Abstract
This chapter provides an overview of research on symmetry dynamics in cultural creations around the world that shed light on human cognition. This is accomplished first of all by paying attention to a range of historical conceptual developments and their methodologies with an initial focus on the origins of mathematical group theory and group theory’s eventual applications within anthropology for the comparative analysis of symmetrical patterns between (and across) cultures. The chapter presents basic group theoretic classification schemes of finite patterns and tiling plane patterns that are enabled by thinking of symmetry as invariant correspondences related by generative transformations along an axis of translation or across an axis of reflection or rotation. This discussion is followed by a survey of key stages in the development of plane pattern analysis applied to folk art designs from material culture and related socio-cultural dynamics. The chapter then shifts to a range of other perspectives on symmetry dynamics at work in areas traditionally studied by the arts (including the humanities, social sciences, and fine arts). Theories of structural semiotics, rhetoric culture, process semiotics, embodied cognition, cognitive semiotics, and semiotic anthropology provide important leverage for understanding the functions and meanings of symmetry relations at work in everything from brand mark designs and literary structures to neurological lateralization and the evolution of upright posture. Recent research on the chiasmus figure receives special attention, along with the role of symmetry (and symmetry breaking) in ritual contexts. Ultimately the chapter suggests that this under-researched area of human culture and cognition holds much promise for understanding the nature and meaning of both cognitive mathematics and human experience.
This is a preview of subscription content, log in via an institution to check access.
References
Blake, W. (1789–1794). Songs of innocence and songs of experience. William Blake.
Boas, F. (1927). Primitive Art. Oslo: H. Aschehoug & Co.
Brainerd, G. W. (1942). Symmetry in primitive conventional design. American Antiquity, 8(2), 164–166. https://doi.org/10.2307/275508
Bråten, S. (Ed.). (1998). Infant learning by altercentric participation: The reverse of egocentric observation in autism. In intersubjective communication and emotion in early ontogeny (pp. 105–124). Cambridge University Press.
Bråten, S. (2009). The intersubjective mirror in infant learning and evolution of speech. John Benjamins.
Cooke, R. L. (2005). The history of mathematics: A brief course (2nd ed.). John Wiley & Sons.
Crowe, D. W. (2004). Introduction to plane symmetries. In D. K. Washburn & D. W. Crowe (Eds.), Symmetry comes of age: The role of pattern in culture (pp. 3–17). University of Washington Press.
Danesi, M. (2009). Opposition theory and the interconnectedness of language, culture and cognition. Sign System Studies, 37(1/2), 11–41.
Danziger, E. (2011). Distinguishing three-dimensional forms from their mirror-images: Whorfian results from users of intrinsic frames of linguistic reference. Language Sciences, 33(6), 853–867. https://doi.org/10.1016/j.langsci.2011.06.008
Darvas, G. (2007). Symmetry: Cultural-historical and ontological aspects of science-arts relations: The natural and man-made world in an interdisciplinary approach. Birkhäuser.
Daugherty, P. (2004). Cognitive structures and body metaphors in the aesthetic and everyday discourse of Turkish-Yörük weavers. In D. K. Washburn & D. W. Crowe (Eds.), Symmetry comes of age: The role of pattern in culture (pp. 304–345). University of Washington Press.
de Almeida, M. W. B. (1990). Symmetry and entropy: Mathematical metaphors in the work of Levi-Strauss. Current Anthropology, 31(4), 367–385.
Douglas, M. (2007). Thinking in circles: An essay on ring composition. Yale University Press.
Ellen, R. F. (1977). Anatomical classification and the semiotics of the body. In J. Blacking (Ed.), The anthropology of the body (pp. 343–373). Academic Press.
Ewins, R. (2004). Symmetry and semiotics: The case of Fijian Barkcloth figuration. In D. K. Washburn (Ed.), Embedded symmetries: Natural and cultural (pp. 160–184). University of New Mexico Press.
Emmer, M. (Ed.). (1993). The visual mind: art and mathematics. Cambridge: MIT Press.
Foolen, A. (2019). The value of left and right. In J. L. Mackenzie & L. Alba-Juez (Eds.), Emotion in discourse (pp. 139–158). John Benjamins. https://doi.org/10.1075/pbns.302.06foo
Frame, M. (2004). Motion pictures: Symmetry as animator, classifier, and syntax in the Nasca embroideries of Peru. In D. K. Washburn & D. W. Crowe (Eds.), Symmetry comes of age: The role of pattern in culture (pp. 133–176). University of Washington Press.
