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TWILL-PLAITED OCTAGONAL DESIGNS
 

Paulus Gerdes

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







Abstract

Well known in many variations — from Egypt to Mexico, from Norway to China, from Turkey to the USA, from Morocco to New Zealand, from Indonesia to Ghana, from Senegal to Brazil — is a decorative motif in the form of an octagonal star or an octagonal flower. 

The octagonal design motif appears woven on textiles, knotted on carpets, laid-in on wood, knitted on pullovers, sewn on quilts, cut out in leather, embroidered on cushions, etc. It appears also on twill-plaited mats and baskets. In this paper, I will present the examples of octagonal designs I encountered so far on twill-plaited mats and baskets. I will propose the hypothesis that one of the possible cultural-historical roots of the decorative motif in the form of an octagonal star — that is one of the possible roots of the imagination the motif embodies — may lie in twill plaited toothed square designs. The paper will be concluded with the presentation of other types of octagonal designs and structures in basketry and a second hypothesis.

Dedication

The paper is dedicated to the late Stieg Mellin-Olsen (1939-1995). He encouraged the study of what he called folk mathematics (cf. Mellin-Olsen, 1986). At the end of the seminar ‘Mathematics and Culture’ he organized in September 1985 in Bergen (Norway), he gave me a knife with a leather cover in which an octagonal star was cut out (Figure 1). In Norway the design is called the ‘eight leaves rose’ and appears also on baskets, dresses, woven or embroidered on tablecloth (see Photograph 1), and in traditional knitting. It is found also on decorations on bone and wood dating back to the age of the Vikings (information from Nora Linden, Stieg’s wife). Stieg asked me where could lie a possible origin of this idea of the octagonal star. I promised him to reflect about the question. The time has come to try to fulfill the promise. 

Photograph 1: Norwegian table cloth

In the books Ethnogeometrie (Gerdes, 1991a), Sobre o Despertar do Pensamento Geométrico (Gerdes, 1991b) and The Awakening of Geometrical Thought in Early Culture (2002), I analyse the origin of several other geometrical concepts in cultural activities.
 
 

Design cut out in leather (Norway)

Figure 1

Textile motif among the Glaoua in the Atlas mountains (Morocco)
[Drawn after Samama p. 50]
a
Quilt design (USA)
[Drawn after Horwitz p. 140.]
b

Figure 2
 

Inlay work on wooden eggs and boxes (Brazil)

a

Inlay work on a wooden box (Egypt)

b

Leather pieces sewn on a hassock (Egypt)

c

Motif on knotted carpets (Turkey)

d 1

Motif on knotted carpets (Turkey)

d 2

Design embroidered on a cushion (Kashmir, India)

e

Some octagonal stars and flowers on objects in the author’s collection

Figure 3




Examples of octagonal designs

Figure 3 presents some octagonal designs on objects in the author’s collection (Brazil, Egypt, India, and Turkey). Figure 4 presents some further design motifs from West India. Figure 5 displays some ‘Kairouan’ and Bedouin knotted carpets motifs design from Tunisia. Figure 6 presents some examples of octagonal designs motifs on Mexican Indian textiles.

a
b

Design motifs from Rajasthan (West India)
[Drawn after Barnard p. 141 and Lehri p. 53. 
Further examples may be seen in Lehri, pp. 56, 101, 103, 131; 
Mookerjee, pp. 15, 121, 127, 141, 160, 166, 183, 190, 194]
Figure 4

       

Carpet designs motifs from Tunisia
[Drawn after Revault pp. 16, 16, 20 ("Kairouan") and 35 (Bedouin)]
Figure 5

Huasteco

a

Nahua

b

Zapotec

c

 
Nahua

d

Chatino

e

 

Design motifs on Mexican Indian textiles
[Drawn after Weitlanger-Johnson pp. 35, 52, 64, 85]
Figure 6

Although the designs from Tunisia and Mexico in Figures 5 and 6 are from carpets and textiles, they may be realised by plaiting mats. The same is true for the octagonal flower designs on textiles from Thailand in Figure 7 and on the coil Yakima basket from the USA in Figure 8. Could it be that they had their origin in the twill plaiting of mats and that the forms were transposed to other materials? And once transposed, could they be liberated from the restrictions imposed by the original context of plaiting in two orthogonal directions, and assume new forms invented by creative artisan-artists?

