Paulus Gerdes

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

Abstract

The paper analyses the spiral structure encountered on twill-plaited ipadge baskets among the Makhuwa of Northeast Mozambique and announces a theorem that expresses the number of spirals and the direction into which they turn in terms of the position of the discontinuities in the weaving directions. It concludes with a comparative analysis of similar spiral structures in other cultures.
 
 

Introduction

In the Makhuwa culture circular baskets of various sizes are used. Twill-plaited trays and winnowing baskets of a relatively small diameter (between 25 and 30 cm) are called epadge (Plural: ipadge). The trays consist of a plaited bottom fastened to a circular wooden strip. Most of the ipadge to be analysed in this paper were made in the district of Namapa and I acquired them at markets in the City of Nampula in the period 1999-2001.

Whereas other types of Makhuwa trays may have circular wooden border strips decorated with designs (Gerdes 2003a), those of ipadge are generally left blank. The basic weave used to plait the bottom is the ‘three over, three under’ twill (3/3), producing ‘diagonal zigzag lines’ that make angles of 45 degrees with the weaving directions (see Figure 1). Discontinuity lines may break the basic weaving structure. A discontinuity line marks the reversal of direction of the diagonal zigzag lines. The diagonals on the two sides of a discontinuity line are perpendicular. Along a discontinuity line the strands go over 1, 3, and 5 strands, respectively (see Figure 2). Distinct positions of the discontinuity lines produce different weaving structures and different designs. Although all strands in both weaving directions are of the natural colour (yellow-sand) the weaving structure of an epadge is emphasised by the play of light. In the figures of this paper different colours will be used for the two weaving directions to reinforce the contrast in the designs. 

Photograph 1 presents an epadge with two parallel discontinuity lines. The discontinuity lines may also be perpendicular.
 
 

Concentric square structure

Photograph 2 presents an epadge whereby the discontinuity lines constitute a cross. The cross divides the bottom into four quadrants whereby a design of successive toothed squares around the centre of the texture appears (see Figure 3). 
 
 

Concentric rectangle structure

I encountered two ipadge with a rectangular, non-square design (see Photograph 3). Figure 4 presents the successive toothed rectangles around their centres. Figure 5 displays the position of the discontinuity lines near the centre of the texture of the first epadge. This time the horizontal discontinuity line is broken into two halves, a left and a right semi-axis. Similarly the vertical discontinuity line is broken into an upper and a lower semi-axis. Instead of a cross we have two hooks, one formed by the left horizontal semi-axis and the upper vertical semi-axis and the other by the right horizontal semi-axis and the lower vertical semi-axis. From the left horizontal semi-axis to the right one we have to go six strands downwards; we may say that the vertical displacement of the horizontal semi-axes is 6 units downwards. In the same way, the horizontal displacement of the vertical semi-axes is 6 units to the right. In the case of the second epadge of this type (Photograph 3b), these displacements are 7 units downwards and 7 units to the right, respectively.
 
 

The diagram in Figure 6 summarises the situation of the ipadge with a concentric square and a concentric rectangle structure, respectively.

Normal position

In order to facilitate the comparison of ipadge with four semi-axes, we will consider them always in the same position as the two ipadge we analysed in the previous section: one hook formed by the left horizontal semi-axis and the upper vertical semi-axis and the other hook by the right horizontal semi-axis and the lower vertical semi-axis. If an epadge seen from above is not in this ‘normal position’ (Figure 7), we always can turn it around its centre until it is in the normal position.
 
 

Triple spiral structure

Photograph 4 presents three ipadge with the same global plaiting structure. Figure 8 illustrates the central part. The horizontal displacement of the vertical semi-axes is 8 units to the right, whereas the vertical displacement of the horizontal semi-axes is 2 units downwards. The resulting structure has three spirals: one spiral of two arms and a pair of single-arm spirals (Figure 9).
 
 

Photograph 5 presents another epadge with three spirals. Figure 10 displays its central part. The horizontal displacement of the vertical semi-axes is 9 units to the right, whereas the vertical displacement of the horizontal semi-axes is 3 units downwards. 
 
 

Quintuple spiral structure

Photograph 6 shows two ipadge with the same global plaiting structure. Figure 11 illustrates the central part. The horizontal displacement of the vertical semi-axes is 10 units to the right, whereas the vertical displacement of the horizontal semi-axes is 2 units upwards. The resulting structure has five spirals: one central spiral of two arms, one pair of two parallel arms and two pairs of single-arm spirals (Figure 12).
 
 

Photograph 7 presents another epadge with five spirals. Figure 13 displays its central part. The horizontal displacement of the vertical semi-axes is 9 units to the right, whereas the vertical displacement of the horizontal semi-axes is 3 units upwards.