Franquemont, E. M., & Franquemont, C. R. (2004). Tanka, Chongo, Kutij: Structure of the world through art. In D. K. Washburn & D. W. Crowe (Eds.), Symmetry comes of age: The role of pattern in culture (pp. 177–214). University of Washington Press.
Gamwell, L. (2016). Mathematics and art: A cultural history. Princeton University Press.
Gasché, R. (1999). Reading Chiasms. In R. Gasché (Ed.), Of minimal things: Studies on the notion of relation (pp. 263–284). Stanford University Press.
Grünbaum, B. (2004). Periodic ornamentation of the fabric plane: Lessons from Peruvian fabrics. In D. K. Washburn (Ed.), Symmetry comes of age: The role of pattern in culture (pp. 18–64). University of Washington Press.
Grünbaum, B., & Shephard, G. C. (1987). Tilings and patterns. W.H. Freeman.
Hargittai, I. (Ed.). (1986). Symmetry: Unifying human understanding. Pergamon Press.
Haüy, R. J. (1815). Memoire sur une loi de cristallisation appelée loi de symmétrie. Mémoires du Muséum d’Histoire naturelle.
Hofstadter. (1979). Gödel, Escher, Bach: An Eternal Golden Braid. Basic Books.
Hon, G., & Goldstein, B. R. (2008). From summetria to symmetry: The making of a revolutionary scientific concept. Springer.
Humphrey, D. (2004). Symmetries in development: The eye is quicker than the hand. In D. K. Washburn (Ed.), Embedded symmetries: Natural and cultural (pp. 7–17). University of New Mexico Press.
Johnson, M. (1987). The body in the mind: The bodily basis of meaning, imagination, and reason. University of Chicago Press.
Klein, F. (1872 [1893]). Vergleichende Betrachtungen über neuere geometrische Forschungen. Mathematische Annalen, 43, 63–100.
Kubovy, M., & Strother, L. (2004). The perception of band patterns: Going beyond geometry. In D. K. Washburn (Ed.), Embedded symmetries: Natural and cultural (pp. 19–26). University of New Mexico Press.
Lanigan, R. L. (2018). The rhetoric of discourse: Chiasm and dialogue in communicology. In R. C. Arnett & F. Cooren (Eds.), Dialogic ethics (pp. 215–243). John Benjamins.
Legendre, A.-M. (1794). Éléments de géométrie. F. Didot.
Levinson, S. C., & Brown, P. (1994). Immanuel Kant among the Tenejapans: Anthropology as empirical philosophy. Ethos, 22(1), 3–41.
Lévi-Strauss, C. (1954). The mathematics of man. International Social Science Bulletin, 6, 581–590.
Lévi-Strauss, C. (1955). The structural study of myth. Journal of American Folklore, 68(270), 428–444.
Lévi-Strauss, C. (1958). Anthropologie structurale. Plon.
Lissner, P. A. (2007). Chi-thinking: Chiasmus and cognition [Ph.D. dissertation]. University of Maryland.
Ljungberg, C. (2007). ‘Damn mad’: Palindromic figurations in literary narratives. In E. Tabakowska, C. Ljungberg, & O. Fischer (Eds.), Insistent Images (Iconicity in Language and Literature 5), pp. 247–265. Amsterdam: John Benjamins.
Ljungberg, C. (2020). Crisscrossing James Joyce’s Ulysses: Chiasmus and cognition. In P. Perniss, O. Fischer, & C. Ljungberg (Eds.), Operationalizing Iconicity (Iconicity in Language and Literature 17), pp. 200–210. Amsterdam: John Benjamins.
Marsden, J., & Thomas, B. (2013). Brand values: Exploring the associations of symmetry within financial brand marks. Design Management Journal, 8(1), 62–71. https://doi.org/10.1111/dmj.12004
McEwen, I. K. (2003). Vitruvius: Writing the body of architecture. MIT Press.
McGilchrist, I. (2019). The master and his emissary: The divided brain and the making of the Western world (2nd ed.). Yale University Press.