Detail of a bed sheet (Nan province, North Thailand)

a

Detail of a bridal bed sheet (Phichit province, Central Thailand)

b

Design on a pillow cover

(north-east Thailand)

c

Octagonal flower designs on textiles from Thailand 
[Drawn after photographs in Conway pp. 46, 127, 132]
Figure 7

Design on a Yakima gathering basket collected in 1940 in Toppenish, Washington
[Drawn after Wyckoff p. 100; see also p. 97]
Figure 8

The second Thai octagonal flower has a diameter of 17 strands

Figure 9




Catalogue

Photograph 2 shows a twill-plaited box in the author’s collection. It comes from China. On its cover appears a toothed square with an octagonal star. Its diameter measures 33 strands. As measure of the diameter we may take the number of vertical (or horizontal) strands needed to plait one copy of the octagonal star design (see the example in Figure 9). Photograph 3 shows a mat from the Tabaru (Halmahera, Indonesia).

Photograph 2: Chinese basket (author’s collection)

Photograph 3: Mat from Halmahera, Indonesia [reproduced from Bezemer 1931, p. 23]

Table 1 presents a list of the toothed squares with octagonal stars I so far found on twill-plaited mats and baskets. These toothed squares come from China, Fiji, Indonesia, New Zealand, Thailand, Tonga, and North America. They are illustrated in Figure 10

a 1
a 2

Design from Sulawesi
[Drawn after Jasper Pl. 57]



 
 

b 1
b 2

Design from Flores
[Drawn after Bossert Pl. 21:7]



 
 

c 1
c 2

Design on a Cherokee basket
[Drawn after Gettys p. 30; Hill p. 311]



 
 

d 1
d 2

Design from the Tebida-Dayak
 [Drawn after Nieuwenhuis T. XV; Jasper Pl. 36, 40; Rooyen pp. 50, 61; Tillema p. 185]



 
 

e 1
e 2

Plaiting design from Sulawesi called ‘permata saogoe’
[Drawn after Jasper p. 236; Rooyen p. 40]



 
 

f 1
f 2

Plaiting design from Sumatra called ‘kembang manggis’
[Drawn after Jasper p. 219; Rooyen p. 37]



 
 

g 1
g 2

Design on a Maori mat
[Drawn after Pendergrast 1984, pp. 126, 127, 130, 131]



 
 

h 1
h 2

Design on a Maori mat
[Drawn after Pendergrast 1984, pp. 128, 129]



 
 

i 1
i 2

Design on a basket from Suva (Fiji)
 [Drawn after LaPlantz p. 138]



 
 

j 1
j 2

Design on a mat from Halmahera
[Drawn after Bezemer p. 19]



 
 

k 1
k 2

Design from Sumatra
[Drawn after Hasselt Pl. LXXVI]


l 1
l 2

Design on a mat from Bali
[Drawn after Eiseman p. 148 h]



 
 

m 1
m 2

Design on a Cherokee rivercane market basket, ca. 1950
[Drawn after Duggan p. 38]



 
 

n 1
n 2

Design on a Chitimacha basket
[Drawn after Mather p. 82]



 
 

o 1
o 2

Design on the bottom of a market basket from Tongatapu
[Drawn after Arbeit Photo 22b, Gerdes 2000, p. 279]



 
 

p 1
p 2

Design on a mat from Halmahera
[Drawn after Bezemer p. 19]



 
 

q 1
q 2

Design on a Cherokee basket
[Drawn after Horwitz p. 100]


r 1
r 2

Design on a Chinese basket
[Drawn after an object in the author’s collection]



 
 

s 1
s 2

Design on a mat from Halmahera
[Drawn after Bezemer p. 23]



 
 

t 1
t 2

Design on a mat from Halmahera
[Drawn after Bezemer p. 23]



 
 

u 1
u 2

Design on a mat from Halmahera
[Drawn after Bezemer p. 23]



 
 

v 1
v 2

Design on a mat from Halmahera
[Drawn after Bezemer p. 26]



 
 

w 1
w 2

Design on a mat from Halmahera
[Drawn after Bezemer p. 26]



 
 

x 1
x 2

Design on a fan from Thailand
[Drawn after Sentance 194]



 
 

y 1
y 2

Design on a basket from Madagascar
[Drawn after Jewell 89]

Catalogue of twill-plaited octagonal stars

Figure 10
 

Table 1

 
Diameter
Country Region People 
a
13
Indonesia Sulawesi  
b
17
Indonesia Flores  
c
21
USA North Carolina Cherokee
d
23
Indonesia Borneo Tebida-Dajak
e     Sulawesi Toradja
f     Sumatra  
g
25
New Zealand   Maori
h   New Zealand   Maori
i
27
Fiji Suva  
j   Indonesia Halmahera Tabaru
k     Sumatra  
l     Bali  
m   USA Georgia, Tennessee, North Carolina Cherokee
n     Louisiana Chitimacha
o
28
Tonga Tongatapu  
p
29
Indonesia Halmahera Tabaru
q   USA Georgia, Tennessee, North Carolina Cherokee
r
33
China    
s   Indonesia Halmahera Tabaru
t     Halmahera Tabaru
u     Halmahera Tabaru
v
39
Indonesia Halmahera Tabaru
w     Halmahera Tabaru
x
41
Thailand    
y
51
Madagascar    

