INTRODUCTION    Paulus Gerdes Mozambican Ethnomathematics  Research Centre, C.P. 915,  Maputo, Mozambique pgerdes@virconn.com     The book Basketry, Geometry, and Symmetry in Africa and the Americas deals with the geometry and symmetries of designs and patterns created by mat and basket weavers in several African and American cultures. The type of basketry discussed is twill basketry (for an introduction to twill basketry, see LaPlantz, 1993). Photograph 1 illustrates a twill woven basket from Oaxaca (Mexico). Twilling consists of the interweaving of two elements of equal thickness and flexibility, whereby at least some strands pass over or under two or more strands in the opposite direction. If all strands go over and under the same number of elements, for instance over two under two (notation 2/2), over three under three (3/3), the weaving is called a balanced (or regular) twill (see Figure 1).  Throughout the book the strands in one direction will often be coloured red in the figures and those in the opposite direction green. The real colours may vary from culture to culture. The representation by red and green guaranties, that the patterns are also visible on a black-and-white printout of the figures. Chapters 1 to 4 analyse aspects of twill woven, circular trays from the Peruvian Amazon and from several regions of Mozambique. The chapters are complementary to the earlier book, entitled The Circle and the Square. Geometric, Artistic and Symbolic Creativity of Basket Weavers from Africa, the Americas, Asia and Oceania (Gerdes, 2000a). Photograph 2 illustrates a twill woven, circular tray from the South of Mozambique. Chapter 1 is a translation of a paper published in the Brazilian Journal for the History of Mathematics (Gerdes, 2003e). Chapter 2 is a reproduction of a paper published in the journal Symmetry: Culture and Science (Gerdes, 2003d).  Chapters 5 to 7 analyse plaited strip and plane patterns, and the colour transformation of plaited strip patterns by Tonga women from the Southeast of Mozambique. The chapters are complementary to the book Sipatsi: Basketry and Geometry in the Tonga culture of Inhambane (Gerdes, 2003a) and the paper (Gerdes, 2003b). Part of chapter 5 was earlier published in the journal Symmetry: Culture and Science(Gerdes, 2003b). Photograph 3 presents an example of a gipatsi handbag. Chapters 8 and 9 analyse cylindrical baskets with a square base from the Brazilian and Peruvian Amazon. Chapter 8 is a translation of a paper originally published in the journals Bolema and Quipu (Gerdes, 1989a, b). Photograph 4 illustrates a cylindrical basket with square base from the Peruvian Amazon. Chapter 10 presents some examples of design transformation. The examples are selected from seventeenth century Kongo baskets. Also from the Kongo cultural area are the Yombe mats analysed in Chapter 11. The chapter deals with twill plaited patterns with spirals, and complements earlier studies of Yombe mats (Gerdes, 1999 [pp. 126-137], 2000b, 2004a) and the spiral patterns may be compared with the central spiral designs interwoven in circular basket trays among the Makhuwa (Mozambique) analysed in Chapter 3. Figure 2 gives an example of a Yombe spiral pattern. Chapter 12 introduces the theme of colourings of regular twill plaited mats and baskets. It presents examples from Mexico and Mozambique, and continues the earlier analysis of a particular type of colourings of Tonga handbags discussed in Chapter 5. Photograph 5 presents an example of a colouring from the Makhuwa in Northeast Mozambique. Chapters 13 and 14 present two short papers on mats from the Mbole in Northeast Congo and on designs that appear on twill plaited baskets and mats from Angola. Chapter 15, divided into four parts, presents four classes of twill-plaited designs and patterns with sets of concentric toothed squares. Examples of their use and distribution around the world are included (cf. also Gerdes, 2002). Photograph 6 presents an example of a pattern with sets of concentric toothed squares. The book is intended as an introduction to the study of geometrical and symmetrical aspects of twill basketry. It is hoped that the studies presented may stimulate further research both of more cultural contexts and in the sense of broadening and deepening the scope of analysis. I dedicate the book to the women and men, who create, as mat and basket weavers, the beautiful designs and patterns, that may fascinate many people, including mathematicians.  I thank all my family members, friends and colleagues who in one way or another have stimulated me to collect mats and baskets, to study baskets and to write the book.  In particular, I thank Slavik Jablan, the editor of Visual Mathematics, Art and Science Electronic Journal, for his continuous interest in my research and the publication of the book in electronic form by Visual Mathematics.     Figure 1 Figure 2 Photograph 1 Photograph 2 Photograph 3 Photograph 4 Photograph 5 Photograph 6 References Gerdes, Paulus (1989a), Sobre aritmética e ornamentação geométrica. Análise de alguns cestos de índios do Brasil, BOLEMA Especial, Universidade Estadual Paulista, Rio Claro, No.1, 11-34. ___ (1989b), Sobre aritmética e ornamentação geométrica. Análise de alguns cestos de índios do Brasil, QUIPU: Revista Latinoamericano de Historia de las Ciencias y la Tecnología, Mexico City, Vol. 6, 171-187. ___ (1999), Geometry from Africa: Mathematical and Educational Explorations, The Mathematical Association of America, Washington DC, 210 pp. ___ (2000a), Le cercle et le carré: Créativité géométrique, artistique, et symbolique de vannières et vanniers d’Afrique, d’Amérique, d’Asie et d’Océanie, L’Harmattan, Paris, 301 pp. ___ (2000b), Gerade und Ungerade – Zu einigen mathematischen Aspekten der Mattenflechterei der Yombe-Frauen am unteren Kongo, in: Jürgen Blankenagel & Wolfgang Spiegel (Eds.), Mathematikdidaktik aus Begeisterung für die Mathematik. Festschrift für Harald Scheid, Ernst Klatt Verlag, Stuttgart, 83-93. ___ (2002), Twill-Plaited Octagonal Designs, Visual Mathematics, 4(3). ___ (2003a), Sipatsi: Cestaria e Geometria na Cultura Tonga de Inhambane, Moçambique Editora, Maputo, 176 pp. ___ (2003b), Exploring Plaited Plane Patterns among the Tonga in Inhambane (Mozambique), Symmetry: Culture and Science, Vol. 12, No. 1-2, Budapest, 115-126. ___ (2003c), Plaited strip patterns on Tonga handbags in Inhambane (Mozambique) – An update, Visual Mathematics, 5(1) ___ (2003d), Symmetry-Geometry Aspects of Mavuku Baskets among the Makhuwa (Mozambique), Symmetry: Culture and Science, Vol. 12, No. 1-2, Budapest, 87-114. ___ (2003e), Níjtyubane — Sobre alguns aspectos geométricos da cestaria Bora na Amazónia peruana, Revista Brasileira de História da Matemática, Vol. 3, No. 6, 3-22. ___ (2004a), Symmetries on mats woven by Yombe women from the Lower Congo area: On the interplay between cultural values and mathematical-technical possibilities, in: Dorothy K. Washburn & Donald W. Crowe (Eds.), Symmetry Comes of Age, The Role of Pattern in Culture, University of Washington Press, Seattle, 81-99. LaPlantz, Shereen (1993), Twill Basketry, A Handbook of Designs, Techniques, and Styles, Lark Books, Asheville NC, 144 pp.