— African ---- — Fractals---- MODERN COMPUTING AND INDIGENOUS DESIGN R O N E G L A S H Contents Acknowledgments ix PART I Introduction CHAPTER J Introduction to fractal geometry 3 CHAPTER 2 Fractals in African settlement architecture 20 CHAPTER 3 Fractals in cross-cultural comparison 39 CHAPTER 4 Intention and invention in design 49 PART II African fractal 7nathematics CHAPTER 5 Geometric algorithms 61 CHAPTER 6 Scaling 71 CHAPTER 7 Numeric systems 86 / CHAPTER 8 Recursion 109 CHAPTER 9 Infinity 147 CHAPTER IO Complexity 151 v iii Contents PART III Implications Ch a p t e r 11 Theoretical frameworks in cultural studies of knowledge 179 c h a p t e r 12 The politics of African fractals 192 c h a p t e r 13 Fractals in the European history of mathematics 203 c h a p t e r 14 Futures for African fractals 216 a p p e n d i x Measuring the fractal dimension of African settlement architecture 23 1 N otes 235 References 243 Index 253 •Acknowledgments Thanks to go first to my wife, Nancy Campbell, who has tolerated my obsessions with grace, and to Evelyn, Albert, and Joanne Eglash, who inspired many of them. I am grateful for the assistance of my professors at UCSC: Ralph Abraham, Steve Caton, James Clifford, Donna Haraway, Gottfried Mayer-Kress, Ken Norris, Heinz- O uo Peiigen, Carolyn Martin Shaw, and Patricia Zavella. Equally important were my fellow UCSC graduate students, in particular David Bain, Julian Bleecker, Peter Broadwell, Kirby Bunas, Claudia Castenada, Giovanna Di Chiro, Joe Dumit, Vincente Diaz, Paul Edwards, Linda Garcia, Jennifer Gonzales, Chris Gray, John Hartigan, Sharon Helsel, Laura Kang, Lorraine Kenny, Matthew Kobbe, Angie Rosga, Warren Sack, Meg Satterthwaite, Sandy Stone, Marita Sturken, Bemt Wahl, and Sarah Williams. Thanks also to Billie Harris, Miranda Hays, Rebecca Lyle, Ken Martin, Sheila Peuse, Adolph Smith, Joshua Weinstein, and Paul Yi. Research funding from the Institute for Intercultural Studies and the Ful- bright program made possible my fieldwork in west and central Africa. As chap­ ter 10 makes clear, 1 owe much to my Senegalese colleagues, Christine Stna Diatta and Nfally Badiane. Also of great help in Senega! were Abdouli Ba, Real Basso, Charles Becker, Kolado Cisse, Ibnou D'tagne, Pathe Diagne, Souleymane Bachir Diagne, Mousse Diop, Waly Coly Faye, Max, Marie-Louise Moreau, Margot Ndiaye, Victor Sagna, Ousinan Sen, Fatou Sow, Yoro Sylla, Sakir Thaim, and Riene Acknowledgments Toje. From the West African Research Center I received the expert advice of American professors Eileen Julien and Janis Mays. Thanks also to Shamira Johnson, Paul and Betsey Harney, Jane Hale,'Lisa McNee, and Liz Mermin. I am also grateful to Issiaka Isaac Drabo and the brilliant Canadian pho­ tography team, Michel et Didi, in Burkina Faso. Thanks also to Amadou Coulibaly, Kalifa Kon£ and Abdoulaye Sylla in Mali. In Cameroon I received the generosity of Ireke Bessike, Ngwa Emmanuel, Noife Meboubo, the late Engel­ bert Mveng, and Edward Njock. My work in Benin would not have been possible without the assistance of Tony Hutchinson; thanks also to Knke Alfred, Natheli Roberts, and Martine de Sousa for their expertise in vodun. In Ghana Michael Orlansky graciously introduced me to the many cultural resources available. Many of the local folks 1 spoke to in west and central Africa, while extending great gen­ erosity and enthusiasm, asked that their names remain unrecorded, and I thank them as well. On my return to the United States 1 received a fellowship from the Cen­ ter for the Humanities at Oregon State University, which also offered the oppor­ tunity to work with anthropologists Joan Gross, David Gross, and Cort Smith, as well as Kamau Sadiki from the Portland Black Educational Center. T hanks also to Michael Roberson for bis geometry advice, and David and Barbara Thomas (now math teachers at Hendersonville High, North Carolina) for investigating owari patcerns. A two month fellowship at the University of Oregon got me through the summer, and into my current position at The Ohio State Univer­ sity. Here I have been thankful for help from Patti Brosnan, Wayne Carlson, Jacqueline Chanda, Cynthia Dillard, David Horn, Lindsay Jones, Okechukwu Odita, Egondu Rosemary Onyejekwe, Robert Ransom, Dan Reff, Rose Kapian, Carolyn Simpson, Daa’iyah I Saleem, Jennifer Terry, Cynthia Tyson, and Manjula Waldron. There are also many collegues, geographically scattered, whose feedback has been invaluable. In particular 1 would like to thank Madeleine Akrich, Jack Alexander, Mary Jo Arnoldi, George Arthur, Marcia Ascher, Jim Barta, Silvio Bedini, TQ Berg, Jean-Paul Bourdier, Geof Bowker, Michael T. Brown, Pat Caplin, Brian Casey, Jennifer Croissant, Don Crowe, Jim Crutchfield, Ubiratan D'Ambrosio, Ronald Bell, Osei Darkwa, Marianne de Laet, Gary Lee Downey, Munroe Eagles, Arturo Escobar, Florence Fasanelli, James Fernandez, Marilyn Frankenstein, Rayvon Fouche, Paulus Gerdes, Chonat Getz, Gloria Gilmer, David Hakken, Turtle Heart, Deborah Heath, David Hess, Stefan Helmreich, Dar- ian Hendricks, David Hughes, Sandy Jones, Esmaeli Knteh, Roger P. Kovach, Gelsa