Cylindrical baskets with a square base from the Bora (Peruvian Amazon) Paulus Gerdes Mozambican
Ethnomathematics
Abstract The paper presents an analysis of six cylindrical baskets with a square base made by Bora basket weavers from the Peruvian Amazon. The baskets were collected in 2000. The basket weavers are aware of the relationship that exists between the period of the decorative motif on the basket wall and the length of the diagonal of the square bottom, as will be shown. Introduction During my stay in 2000 in Iquitos, the capital of the Peruvian Amazon, I gathered six baskets from the Bora that have a square base, a cylindrical wall and a circular border. I invited the participants in the ethnomathematics workshop I conducted to analyse mathematical aspects of the baskets, giving them some hints. Photograph 1 displays some workshop participants analysing one of the baskets. Photograph 2 displays the six baskets, three of them upsidedown to give a clear view of the their shape. On the outside of the baskets, dark and light strands are seen. The darkbrown colour is the natural colour of one of the faces of the leaves of the bájyuhba plant. The light coloured, darkyellow strands resulted from scraping off the natural darkbrown colour. On the inside of the baskets all strands are darkyellow corresponding to the natural colour of the other face of the leaves of the bájyuhba plant. Production process To produce this type of basket (cf. Gerdes, 2004b), the basket weaver starts at the centre of the bottom. A square mat is woven in such a way that the middle lines are well visible and that large strand parts stay out at its sides (see the scheme in Figure 1). The basket weaver takes the mat in the open hands and folds it slightly in such a way that the outstanding strand parts on the left and the outstanding parts on the right of the same side of the square are superimposed (see the scheme in Figure 2a). These strand parts are interwoven until a smaller square is completed (Figure 2b). Doing the same on the other sides of the initial square mat, the four triangles marked by 1, 2, 3, and 4, and formed by the vertices and midpoints of the sides of the initial square mat, are folded upwards. They integrate themselves together with the newly woven squares into the wall of the basket. As the basket becomes higher and higher the material forces the wall to take on a circular cylindrical shape. Once the desired height of the basket is achieved, the basket weaver fastens a small circular rim and gives the finishing touches to the basket. Figure 3 summarises the whole process. In the diagram the large outstanding strand parts are represented by small strokes to show clearly (1) which triangles are effectively folded upwards and (2) that the vertices of the square base of the final basket are the midpoints of the sides of the initial square mat. It may be noted that the strands climb up along the cylindrical wall, leftwards and rightwards, in two families of helices, making angles of 45 degrees with the horizontal plane (See Figure 3c). A detailed analysis of the decoration of each of the six baskets will show that the length of the diagonal of the square bottom cannot be chosen arbitrarily. Obviously this length depends on the dimensions of the basket as pretended by the basket weaver. However, the decoration the artisan intends to introduce on the basket wall is the second factor the basket maker has to take into account. The basket weavers know that the length of the diagonal, expressed in the number of parallel strands to form the bottom, has to be a multiple of the period of the decorative motif on the basket wall. In the following I will describe and analyse each of the baskets in our collection. Basket 1 The cylindrical wall of the first basket (Photograph 3) is plaited "over four – under four" (4/4 ). The period of the wall decoration is 4: pairs of darkbrown strands and pairs of darkyellow strands alternate in both perpendicular weaving directions (Figure 5). The square bottom of the basket has a rotational symmetry of 90 degrees: the four quadrants are congruent (Figure 6). Each quadrant (see Figure 7) presents the structure (3, 2, 4, 4, 41): the (1) of (41) corresponds to the localisation of the four corners of the square bottom. As 3+2+4+4+41 = 16 and 16 is a multiple of the period 4, the periodicity of the cylindrical wall was guaranteed. Basket 2 The cylindrical wall of the second basket (Photograph 4) is plaited 4/4 . The period of the wall decoration is 4: sets of three darkyellow strands and of one darkbrown strand alternate in the perpendicular weaving directions (Figure 8). The square bottom of the basket has a rotational symmetry of order 4: the four quadrants are congruent (Figure 9). Each quadrant (see Figure 10) presents the structure (3, 3, 4, 4, 42): the (2) of (42) corresponds to the localisation of the four corners of the square bottom. As 3+3+4+4+42 = 16 and 16 is a multiple of the period 4, the periodicity of the cylindrical wall was guaranteed. Basket 3 The cylindrical wall of the third basket (Photograph 5) is plaited once more 4/4 . This time the period of the wall decoration is 2: darkyellow strands and darkbrown strands alternate in both weaving directions (Figure 11). The square bottom of the basket has a rotational symmetry of order 4 (Figure 12). Each quadrant (see Figure 13) presents the structure (4, 4, 4), being the centre