|
Regular square plane patterns on
Paulus Gerdes Mozambican Ethnomathematics
Research Centre, C.P. 915,
Abstract The paper defines and analyses a
class and several subclasses of regular patterns appearing on twill plaited
mats and baskets. The patterns are composed of sets of congruent concentric
toothed squares that are at equal distances in both weaving directions.
A comparative overview and examples of their regional distribution are
given.
In this paper I present a comparative overview and classification of regular square plane patterns as they appear on twill plaited mats and baskets from several cultures around the world (see the examples in Photographs 1 and 2). Under consideration are woven mats (or plane parts of baskets) whereby (1) the two weaving directions are perpendicular to each other, (2) all strands have the same width, (3) all strands in the same direction have the same colour, (4) strands may pass over and under more than one strand in the opposite direction (twill weave). Square plane patterns on twill plaited mats consist of congruent sets of concentric toothed squares at constant distance from each other. These patterns will be called regular if (1) for the weaving of all concentric toothed squares always the same average number of strands are passed over or under. Figure 1 gives an example of a regular set of concentric toothed squares: The concentric toothed squares are formed by passing over, in one of the weaving directions, 1, 3, 2, 2, … strands in the opposite direction. Each time the average is 2. Figure 2 displays an example of a not regular set of concentric toothed squares: The last concentric toothed square is formed by passing over, in the horizontal direction, 1, 3, 5, 3, 3, … strands in the vertical direction. The average is 3, whereas for the before last toothed square the average is 2. (2) the distance between two horizontally neighbour sets of concentric toothed squares is always the same as the distance between two vertically neighbour sets of concentric toothed squares. Figure 3 gives an example of a not regular set of concentric toothed squares: The horizontal distance is 3, whereas the vertical distance is 1. Figure 4 presents an example of a regular set of concentric toothed squares: Both distances are equal to 3. The second characteristic of regular square plane patterns on twill plaited mats guarantees that the two-colour image of such a pattern has axes of symmetry in four directions. In other words, these images belong to symmetry class p4m (see the example in Figure 5). Regular square plane patterns on twill plaited mats may be characterised by a set of four numbers (p,q,r,s) as follows: p denotes the diameter of the smallest toothed square; p=1 in the case that the smallest toothed square is a real square of unit length (the unit of measurement is the constant width of the strands) (Figure 6 presents the examples p=3 and p=1); q denotes the number of successive concentric toothed squares including the smallest square at the centre; the qth and last toothed square will be called an external toothed square (Figure 7 presents examples where q=1 and q=4); r denotes the twill weave r/r used to build up the successive concentric toothed squares; in other words r is the average number of strands over which the strands of the same colour pass to make a toothed square visible. In the case q=1, r will be 0 (Figure 8 presents examples where q=1 and q=4); s is the distance between two horizontally or vertically neighbouring external toothed squares; in other words s is also the weaving distance between two diagonally neighbouring external toothed squares (Figure 9 presents examples where s=1 and s=3). In the following I will present the classes of regular square plane patterns so far encountered by me on twill plaited mats and baskets from several cultures around the world. The list is by class (p,q,r,s), by country, by people (linguistic or ethnic population group) [or region], the source of the book or paper, or of the museum in which I saw a photograph of a mat or a basket with the regular square plane pattern. In the case of a mat or basket belonging to my personal collection I indicate as source pg, followed by the place and date that I acquired the object. In the case that only (a few) congruent sets of concentric toothed squares are woven as a strip pattern, I indicate ‘strip’, supposing that in the same culture the plane pattern may be known too (or imaginable in the head of the mat weaver). In the cases found so far, p and s are always odd numbers. Other classes of square plane patterns
on twill plaited mats and baskets will be analysed in future papers.