Meaney, T., Trinick, T., & Allen, P. (2022). Ethnomathematics in education: The need for cultural symmetry. In M. Danesi (Ed.), Handbook of cognitive mathematics (pp. 1–29). Springer International Publishing. https://doi.org/10.1007/978-3-030-44982-7_4-1
Merleau-Ponty, M. ([c.1960]1968). The intertwining—the chiasm. In Le Visible et l’invisible: The Visible and the Invisible, C. Lefort (Ed.), trans. A. Lingis, pp. 130–155. Evanston: Northwestern University Press.
Morava, J. (2003–2020). On the canonical formula of C. Lévi-Strauss I–II. arXiv preprint math/0306174v2, and arXiv preprint/2002.12813v1.
Nänny, M. (1988). Chiasmus in literature: Ornament or function? Word & Image, 4(1), 51–59.
Needham, R. (Ed.). (1973). Right & left: Essays on dual symbolic classification. University of Chicago Press.
Nöth, W. (1994). Opposition at the root of semiosis. In W. Nöth (Ed.), Origins of semiosis: Sign evolution in nature and culture (pp. 37–60). Mouton de Gruyter.
Palka, J. W. (2002). Left/right symbolism and the body in ancient Maya iconography and culture. Latin American Antiquity, 13(4), 419–443. https://doi.org/10.2307/972224
Paul, A. (2009). When fair is foul and foul is fair: Lessons from Macbeth. In I. Strecker & S. Tyler (Eds.), Culture and Rhetoric (pp. 104–114). Berghahn Books.
Peirce, C. S. ([1861–1913] 1931–1958). In C. Hartshorne, P. Weiss, & A. W. Burks (Eds.), Collected Papers of Charles Sanders Peirce (8 Vols). Cambridge, MA: Harvard University Press (cited as CP).
Pelkey, J. (2013a). Cognitive chiasmus: Embodied phenomenology in Dylan Thomas. Journal of Literary Semantics, 42(1), 79–114. https://doi.org/10.1515/jls-2013-0005
Pelkey, J. (2013b). Chiastic Antisymmetry in language evolution. The American Journal of Semiotics, 29(1), 39–68. https://doi.org/10.5840/ajs2013291-43
Pelkey, J. (2016). Symbiotic modeling: Linguistic anthropology and the promise of chiasmus. Reviews in Anthropology, 45(1), 22–50. https://doi.org/10.1080/00938157.2016.1142294
Pelkey, J. (2017a). Greimas embodied: How kinesthetic opposition grounds the semiotic square. Semiotica, 2017(214), 277–305. https://doi.org/10.1515/sem-2016-0188
Pelkey, J. (2017b). The semiotics of X: Chiasmus, cognition, and extreme body memory. Bloomsbury Academic.
Pelkey, J. (2018). Upright posture and the meaning of meronymy: A synthesis of metaphoric and analytic accounts. Cognitive Semiotics, 11(1) https://doi.org/10.1515/cogsem-2018-0003
Pelkey, J. (2020). Intercultural parallax: Comparative modeling, ethnic taxonomy, and the dynamic object. Semiotica, 2020(232), 147–185. https://doi.org/10.1515/sem-2019-0075
Pelkey, J. (2021a). Embodied chiasmus: From alienation to participation. In J. Abbink & S. LaTosky (Eds.), Rhetoric in Social Relations: Dialectics of Bonding and Contestation, pp. 30–54 (Studies in Rhetoric and Culture 8). Berghahn.
Pelkey, J. (2022, in press). Tonal iconicity and narrative transformation: Transverse embodied chiasmus in Sylvia Plath and Dolly Parton. In O. Fischer, C. Ljungberg, E. Tabakowska and S. Lenninger (Eds.), Iconicity in Language and Literature 18. Amsterdam: John Benjamins.
Pelkey, J. (2021b). Zhuangzi, Peirce, and the butterfly dreamscape: Concentric meaning in the Qiwulun 齊物論. Chinese Semiotic Studies, 17(2), 255–287. https://doi.org/10.1515/css-2021-0013
Schattschneider, D. (1978). The plane symmetry groups: Their recognition and notation. The American Mathematical Monthly, 85(6), 439–450. https://doi.org/10.2307/2320063
Sheets-Johnstone, M. (2003). Kinesthetic memory. Theoria et Historia Scientiarum, 7(1), 69–92. https://doi.org/10.12775/ths.2003.005
Shepard, A. O. (1948). The symmetry of abstract design with special reference to ceramic decoration. Carnegie Institution of Washington.