constituted by a square of dimensions 8×8. As 4+4+4 = 12 and 12 is a multiple of the period 2, the periodicity of the cylindrical wall was guaranteed. Basket 4 The cylindrical wall of the fourth basket (Photograph 6) is woven 5/5 (see Figure 13). The period of the wall decoration is 10: sets of five darkyellow strands and sets of five darkbrown strands alternate in both weaving directions (Figure 14). The square bottom of the basket has a rotational symmetry of order 4 (Figure 15). Each quadrant presents the structure (4, 5, 5, 5, 54). As 4+5+5+5+54 = 20 and 20 is a multiple of the period 10, the periodicity of the cylindrical wall was guaranteed. Basket 5 The cylindrical wall of the fifth basket (Photograph 7) is woven 5/5. The period of the wall decoration is 5: pairs of darkyellow strands and sets of three darkbrown strands alternate in the perpendicular weaving directions (Figure 16). The square bottom of the basket has a rotational symmetry of order 4 (Figure 17). Each quadrant presents the structure (4, 2, 5, 5, 51). As 4+2+5+5+51 = 20 and 20 is a multiple of the period 5, the periodicity of the cylindrical wall was guaranteed. The lid of basket 6 The cylindrical wall of the lid of the sixth basket (Photograph 8) is woven 5/5. The period of the wall decoration is 2: darkyellow strands and darkbrown strands alternate in both weaving directions. The square bottom of the basket lid has a rotational symmetry of order 4. Each quadrant presents the structure (4, 5, 5, 5, 5) (see Figure 18). As 4+5+5+5+5 = 24 and 24 is a multiple of the period 2, the periodicity of the cylindrical wall was guaranteed. Basket 6 The cylindrical wall of the sixth basket (Photograph 9) is woven 6/6 (see Figure 19). The period of the wall decoration is 2: darkyellow strands and darkbrown strands alternate in the perpendicular weaving directions. The square bottom of the basket has a rotational symmetry of order 4. Each quadrant presents the structure (5, 6, 6, 6, 65); the (5) of (65) corresponds to the localisation of the four corners of the square bottom, indicating that the basket weaver folded five units of each strand upwards when leaving the basket bottom (see Figure 20). As 5+6+6+6+65 = 24 and 24 is a multiple of the period 2, the periodicity of the cylindrical wall was guaranteed. Square bases Figures 21 to 27 display the beautiful rotational symmetry of the visual impression of each of the bases of the six Bora baskets. Concluding remarks The analysis of the decoration of the six baskets showed that the length of the diagonal of the square bottom could not be chosen arbitrarily. The basket maker has to take into account the decoration the artisan intends to introduce on the basket wall. The basket weavers are aware of the fact that the length of the diagonal, expressed in the number of parallel strands to form the bottom, has to be a multiple of the period of the decorative motif on the basket wall. Although the type of decoration is very different from the strip patterns on cylindrical baskets from the Brazilian Amazon discussed in (Gerdes 2004b), the essence of this knowledge is the same: each quadrant of the base produces a quarter of the wall decoration. In the discussed Peruvian case the decoration results from the use of darkbrown and darkyellow strands in both weaving directions on the wall. In the cases analysed from Brazil the decoration, however, stems from the use of darkbrown strand parts in one direction and darkyellow strand parts in the other direction around the cylindrical wall. References Gerdes, Paulus (2003), Geometria y Cestaria de los Bora en la Amazonia peruana, Iquitos (in press) Gerdes, Paulus (2004a), Geometrical aspects of Bora basketry in the Peruvian Amazon, Visual Mathematics (included in this Ebook) Gerdes, Paulus (2004b), Geometric
ornamentation and arithmetic in the Brazilian Amazon: analysis of cylindrical
baskets with a square base, Visual Mathematics (included in this
Ebook)
A square mat with outstanding strand parts Figure 1
Folding slight along the middle line
Interweaving the small square at the ‘top’ side b Figure 2
Square mat with outstanding strand parts a
Four triangles
Folding the triangles upwards c
Final shape of the basket d Diagram Figure 3
4/4 weaving Figure 4
Decoration, with period 4, in zigzags Figure 5
The square bottom of basket 1 Figure 6
One quadrant of the bottom of basket 1 Figure 7 Decoration, with period 4, in zigzags Figure 8 The square bottom of basket 2
One quadrant of the bottom of basket 2 Decoration of period in zigzag The square bottom of basket 3
Regular 5/5 weave Decoration of period 10 The square bottom of basket 4 Decoration of period 5 The square bottom of basket 5 One quadrant of the bottom of the lid of basket 6 The regular 6/6 weave The square bottom of basket 6 Visual impression of the outside face of the base of basket 1 Visual impression of the outside face of the base of basket 2 Visual impression of the outside face of the base of basket 3 Visual impression of the outside face of the base of basket 4 Visual impression of the outside face of the base of basket 5 Visual impression of the outside face of the lid of basket 6 Visual impression of the outside face of the base of basket 6
Some workshop participants analysing basket 1 Six cylindrical baskets with square base (a)
(b) Basket 1
(a)
(b) Basket 2 (a)
(b) Basket 3
(a)
(b) Basket 4 (a)
(b) Basket 5 Lid of basket 6 (a)
(b) Basket 6 Photograph 9
