Regular square plane patterns on

Paulus Gerdes

Mozambican Ethnomathematics Research Centre, C.P. 915,
Maputo, Mozambique
pgerdes@virconn.com

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)

 Country People Source Brazil [Amazonia] pg São Paulo (Brazil) May 1988 (strip) Indonesia Rooyen 38 (strip) [Halmahera] Bezemer 18 (strip) Mexico [Oaxaca] pg Mexico Nov. 2000, July 2001 Mozambique Makhuwa pg Nampula (Mozambique) Mar.2000 (strip) Philippines [Palawan] MH 973.35.43 (strip)

Class (1,2,2,1)

 Country People Source Congo Kongo Sentance 111 Indonesia [Sulawesi] Rooyen 74

Class (1,2,2,3) (Figure 11)

 Country People Source Angola [Lunda] MNEP Brazil Wayana Aubert 44 (strip), Ferreira 46 (strip), Velthem 153, 158, 159 (strip) [Estado do Amazonas] Pinheiro 66 China LaPlantz 99, pg ? Colombia Embera-Chami Uribe 116 Congo Kongo RMCA G5053 Yombe RMCA G4276, Gerdes [1] 127, 128 Guatemala LaPlantz 11 (strip) Indonesia [Lombok] LaPlantz 65, 115 (strip) Punan [Borneo] Tillema 133 (strip) [Sumatra] Hoop 19 [d] Mexico [Oaxaca] Orellana 19, pg Mexico Nov. 2000 (Photograph 1) [Puebla] pg Mexico Nov. 2000 (strip) Mozambique Mwani [Cabo Delgado] pg foto 21.04.01 (strip) Philippines [Palawan] MH 973.35.43, MH 973.35.39, MH 973.35.38, MH 973.35.48 (strip) LaPlantz 192 USA Cherokee Duggan 43 (strip), Gettys 33, Mowat 13, Wyckoff 186, i Chitimacha LaPlantz 63, Mather 82, Mason Pl. 132, 133, Taylor 18 Vietnam Sédang Gerdes [2] 246 Lachi Nguyen 41 (strip) --- pg Loja Tribo, Lisbon (Portugal), Sep. 2000

Class (1,2,3,1) (Figure 12)

 Country People Source Indonesia Rooyen 75 [Sumatra] Hoop 19 [d] Mozambique Makhuwa pg Nampula (Mozambique) Mar.2000 (strip)

Class (1,2,3,3) (Figure 13)

 Country People Source Angola Chokwe Bastin 116 Kongo Herreman 59 Colombia Desana Reichel 63 (strip) Congo Kuba (?) Aubert 188 Yombe Gerdes [1] 128, RMCA G4280 Sundi RMCA G3940 Japan [Japan] 25, 79 Kenya pg Nairobi (Kenya) 1986, Gerdes [2] 139 Mexico [Oaxaca] pg Mexico Nov 2000 Thailand LaPlantz 87 USA Cherokee Duggan 28 (strip), Hill 55 (strip), Leftwich 50, 55 Hopi Teiwes 8

Class (1,2,5,1) (Figure 14)

 Country People Source Indonesia Tabaru, [Halmahera] Bezemer 23, Nieuwenhuis Pl. XXXI

Class (1,3,2,1) (Figure 15)

 Country People Source Indonesia [Island Sulawesi] Rooyen 40, Jasper 236 Malaysia Sakai MH 33.61.132

Class (1,3,2,3) (Figure 16)

 Country People Source Indonesia Rooyen 75 (strip) Mexico [Oaxaca] pg Mexico City, July 2001

Class (1,3,3,3) (Figure 17)

 Country People Source Brazil Bakairí Schmidt [1] 503, Schmidt [3] 350 Colombia Desana Reichel 54, Gerdes [2] 191 Congo Sundi RMCA cl. G3940 Guyana Ribeiro [3] Pl. IV-21 Philippines [Bohol] Lane 164 USA Hopi Teiwes 30, 31

Class (1,4,2,3) (Figure 18)

 Country People Source Indonesia Punan [Borneo] Tillema 133 (strip) Mexico [Oaxaca] Orellana 19 (strip); pg Mexico City, Nov. 2000, July 2001 (strip)

Class (1,5,2,3)

 Country People Source Mexico [Oaxaca] pg Mexico City, July 2001 (strip)

Class (1,6,2,3)

 Country People Source Mexico [Oaxaca] pg Mexico City, July 2001 (strip)

Class (1,7,3,3)

 Country People Source Colombia Waunana Gomez 73

Class (1,8,3,3)

 Country People Source Colombia Waunana Gomez 73

Class (1,14,2,3)

 Country People Source New Zealand Maori Pendergrast 125

Class (3,1,0,1) (Figure 19)

 Country People Source Ghana Ashanti Gerdes [1].25

Class (3,1,0,3) (Figure 20)