Class (1,2,1,3) (Figure 10)
Class (1,2,2,1)
Class (1,2,2,3) (Figure 11)
Class (1,2,3,1) (Figure 12)
Class (1,2,3,3) (Figure 13)
Class (1,2,5,1) (Figure 14)
Class (1,3,2,1) (Figure 15)
Class (1,3,2,3) (Figure 16)
Class (1,3,3,3) (Figure 17)
Class (1,4,2,3) (Figure 18)
Class (1,5,2,3)
Class (1,6,2,3)
Class (1,7,3,3)
Class (1,8,3,3)
Class (1,14,2,3)
Class (3,1,0,1) (Figure 19)
Class (3,1,0,3) (Figure 20)
Class (3,2,2,3) (Figure 21)
Class (3,2,3,3) (Figure 22)
Class (3,2,4,3)
Class (3,2,4,5)
Class (3,3,3,3) (Figure 23)
Class (3,4,3,3) (Figure 24)
Class (3,6,2,3) (Figure 25)
Class (3,6,3,3)
Class (5,1,0,1)
Class (5,1,0,3)
Class (5,1,0,5) (Figure 26)
Class (5,1,3,1)
Class (5,2,3,3) (Figure 27)
Class (5,2,4,3)
Class (5,3,3,3)
Class (5,3,4,5)
Detail of a basket from Oaxaca (Mexico) with the regular
(1,2,2,3) pattern
Circular basket tray from the Brazilian Amazon with the
regular (3,3,3,3) pattern
Example of a regular set of concentric toothed squares
Example of a not regular set of concentric toothed squares
Example of a not regular set of concentric toothed squares
Example of a regular set of concentric toothed squares
Axes of symmetry in four directions
a. p=3, b. p=1
a. p=5, q=1, b. p=1, q=4
a. p=1, q=4, r=2 , b. p=3,
q=2,
r=3
a. p=1, q=2, r=3, s=1,
b. p=3, q=1, r=0, s=3
Museums: Coimbra: Museu Antropológico da Universidade de Coimbra, Portugal MH: Musée de l’Homme, Paris, France MNEP: Museu Nacional de Etnologia, Lisboa, Portugal RMCA: Royal Museum for Central Africa, Tervuren, Belgium References Bastin, Marie-Louise, Art décoratif Tshokwé, Publicações Culturais da Companhia de Diamantes de Angola, Lisbon, 1961 Bezemer, T. J., Indonesian Arts and Crafts, Colonial Institute, Amsterdam, n.d. Brito, Joaquim Pais de, Os índios, nós, Museu Nacional de Etnologia, Lisboa, 2000 Cáurio, Rita, Artêxtil no Brasil: Viagem pelo Mundo da Tapeçaria, Funarte, São Paulo, 1985 Duggan, Betty J. & Riggs, Brett H., Studies in Cherokee Basketry with a Reprint of ‘Decorative Art and basketry of the Cherokee’ by Frank G. Speck, The Frank H. McClung Museum, The University of Tennessee, Knoxville, 1991 Eiseman, Fred B., Ulat-ulatan, Traditional basketry in Bali, White Lotus, Bangkok, 1999 Ferreira, Maria da Glória (Ed.), Arte e corpo: Pintura sobre a pele e adornos de povos indígenas brasileiros, Funarte, Rio de Janeiro, 1985 Ferry, Marie-Paule, Bedik, Images de leur savoir-faire, Éditions Sépia, Saint-Maur, 1997 Gerdes, Paulus [1], Geometry from Africa: Mathematical and Educational Explorations, The Mathematical Association of America, Washington, 1999 Gerdes, Paulus [2], Le cercle et le carré: Créativité géométrique, artistique, et symbolique de vannières et vanniers d’Afrique, d’Amérique, d’Asie et d’Océanie, L’Harmattan, Paris, 2000 Gettys, Marshall (Ed.), Basketry of Southeastern Indians, Museum of the Red River, Idabel, Oklahoma, 1984 Gomez, Marta Lucia Bustos, Objectos Textiles en el Departamento del Choco, Instituto Andino de Artes Populares, Quito, 1994 Guss, David M., To weave and sing: Art, symbol, and narrative in the South American rain forest, University of Californa Press, Berkeley, 1989 Hill, Sarah H., Weaving New Worlds: Southeastern Cherokee Women and their Basketry, The University of North Carolina Press, Chapel Hill, 1997 Hoop, A. van der, Indonesische Siermotieven – Ragam-ragam Perhiasan Indonesia – Indonesian Ornamental Design, Koninklijk Bataviaasch Genootschap van Kunsten en Wetenschappen, 1949 [Japan] Instruction book on basket weaving (in Japanese) [ISBN4-8377-0485-9, acquired in May 1988 in São Paulo, Brazil] Jasper, J. & Pirngadie, Mas, De inlandsche kunstnijverheid in Nederlandsch Indië, Vol. 1, Het vlechtwerk, Mouton, ‘s-Gravenhage, 1912 Lane, Robert F., Philippine basketry, an appreciation, Bookmark