Knijnik, Bruno Latour, Murray Leaf, Bea Lumpkin, Robin Mackay, Caro! Malloy, Benoit Mandelbrot, Mike Marinacci, Joanna Masingiia, Lynn McGee, James Ac/cnowledgnien is Morrow, David Mosimege, Brian M Murphy, Diana Baird N ’Diaye, Nancy Nooter, Karen Norwood, Spurgeon Ekundayo Parker, Clifford Pickover, Patricia Poole, Arthur Powell, Dean Preble, Dan Regan, J i'm Rauff, Sal Restivo, Pierre Rondeau, John Rosewall, Rudy Rucker, Nora Sabelli, Jnron Sampson, Doug Schuler, Patrick (Rick) Scott, Rob Shaw, Enid Schildkrout, David Williamson Shaffer, Larry Shirley, Dennis Smith, George Spies, Susan Leigh Star, Paul Stoller, Peter Tay­ lor, Agnes Tuska, Gary Van Wyk, D onnell W alton, D M W arren, Dorothy W ash­ burn, Helen Watson-Verran, Mark W. Wessels, Patricia S. Wilson, and Claudia Zaslavsky. Last and not least, thanks to my editors at Rutgers, Doreen Valentine and Martha Heller. PART -Introduction -----------------— CHAPTER Introduction ------------------------------------------- to --------------------------- :------------------- — fractal------------------------------------------------ —geometry -----------------:---------------------------- Fractal geometry has emerged as one of the most exciting frontiers in the fusion between mathematics and information technology. Fractals can be seen in many of the swirling patterns produced by computer graphics, and they have become an important new tool for modeling in biology, geology, and other nat­ ural sciences. While fractal geometry can indeed take ur into the far reaches. of high-rech science, its patterns are surprisingly common in traditional African designs, and some of its basic concepts are fundamental to African knowledge systems. This book will provide an easy introduction to fractal geometry for people without any mathematics background, and it will show how these same categories of geometric pattern, calculation, and theory are expressed in African cultures. Mathematics and culture For many years anthropologists have observed that the patterns produced in dif­ ferent cultures can be characterized by specific design therr.es. In Europe and A m er­ ica, for example, we often see cities laid out in a grid pattern of straight streets and right-angle corners. Another grid, the Cartesian coordinate system, has long been a foundation for the mathematics used in these societies. In many works Introductton of Chinese art we find hexagons used with extraordinary geometric precision— a choice that might seem arbitrary were it not for the importance of the num­ ber six in the hexagrams of their fortunetelling system (the I Ching), in the anatomy charts for acupuncture (liu-qi or “six spirits”), and even in C hinese architecture.1 Shape and number are not only the universal rules of measurement and logic; they are also cultural tools that can be used for expressing particular social ideas and linking different areas of life. T hey are, as C laude Levi-Strauss would put it, “good to think with.” Design themes are like threads running through the social fabric; they are less a commanding force than something we command, weaving these strands into many different patterns of meaning. The ancient Chinese empires, for example, used a base-io counting system, and they even began the first univer­ sal metric system.^ So the frequent use of the number 60 in Chinese knowledge systems can be linked to the combination of this official base 10 notation with their sacred number six. In some A m erican cities we find that the streets are num ­ bered like Cartesian coordinates, but in others they are named after historical figures, and still others combine the two. These city differences typically corre­ spond to different sociat meanings— an emphasis on history versus efficiency, for example. Suppose that visitors from another world were to view the grid of an American city. For a city with numbered streets, the visitors (assuming they could read our numbers) could safely conclude that Americans made use of a coordi- ■ nate structure. But do these Americans actually understand coordinate mathe­ matics? Can they use a coordinate grid to graph equations? just how sophisticated is their mathematical understanding? In the following chapter, we will find our­ selves in a similar position, for African settlement architecture is filled with remark­ able examples of fractal structure. Did precolonial Africans actually understand and apply fractal geometry? As I will explain in this chapter, fractals are characterized by the repeti­ tion of similar patterns at ever-diminishing scales. Traditional African settle­ ments typically show this “self-similar” characteristic: circles of circles of circular dwellings, rectangular walls enclosing ever-smaller rectangles, and streets in which broad avenues branch down to tiny footpaths with striking geo­ metric repetition. The fractal structure will be easily identified when we com­ pare aerial views of these African villages and cities with corresponding fractal graphics simulations. W hat are we to make of this comparison? Let’s put ourselves back in the shoes of the visitors from another planet.