Africa + Mathematics: Summary

Origins of the Illustrations

Drawings made by the author are not mentioned separately.

Number Description Origin 3 The round universe … Reproduced with approval of Acquier Jean-Louis. 6 Hawkins’ poster “African Reproduction with approval of author Bill Hawkins: and African-American [email protected]. Pioneers in Mathematics.” 7 Evolution of ground plans Reproduced with approval of Acquier Jean-Louis. in Burundi 9 Title page of a text about From old journals held at Africa Museum Tervuren oral history in Rwanda. (Belgium). 12–13 Map of eclipses. Map of Jean Meeus. 14–15 A walk through a Reproduced with approval of Acquier Jean-Louis. landscape … 16 Interpretation by Marshack Drawings by Frederic Delannoy. … 17 A poetic representation … From Kagame. 18 In the “rugo” of a student’s Reproduced with permission from Acquier Jean-Louis. … 19 This entrance to a school Photo art school Nyundo, photo C. Blondeel. in 21 J.-P. Harroy had a high Photo Wikimedia Commons, author function … Shabanmasengesho. 28 The large drums had Phtoto Wikimedia Commons. 35–36 Announcement in a References Africa Museum Tervuren; Answer to 6134 colonial... of July 30 and to 6425 of August 20, 1909, and entitled “Report about native money, and the counting systems, in use in the district around ‘Lake Leopold II’”. Ethnographic File 126. Collection Royal Museum for Central Africa ©. (continued)

Number Description Origin 42 Table of… See Mubumbila, 1988. 43 Until today a Dozen Internet publication of de Dozenal Society, http://www. Society exists… dozenalsociety.org.uk/ accessed on March 25, 2016. 47, 52–58 Patterns and matrices. P. Gerdes. 61 Neroman’s study of... Left: illustration Neroman; right: drawing Frederic Delannoy. 62 Examples of reasoned … Drawing by Frederic Delannoy, and work of C. Blondeel’s group, Zwarte Zusterstraat Gent (Belgium). 63 Two illustrations from a Drawing by Frederic Delannoy, after illustrations from mathematical magazine. The Mathematics Teacher. 64 Weaver… Drawings by Frederic Delannoy. 65 A modern piece… Photo Tijl Beyl. 66 Interior of a Rwanda hut… Drawing by Frederic Delannoy. 67 Pauwels’ list … From Pauwels. 68–69 Traditional imigongo from From Celis, reproduced with permission of authors. Rwanda. 70 In Benin one finds all 7 … Drawing by Frederic Delannoy, after Crowe. 71 This fractal was named.... Universal mathematical fractal drawing. 72–74 Illustrations African Drawing by Frederic Delannoy after work of Ron Fractals. Eglash, with permission from author. 75 Catholic, Hindu, … Author’s picture (left) and Wikimedia Commons (middle, right). 76 Gerdes’ fractal… Author drawing, after Gerdes. 78 Standard illustration about Drawing by Frederic Delannoy. quipu’s. 79 Counting strings. References Africa Museum Tervuren: Left, cordelette à nœuds, 1960 5/5 R.IV.D.a.6, Doss. ethno. 126, dossier report; 74, May 3, 1920. right: cordelette à nœuds; 1960 4/5; R.IV.D.a.5; Doss. ethno. 126; dossier report; 74; May 3, 1920. 80 All kinds of counting References Africa Museum Tervuren (in order): strings corde à nœuds, bicolore, 10,563; corde à nœuds, 1536, R.VII.D.a.4; corde à nœuds, 1960 1/5, R.IV.D.a.2; Doss. ethno. 126; dossier rapport 74, May 3, 1920; corde à compter (grande), 2025, R.VII.D.a.5; corde à nœuds, 1816, R.V.D.a.5. 81 Lagercrantz’ map… Right: drawing by Frederic Delannoy. 82 A string with… References Africa Museum Tervuren: Left: corde à nœuds (152 nœuds ou mosolo); 884; R. IV.D.a.1; Doss. ethno. 77; dossier report 2; région de Basoko; February 7, 1910; Right: corde à nœuds 1816; R.V.D.a.5. (continued) Origins of the Illustrations 219

