Nested toothed square designs on twill plaited mats and baskets

Paulus Gerdes

Mozambican Ethnomathematics Research Centre, C.P. 915,
Maputo, Mozambique


The paper introduces the concept of nested toothed square and of double nested toothed square designs. It presents an overview and examples of the use of these designs by mat and basket weavers from around the world

In this paper I present some examples of nested toothed squares as they appear on twill plaited mats and baskets from several cultures around the world. Photographs 1 and 2 present examples. A nested toothed square will be defined as a regularly built up toothed square whose centre is composed of a square number of equally sized and regularly placed toothed squares. As the centre may be considered part of a plane pattern, the notation (p,q,r,s) may be used to represent the respective regular square plane pattern; the centre itself may be designated by t × (p,q,r,s), whereby t is the square number of toothed squares involved. In this way, the nested toothed square may be designated by the notation for a toothed square [x,y,z], being the x substituted by the notation for its centre. In other words, the following notation for a nested toothed square may be used: [t × (p,q,r,s), y, z]. For example, [4 × (1,2,2,3),2,2] represents the toothed square in Figure 1a, observed on a plaited purse from Oaxaca (Mexico) in the authorís collection:

In Table 1 I present the nested toothed squares on twill plaited mats and baskets I had the chance to observe so far.

Table 1

Nested toothed squares

Class Country People Source
[4 × (1,2,2,3),2,2] Mexico [Oaxaca] pg Mexico City Nov 2000
[4 × (1,2,2,3),14,2] China   pg Athens (USA) 1997
[4 × (1,2,3,3),2,3] Thailand   LaPlantz 87
[4 × (3,1,0,3),1,3] Brazil [Amazonia] pg São Paulo [Brazil] Nov 1999
  Cameroon Bamileke Gerdes [1] 33, Guidoni 127
[4 × (3,2,3,3),3,3] USA Cherokee Mowat 12
[4 × (3,3,3,3),6,3] Brazil [Amazonia] pg São Paulo [Brazil] Nov 1999
[9 × (1,2,2,3),1,3] China   pg Athens GA 1997 (see Photo 1)
[9 × (1,2,3,3),2,3] Angola Chokwe Bastin 116
[9 × (3,1,0,3),2,3] USA Chitimacha Mather 82
[9 × (3,1,0,3),4,3] Cameroon Bamileke Gerdes [1] 33, Guidoni 127
[16 × (1,2,2,3),1,2] Mexico [Oaxaca] pg Mexico City July 2001

Sometimes double nested toothed squares may be observed as in the following example from the Brazilian Amazon (Photograph 3 and Figure 3). This double nested toothed square is based on the nested toothed square [4 × (3,1,0,3),1,3] (Figure 2).

[4 × (1,2,2,3),2,2]

Visual image

Oaxaca, Mexico
Figure 1

[4 × (3,1,0,3),1,3]
Figure 2

Double nested toothed square design
Brazilian Amazon
Figure 3


Photograph 1: Inner part of a purse from Oaxaca (Mexico) (authorís collection)

Photograph 2: Circular basket from China (authorís collection)

Photograph 3: Circular basket from the Brazilian Amazon (authorís collection)


pg: Authorís collection


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Guidoni, Enrico, Primitive Architecture, Rizzoli, New York, 1987

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