Inc., Makati, 1986 LaPlantz, Shereen, Twill basketry: A handbook of designs, techniques, and styles, Lark Books, Asheville, 1993 Leftwich, Rodney L., Arts and Crafts of the Cherokee, Land-of-the-Sky Press, Cullowhee NC, 1970 Mason, Otis Tufton, American Indian Basketry, Dover, New York, [1904] 1988 Mather, Christine, Native America: Art, traditions, and celebrations, Clarkson Potter, New York, 1990 Mowat, Linda; Morphy, Howard & Dransart, Penny, Basketmakers: Meaning and Form in Native American Baskets, Pitt Rivers Museum / University of Oxford, Oxford, 1992 Nieuwenhuis, A. W., Die Veranlagung der Malaiischen Völker des Ost-Indischen Archipels erläutert an ihren industriellen Erzeugnissen, Brill, Leiden, 1913 Nguyen, Van Huy (Ed.), Musée d’Ethnographie du Vietnam, Musée d’Ethnographie du Vietnam, Ho Chi Minh Ville, 1997 Orellana, Margarita de (Ed.), Cestaría, Artes de México, No. 38, México D.F., 1997 Panyella, August (Ed.), Artesania y Arte Popular de las Americas, Editorial Blume, Barcelona, 1981 Pendergrast, Mick, Raranga Whakairo: Maori Plaiting Patterns, Reed Books, Auckland, 1984 Pinheiro, Geraldo Sá Peixoto (Ed.), Arumã: II mostra antropológica de cultura material (trançado do Amazonas), Museu Amazônico – FUA, Manaus, 1994 Reichel-Dolmatoff, Gerardo, Basketry as metaphor: Arts and crafts of the Desana indians of the Northwest Amazon, University of California Musuem of Cultural History, Los Angeles, 1985 Ribeiro, Berta G. [1], A arte do trançado dos índios do Brasil. Um estudo taxonômico, Museu Paraense Emílio Goeldi, Belém, 1985 Ribeiro, Berta G. (Ed.) [2], Tecnologia Indígena, Suma Etnológica Brasileira, Vol. 2, Vozes, Petrópolis, 1987 Ribeiro, Berta G. [3], Dicionário do Artesanato Indígena, Editora Itatiaia, Belo Horizonte, 1988 Rooyen, Pepin van (Ed.), Indonesian Ornamental Design, The Pepin Press, Amsterdam, 1998 Roth. Walter Edmund [1], An introductory study of the arts, crafts, and customs of the Guiana indians, Government Printing Office, Washington, 1924 Roth. Walter Edmund [2], Additional Studies of the Arts, Crafts, and Customs of the Guiana Indians with special reference to those of Southern British Guiana, Government Printing Office, Washington, 1929 Schmidt, Max [1], Ableitung südamerikanischer Geflechtmuster aus der Technik des Flechtens, Zeitschrift für Ethnologie, 1904, 34, 490-512 Schmidt, Max [2], Die technischen Voraussetzungen in der Ornamentik der Eingeborenen Südamerikas, Jahrbuch für prähistorische und ethnographische Kunst, 1926, 142-174 Schmidt, Max [3], Indianerstudien in Zentralbrasilien; Erlebnisse und ethnologische Ergebnisse einer Reise in den Jahren 1900 bis 1901, Dietrich Reimer, Berlin, 1905 Sentance, Bryan, Art of the basket, Traditional basketry from around the world, Thames & Hudson, London, 2000 Taylor, Colin (Ed.), Native American Arts and Crafts, Salamander Books, London, 1995 Teiwes, Helga, Hopi Basket Weaving. Artistry in Natural Fibers, The University of Arizona press, Tucson, 1996 Tillema, Hendrik, A Journey among the Peoples of Central Borneo in Word and Picture, Oxford University Press, Oxford, 1989 Turnbaugh, Sarah Peabody & Turnbaugh, William A., Indian Baskets, Schiffer, West Chester, 1986 Uribe, Luis Guillermo Vasco, Semejantes a los Dioses: Ceramica y Cestaria Embera-Chami, Universidad Nacional de Colombia, Bogotá, 1991 Velthem, Lúcia H. van, A pele de Tuluperê, uma etnografia dos trançados Wayana, Museu Paraense Emílio Goeldi, belém, 1998 Verswijver, Gustaaf (Ed.), Kaiapó-Amazonia. The art of body decoration, Royal Museum for Central Africa, Tervuren, 1992 Wyckoff, Lydia (Ed.), Woven Worlds: Basketry from the Clark Field Collection, The Philbrook Museum of Art, Tulsa, 2001 Yde, Jens, Material Culture of the
Waiwái, The National Museum of Copenhagen, 1965
|