Number Description Origin 83 Counting sticks of… References Africa Museum Tervuren (in order): bâton à compter (petit avec biseau), 1812, étiquette “n° 2 Uoira” ou “Noira”; bâton encoché, 2144; bâtonnet à compter (grand), 1538, R.VII.D.a.1, Long; 600 mm, région d’Ubangala, ex-Zaire; au nord de Kisangani (territoire de Banalia) et au sud-est de Buta; province du Haut-Zaire/Congo, Ababoa ou Boa, 23/03/1910. 84 Sticks to count to… References Africa Museum Tervuren (in order): Tally stick, 2111 3/2, R.II.D.a.2, cfr 67.63.5695, 2/2, Rég.; Cabinda, administrative subdivision: Tchimbuande, Kakongo; tali stick (roseau formant une baguette et maintenu par des fruits secs ou épices), 65.59.96, “Ami du Musée”, 440 × 19 mm, Rhodésie du Nord, Bangweolo, October 12, 1965; tally stick, 67.63.5695 2/1, ET.38.20.847 ½, 449 × 45 mm, Rég.; Cabinda, subdivision administrative: Tchimbuande, Kakongo May 1938. 90 An old mancala version … Drawing by Frederic Delannoy. 98 A traditional scene Drawing by Frederic Delannoy. 113 A 50-year-old poetry From Kagame, over 50 years old. collection… 127 Views of Semliki Valley Photos © ADIA RBINS Brussels. 128–133 Photos about the Ishango Photos from de Heinzelin’s report of 1957. excavations, and the site. 134 Six views of Ishango rod Ishango rod, 6 views: photos and infography: Patrick Semal, 15/06/2001; photos © ADIA RBINS Brussels. 135–136 Photos of Ishango rod and Photos from de Heinzelin’s report of 1957. the site. 137 Official photo of KBIN. Ishango rod: photo: Marcel Springaer; infography: Ivan Jadin, 23/12/1999; © ADIA RBINS Brussels. 138–142 By cutting out… Drawings from de Heinzelin’s report of 1957. 143–144 Marshack’s drawings. Author’s drawings, after Marshack. 145 A part of Pletser’s study of Drawing by V. Pletser. the carvings. 148–149 Map with de distribution of From de Heinzelin’s report of 1957. harpoons … 154–155 Illustrations of Noguera: a Drawing by Frederic Delannoy aftr Noguera. Peul woman, …. 156 Cultural relations … Author’s drawing, with illustrations from Noguera. 157 Data analysis diagram … Reproduced with approval of Arnaiz. 159 Lagercrantz’s map… Wikimedia Commons, “Lissala man with tattooes.” Reference Africa Museum Tervuren: bambou encoché, 34,992. 164 Cartoons … Drawing by Frederic Delannoy. 170 Sign at … Photo by Magloire Paluku, reproduced with permission. (continued) 220 Origins of the Illustrations

Number Description Origin 171 Ishango rod in Kenya… Left: photo by Patricia Mergen, Liaison Officer, Legal Entity Appointed Representative (LEAR) at European Commission, Royal Museum for Central Africa, Tervuren, Belgium. 172 3D scans of… Photos © ADIA RBINS Brussels; TNT project, NESPOS Society. 173–174 Three views of… Photos © ADIA RBINS Brussels. 175–177 Neolithic patterns Drawings and photos by Slavik Jablan, Belgrade, Serbia; reproduced with permission. 178–180 Museum for Central Photos taken by author. Africa, Tervuren, Belgium 181 Patterns on… References Africa Museum Tervuren (in order): coupe/ schaal, 0.0.33710; coiffe/hoofdtooi, 1949.21.1; paravent, 0.0.41670. 182 Patterns on carpets References Africa Museum Tervuren (in order): tapis/ tapijt, 1968.39.3; tapis/tapijt, 1980.31.2. 183 Fabric patterns. References Africa Museum Tervuren (in order): 0.0.28094; 0.0.25651; 0.0.27757–37; 0.0.28089; 1959.21.672; 0.0.16482 3; 1950.24.135; 0.0.45084; 0.0.45107; 1950.24.150. 186 At right, the above rope Reference Africa Museum Tervuren: 1975.48.1. says 187 Story board… References Africa Museum Tervuren: 0.0.42871; 0.0.27312. 188 Strategy game… Reference Africa Museum Tervuren: 1969.59.636 (négatif G 4878). 189 Another strategy game… References Africa Museum Tervuren: 1974.54.37 1–9; 1974.54.37 1. 193 Illustration… Drawing by Frederic Delannoy. 194 All but one… From Celis, reproduced with permission of authors. 202–204 Ashanti stamps and Drawings by Frederic Delannoy, after Delene Perley, of patterns Millikin University. 205 Diagrams… Drawings Sophie Ponsaert; reproduced with permission. References