 Country People Source Brazil [Amazonia] pg São Paulo (Brazil) Nov. 1999, Gerdes [2] 230 Apalai Mowat 152 Baniwa Ricardo 57, 58 Desana Ribeiro [1] 83 Wayana Aubert 8, Cáurio 49, Velthem 141, 144, 156 Waiwai Ribeiro [2] 282, 310, Roth [2] Pl. 13b, 14; Yde 259, 261, 262 (strip), pg São Paulo (Brazil) May. 1988 Cameroon Bamileke Gerdes [1]33 China LaPlantz 99, pg Athens (USA) Aug 1997 Colombia Desana Reichel 51 (strip) French Guyana Emerillon MH 32.9.28 Guyana Roth [1] 319, 358, 360-363, 366-368, Gerdes [2] 182, 183 Akawaio Mowat 152 Panama Cuna Panyella 155 (strip) Surinam Schmidt [2] pl. 51 USA Cherokee Duggan 22 (strip), Leftwich 26 Chitimacha Mather 82 Venezuela Yekuana Guss 186, 194, 196, 202, 204, 209, 210

Class (3,2,2,3) (Figure 21)

 Country People Source Indonesia [Borneo] LaPlantz 6 (strip) USA Chitimacha Taylor 18 (strip), Wyckoff 173

Class (3,2,3,3) (Figure 22)

 Country People Source Angola Chokwe Bastin 104, Gerdes [2] 143, Coimbra 89.1.184, 89.1.293 Brazil [Amazonia] Coimbra Br. 25 (18th century), Brito 80, pg São Paulo (Brazil) Nov. 1999, pg Rio Claro (Brazil), May 1988 (strip), Baniwa MNEP, Ricardo 12, 56, 58, 59 China pg Bergen (Norway) Sept. 1985 Colombia Cuna Mowat 121 Desana Reichel 36, 55, 63, Gerdes [2] 192 Tucano Panyella 176 (strip) Guyana Akawaio Mowat 152 Warao Roth [1] 319, 355 (strip), Gerdes [2] 184 Japan [Japan] 22, 72, 73 Mozambique Makhuwa ? Gerdes [2] 126 Panama Kuna Panyella 155 (strip) Philippines [Bohol] Lane 164 USA Cherokee Duggan v, 17, 22, 41, 44, Gettys 27, Leftwich 56, Mowat 12, Hill 102, 302 Chitimacha Turnbaugh 16 Venezuela Yekuana Guss 217 (strip)

Class (3,2,4,3)

 Country People Source Brazil [Estado do Amazonas] Pinheiro 66 USA Cherokee Duggan 28 (strip), Hill 101(strip)

Class (3,2,4,5)

 Country People Source Brazil Wayó MNEP

Class (3,3,3,3) (Figure 23)

 Country People Source Brazil [Amazonia] pg Rio Claro (Brazil), Maio 1988, Gerdes [2] 231, pg São Paulo (Brazil), Nov. 1999 (Photograph 2) Guyana Warao Roth [1] 321, 357, Gerdes [2] 185 Indonesia [Bali] Eiseman 148 Philippines [Bohol] Lane 164 Senegal Bedik Ferry 25 I, Gerdes [2] 68

Class (3,4,3,3) (Figure 24)

 Country People Source Colombia Desana Reichel 58 Guyana Warao Roth [1] 320, Gerdes [2] 181 Peru Bora pg Iquitos May 2000 Philippines [Bohol] Lane 164 USA Cherokee Duggan 22

Class (3,6,2,3) (Figure 25)

 Country People Source New Zealand Maori Pendergrast 124 (strip)

Class (3,6,3,3)

 Country People Source Brazil Baniwa MNEP

Class (5,1,0,1)

 Country People Source Indonesia [Sulawesi] Rooyen 74

Class (5,1,0,3)

 Country People Source Guyana Roth [1] 362 Vietnam Jorai Gerdes [2] 244

Class (5,1,0,5) (Figure 26)

 Country People Source Venezuela Yekuana Guss 195 (strip)

Class (5,1,3,1)

 Country People Source Brazil Kaiapó Verswijver 129

Class (5,2,3,3) (Figure 27)

 Country People Source Colombia Desana Reichel 63 (strip) Vietnam Jorai Gerdes [2] 245

Class (5,2,4,3)

 Country People Source Brazil Wayó MNEP

Class (5,3,3,3)

 Country People Source Angola Chokwe Bastin 116 (strip) Brazil Baniwa Coimbra 94.5.107

Class (5,3,4,5)

 Country People Source Brazil Wayó MNEP

Detail of a basket from Oaxaca (Mexico) with the regular (1,2,2,3) pattern
Photograph 1

Circular basket tray from the Brazilian Amazon with the regular (3,3,3,3) pattern
Photograph 2

Example of a regular set of concentric toothed squares
Figure 1

Example of a not regular set of concentric toothed squares
Figure 2

Example of a not regular set of concentric toothed squares
Figure 3

Example of a regular set of concentric toothed squares
Figure 4

Axes of symmetry in four directions
Figure 5

a. p=3, b. p=1
Examples
Figure 6

a. p=5, q=1, b. p=1, q=4
Examples
Figure 7

a. p=1, q=4, r=2 , b. p=3, q=2, r=3
Examples
Figure 8

a. p=1, q=2, r=3, s=1, b. p=3, q=1, r=0, s=3
Examples
Figure 9

Class (1,2,1,3)
Figure 10

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 (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 (3,3,3,3)
Figure 23

Class (3,4,3,3)
Figure 24

Class (3,6,2,3)
Figure 25

Class (5,1,0,5)
Figure 26

Class (5,2,3,3)
Figure 27

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

pg: Author’s collection

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