First Hypothesis

Let me now try to sketch a hypothesis of how these and other octagonal star and flower motifs might have been inventively imagined in the context of twill plaiting. Consider a nested toothed square composed of, for example, 9 smaller equal sized toothed squares (schematically represented in Figure 11a). Reflecting about such a nested toothed square, an artisan may imagine a toothed square with equally sized toothed squares at its corners (see Figure 11b).

Now raises the question of how to "fill up" the rest of the initial toothed square. The artisan-geometer may start with the weaving of triangles on the sides of the larger toothed square (see the schema in Figure 11c). 

The next step in the creative process might be ‘making visible’ in one way or another the diagonals and midlines of the larger toothed square (Figure 12a) or by extending the right angle sides of the triangles (Figure 12b), leading to the imagination of the octagonal designs in Figure 13.

a
b
c

Schematic representation of the first phase in the genesis of octagonal star designs

Figure 11

a
b

Schematic representation of the second phase in the genesis of octagonal star designs

Figure 12

a
b

Octagonal design structures

Figure 13

The process described is a reconstructive sketch of a possible cultural-historical genesis of some octagonal forms that then may be interpreted as stars or flowers or whatever. And looking to the examples of twill plaiting and historically younger decorative contexts as textiles, carpet knotting, wood work, that are presented in this paper, artisans from diverse parts of the Earth were engaged in comparable activities of imagination, application and interpretation.
 

Other cultural contexts

This hypothesis should not be considered absolute and the only possible. Other cultural contexts may have stimulated the emergence of the visualisation and conceptualisation of octagonal design motifs. 

These other cultural contexts might even be other instances of basket weaving. Photograph 4 shows two coiled baskets from Swaziland. Photograph 5 shows an octagonal star interlaced in the way as has traditionally been done on Malaita in the Solomon Islands (see Pendergrast, 1987, p. 44). Maria Dedò, professor of mathematics at the University of Milan (Italy), showed me how she had learned as a child to interlace another octagonal star (see Photograph 6). Photograph 7 presents an interlaced octagonal buttom on a Copi basket from the South of Mozambique.

Photograph 4: Coiled basket from Swaziland (author’s collection)


Photograph 5: Interlaced octagonal star (Solomon Islands)


Photograph 6: Interlaced octagonal star (Italy)


a: Octagonal knot on a Copi basket (author’s collection)
 

b: Octagonal knot made out of coloured card board strips

Photograph 7

Also other types of twill-plaited octagonal designs are possible. As a first example, Figure 14 displays the centre of a circular plate I bought in the 1982 at a market in Nairobi (Kenya). 

Centre of a circular plaited tray (Kenya)

Figure 14

Second hypothesis

Figure 15 displays a traditional octagonal-circular design on Chitimacha baskets (Louissiana, USA). If we look the the distribution of the small unit squares, i.e. the places where the horizontal strands jump over only one vertical strip, the regular octagonal structure is very visible (Figure 16). To go from one unit square to the next, one moves like a horse on a chess board. 


Chitimacha design
[Drawn after Mason p. 167; Hoover p. 63; Taylor p.18]

Figure 15

 

Distribution of unit squares

Figure 16

Figure 17 shows an octagonal design on a mat from Aceh (North Sumatra, Indonesia). A ‘chess-horse-distribution’ of unit squares is also visible on several designs from Borneo (see Figure 18). 


Design from Aceh (North Sumatra, Indonesia)

Figure 17

Design from the Kabahan Dayak
[Drawn after Van E. p. 30; Nieuwenhuis Pl. XIV; basket in the author’s collection]
a
Design from Central Borneo
[Drawn after Tillema p. 184]
b

Design from Borneo
[Drawn after Jasper Pl. 50; basket in the author’s collection]
c

Figure 18

Photograph 8 displays part of a basket from Borneo, where in each woven triangle (Figure 19) a unit square, a toothed square (Figure 20) and an octagonal design (Figure 18a) appear together. This particular situation suggests a second hypothesis: the emergence of octagonal designs from squares via toothed squares.


 

Photograph 8: 
a and b: Details of baskets from Borneo (author’s collection)
 


Plaited triangle on basket from Borneo

Figure 19

Toothed square

Figure 20



References

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Barnard, Nicholas (1994), Artisans de l’Inde, Flammarion, Paris

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