Speiser, A. (1927). Die Theorie der Gruppen von Endlicher Ordnung: Mit Anwendungen auf Algebraische Zahlen und Gleichungen Sowie auf die Kristallographie. Springer.
Stewart, I. (2014). Fearful symmetry. In The best writing on mathematics 2013 (pp. 23–31). Princeton University Press. https://doi.org/10.1515/9781400847990-004
Strecker, I. A. (2011). Ethnographic chiasmus: Essays on culture, conflict, and rhetoric. Michigan State University Press.
Stjernfelt, F. (2007). Diagrammatology. Springer.
Strecker, I., & Tyler, S. (Eds.). (2009). Culture and rhetoric. Berghahn.
Uexküll, J. V. (1921). Umwelt und Innenwelt der Tiere (2nd ed.). Verlag von Julius Springer.
Van Lier, H. (2003). Around Homo in 80 theses: A fundamental anthropogeny (M. Lo, Ed.; P. Lottefier, Trans.). Anthropogenie.com.
Van Lier, H. (2010). Anthropogénie. Impressions nouvelles.
Vitruvius Polio, M. ([c.25BCE] 1826). de Architectura: The Architecture of Marcus Vitruvius Pollio in Ten Books. Priestly and Weale.
Vahedi, Z., Pelkey, J., Park, S., & Walsh Matthews, S. (2021). Testing Symmetrical Knot Tracing for Cognitive Priming Effects Rules out Analytic Analogy. Symmetry, 13(1), 34. https://doi.org/10.3390/sym13010034
Washburn, D. K. (Ed.). (2004). Embedded symmetries: Natural and cultural. University of New Mexico Press.
Washburn, D. K. (1986). Pattern symmetry and colored repetition in cultural contexts. Computers & Mathematics with Applications, 12B(3–4), 767–781.
Washburn, D. K. (1988). Analysis of pattern structure by geometric symmetries. In J. Volmer (Ed.), Textiles as primary sources: Proceedings of the first symposium of the textile Society of America (pp. 40–49). Minneapolis Institute of Art.
Washburn, D. K. (1999). Perceptual anthropology: The cultural salience of symmetry. American Anthropologist, 101(3), 547–562. https://doi.org/10.1525/aa.1999.101.3.547
Washburn, D. K. (2006). Aesthetic universals in cultural perception and practice. In New directions in aesthetics, creativity and the arts (pp. 59–75). Baywood Publishing Co.
Washburn, D. K., & Crowe, D. W. (Eds.). (2004). Symmetry comes of age: The role of pattern in culture. University of Washington Press.
Washburn, D. K., & Crowe, D. W. (Eds.). (1988). Symmetries of culture: Theory and practice of plane pattern analysis. University of Washington Press.
Weyl, H. (1952). Symmetry. Princeton University Press.
Wiseman, B. (2014). Chiasmus, mythical creation and H.C. Andersen’s ‘the shadow’. In B. Wiseman & A. Paul (Eds.), Chiasmus and culture (pp. 219–238). Berghahn Books.
Wiseman, B., & Paul, A. (Eds.). (2014). Chiasmus and culture. Berghahn.
Wiseman, B. (2009). Chiastic thought and culture: A reading of Claude Lévi-Strauss. In I. Strecker & S. Tyler (Eds.), Culture and rhetoric (pp. 85–103). Berghahn Books.
Wynn, T. (2004). Evolutionary developments in the cognition of symmetry. In D. K. Washburn (Ed.), Embedded symmetries: Natural and cultural (pp. 47–57). University of New Mexico Press.
Zirker, A. (2017). Performative iconicity. In A. Zirker, M. Bauer, O. Fischer, & C. Ljungberg (Eds.), Dimensions of Iconicity (pp. 287–300). John Benjamins. https://benjamins.com/catalog/ill.15.16zir
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Section Editor information
Rights and permissions
Copyright information
© 2022 Springer Nature Switzerland AG
About this entry
Cite this entry
Pelkey, J. (2022). Cultural Symmetry: From Group Theory to Semiotics. In: Danesi, M. (eds) Handbook of Cognitive Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-44982-7_46-1
Download citation
DOI: https://doi.org/10.1007/978-3-030-44982-7_46-1
Received:
Accepted:
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-44982-7
Online ISBN: 978-3-030-44982-7
eBook Packages: Springer Reference MathematicsReference Module Computer Science and Engineering