Photograph 8 presents a fourth epadge with five spirals. Figure 14 displays its central part. The horizontal displacement of the vertical semi-axes is 8 units to the right, whereas the vertical displacement of the horizontal semi-axes is 4 units upwards. This epadge is the only one with a spiral structure for which coloured strands were used. In one of the weaving directions the basket weaver used alternately browned and naturally coloured strands. Figure 15 displays the visual image of this epadge in the normal position.

A common property

When we observe these multiple spiral structures and count the displacements of the semi-axes, we may ask what relationship exists between the number of spirals and the values of the displacements.

We used the normal position as starting point for the measurement of the displacements of the semi-axes. The displacement of the horizontal and vertical semi-axes corresponds to the displacement of the vertex of the hook on the upper left to the vertex of the hook on the lower right. This displacement may be characterised by a pair of numbers (m, n). The first number indicates the horizontal displacement of the vertical semi-axes; the second number indicates the vertical displacement of the horizontal semi-axes. The whole numbers m and n can be positive, zero or negative.

The pair (0,0) corresponds to the ipadge that present a structure of concentric toothed squares. The pairs (6, -6) and (7, -7) describe the ipadge in Photograph 3 with a structure of concentric toothed rectangles. The pairs (8, -2) and (9, -3) describe the structure of the ipadge in Photographs 4 and 5 with three spirals, whereas the pairs (10, 2), (9, 3) and (8, 4) describe the structure of the ipadge in Photographs 6, 7 and 8 with five spirals.

We may notice that in the ipadge observed we have the general following regularity: 

Extrapolation on the basis of the data obtained from the analysis of ipadge may lead us to the formulate the following hypothesis:

The reader is invited to prove the theorem (see Gerdes 2004).

Analogously, for right-turning spirals, it holds that

Comparative analysis

So far I did not find a spiral weaving structure on other Makhuwa circular baskets. In particular, I did not see it on any of the mavuku containers analysed in (Gerdes 2003). Although I have not seen any baskets with a spiral design in the collection of Mozambique’s National Ethnographic Museum in the City of Nampula, I had the opportunity to analyse one in the Museum of the Institute of Anthropology of the University of Coimbra (Portugal). Object M.L.A. 249 is a plaited container of the mavuku type. Its lid presents a triple spiral structure (Figure 16) with (m, n) = (0, -6), whereas its base has a double spiral structure (Figure 17) with (m, n) = (0, -3). Figure 18 presents the visual impression of the base if different colours would have been used in the two plaiting directions. The 1955 catalogue informs that Lacmichaud Primogy who lived on Mozambique Island on the coast of today’s Nampula Province gave the basket container in 1896 (Pacheco de Amorim, p. 377). More recent catalogues suppose that a Makonde basket weaver made the container (Rodrigues Martins, p. 26; Marques Abreu, p. 119). The Makonde are the northern neighbours of the Makhuwa. Whether Makonde or Makhuwa, it is sure that plaited, triple and double spiral structures were known in the 19th century in the Northeast of Mozambique.

In (Gerdes 2000, pp. 151-175) the only other plaited, circular baskets with a central spiral structure I saw so far are analysed. Anasazi basket makers produced them between 1100 and 1400 (Morris & Burgh). The Anasazi from whom the Hopi descent lived in North America, in today’s states of Utah, Colorado, Arizona and New Mexico. The exceptional climatic conditions of their habitat made the preservation of some of their baskets possible, four of them with a spiral structure: two with a triple, one with a quintuple and one with a double spiral structure. Figure 19 displays their central parts. The first one is left turning [(m, n) = (6, 0)] while the others are right turning [(m, n) = (4, -10), (3, -15), (3, -6)]. Each time the relationship between (m, n) and the number and orientation of the spirals is in agreement with the above mentioned theorems.

It seems to me very interesting that basket weavers belonging to cultures from North America and Southeast Africa invented similar central spiral structures on circular baskets. In (Gerdes 1990, 2003b) I present some reflections relative to the similarity of geometric shapes in diverse human cultures.

Structures that present a series of interlocked plaited spirals appear in other cultural environments. Figure 20 presents details of two baskets from the Desana (Colombia) and the Yekuana (Venezuela). Around the local spiral centres, the relationships between the displacements of the discontinuity lines (segments) and the number of concentric spirals are the same before.
 

Note: A one-spiral structure?

We saw that a structure of concentric toothed squares or rectangles corresponds to m+n = 0. If the theorem is applicable, it follows that S-1 = 0. In other words, we are in the ‘extreme’ situation of only one spiral. As zero is neither positive nor negative, the spiral turns neither left or rightwards. In this sense we might call a weaving structure of concentric toothed squares or rectangles a one-spiral structure. 
 
 

References