using paper and pencil), butthey kept theirdiaries and notebookswell hidden in the closet.But stand by for a paradigm shift:Thanks to ItsOmnipotence theComputer,math?that last stronghold of dearPlato?is becoming (overtly!) experi

mental, a posteriori nd even contingent.But what are poor pure mathematicians to do? Their professional Weltanschauung?in other words, theirphilosophy?and more important,theirworking habits?in otherwords,theirmethodology?never preparedthem forserving thisnew siliconmaster.Some of them, like the conceptualgenius Alexander Grothendieck, evenconsider the computer (seriously!) thedevil. But although many pure mathematicians stronglydislike and mistrust

thecomputer, some others have alreadystarted to see the light.For example, thegreat noncommutative geometer AlainCormes stated in a recent talk thathiscomputer had confirmed a certain conjecture of his for30 special cases, andconsequently he is absolutely certainthat thegeneral conjecture is correct.Mathematicians who want to jumpon thisbandwagon (which,unlike mostbandwagons, ishere to stay) had betterread bothMathematics byExperiment ndExperimentationn athematics. Traditionalistsmay get annoyed, since theauthors(Jonathan orwein,David Bailey andRolandGirgensohn) don'tmake any bonesabout "math by experiment" being trulya paradigm shift.They even dedicate awhole section to theKuhnian notion ofparadigm shift,uotingMax Planck ("thetransfer f allegiance fromparadigm toparadigm isa conversion experience thatcannot be forced") tomake thepoint thatwe can't hasten acceptance of thenewperspective,we can only be patient andwait for theold guard todie.These are such funbooks to read! Actually, calling them booksdoes not do

them justice. They have the livelinessand feel of greatWeb sites,with theirbite-size fascinating factoids and theirmany human- andmath-interest storiesand other gems. But do not be fooledbythe lighthearted, immensely entertaining style.You are going to learnmoremath (experimental or otherwise) thanyou ever did from any two single volumes. Not only that,you will learnbyosmosis how tobecome an experimentalmathematician.One of themany highlights is a detailed behind-the-scenes account ofthediscovery of the amazing BorweinBa?ey-Plouffe (BBP) formula for :

7t~??16/U/ l 8/ 4 8/ 5 8/ 6(By theway, theBailey isDavid, but theBorwein is Jonathan's brother Peter. Simon Plouffe, a latter-dayRamanujan, isthewebmaster of the celebrated InverseSymbolic Calculator site.)The BBP formula allows one to compute the bilHon-and-first digit of (inbase 2)without computing the first illiondigits. Itwas discovered with theaid

This figure plots all roots of polynomials, ?jv,with coefficients in {0,1, -1} up to degree N=1S.The zeroes are colored by their local density normalized to the range of densities, from red(low) to yellow (high). The fractal structures and holes around the roots come in differentshapes and have precise locations. From Experimentation inMathematics.

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