Acquier J.-L., Le Burundi, Collection Architectures Traditionnelles, Editions Parenthèses, 1986. Apostel L., Towards the formal study of models in the non-formal sciences. In H. Freudenthal (ed.), The Concept and the Role of the Model in Mathematics and Natural and Social Sciences, Dordrecht: Reidel, 1961. Arnold M.-M., Jean de Heinzelin, le “père” du baton d’Ishango, Athena 153, magazine de la D.G.T.R.E. de la Région Française de Belgique, September 1999. Arnaiz-Villena A., Gomez-Casado E., Martinez-Laso J., Population genetic relationships between Mediterranean populations determined by HLA allele distribution and a historic perspective, Tissue antigens: histocompatibility and immunogenetics, p. 111–121, Vol. 60, 2002. Ascher M., Graphs in Cultures: A study in Ethnomathematics, Historia Mathematica 15 201–227, 1988. Ascher M., Ethnomathematics: A Multicultural View of Mathematical Ideas, CRC Press 1994. Atkins W. J., Notes on the Concords and Classes of Bantu Numerals, African Languages Studies, Vol. 2, London, 1961. Ball J., Games from around the world, Wetherby MBA Publishing 1989. Barrow J. D., Pi in the Sky, Counting, thinking and being, Oxford University Press, published as Penguin Book 1993. Bell R. C., Board Games around the World, Cambridge : Cambridge University Press, 1988. Bernal M., Black Athena, Volume 1: The Afroasiatic Roots of Classical Civilisation, The Fabrication of Ancient Greece, New Brunswick, Rutgers University 1987. Bernal M., Black Athena, Volume II: The Archaeological and Documentary Evidence, Rutgers University 1991. Bigirumwami A., Proverbes, dictons, devinettes, Nyundo (Rwanda) 57p. (1967). Bogoshi J., Naidoo K., Webb J., The oldest mathematical artifact, Mathematical Gazette, 71, 458, 294, 1987. Bouquiaux L., A propos de numération: L’emploi du système décimal et du systéme duodécimal dans la langue birom (Nigéria septentrional), Africana Linguistica, Tervuren, 1962, 7–10. Boucharlat A., Le commencement de la Sagesse, Société d’Etudes Linguistiques et Anthropologiques de France SELAF Paris (1975). Brandel R., The Music of Central Africa, An Ethnomusical Study, Martinus Nijhof, The Hague, 1961. Brooks A. S. et al., Dating and context of three middle stone age sites with bone points in the Upper Semliki Valley, Zaire, Science 268, 548–553, 1995. Broline D. M., Loeb D. E., The Combinatorics of Mancala-Type Games: Ayo, Tchoukaillon and 1/ pi, arXiv: math.CO/9502225 v1 9 Feb 1995.

Buekenhout F., Les Polyèdres: de Lucy à Jacques Tits, 6ième congrès de l’Association des Cercles francophones d’Histoire et d’Archéologie de Belgique, Mons-Bergen, 24–27 août 2000. Celis G., Celis Th., Les Peintures Murales Intérieures des Habitations du Migongo, Rwanda, Africa-Tervuren XVII-2, 1972. Coupez A., Kamanzi T., Récits historiques Rwanda, Tervuren (Annales du Musée Royal de l’ Afrique Centrale) série in 80 n 43 328 p., 1962. Coupez A., Kamanzi T., Littérature de Cour au Rwanda, Oxford, Clarendon Press, 1970. Coupez A., Kamanzi T., Rythmes Quantitatifs en Poésie Rwanda, Sciences Humaines, Folia Scientifica Africa Centralis III, 3, 8/60, 1957. Coupez A., d’Hertefelt M., La Royauté Sacrée de l’ ancien Rwanda, Koninlijk Museum voor Midden-Afrika, Annalen reeks IN-8, nr 52, Tervuren, 1964. Coupez A., Grammaire Rwanda simplifiée, Notes de Phonologie Ruanda, Institut pour la Recherche Scientifique en Afrique Centrale, Astride, 1960. Coupez A., Benda V., Terminologie du jeu d' igisoro au Rwanda, Africa Tervuren IX, 2, 37/41, 1963. Crowe D. W., The Geometry of African Art, Historia Mathematica 2, pp. 253–271, 1975. De Heinzelin de Braucourt J., Exploration du Parc National Albert : Les fouilles d’Ishango, Fascicule 2, Institut des Parcs Nationaux du Congo Belge, (“Exploration of the National Park Albert: The Ishango excavation”, Volume 2, Institute of National Parks of the Belgian Congo), Brussels, 1957. De Heinzelin de Braucourt J., Ishango, Scientific American, 206-6, June, 105–116, 1962. Delafosse M., La numération chez les Nègres Africa, Journal of the International Institute of African Languages and Cultures, 1, (1928), p. 387–390. Delano C. I. O., Conversations in Yoruba and English, New York, Frederick A. Praeger, 1963. Desouter S., Abrégé agro-pastoral, Agence de Coopération culturelle et technique, Paris, 1982. Dion G. M., Devinettes au Rwanda, Ibisakuzo Editions Universitaires du Rwanda coll Muntu 2, Butare (Rwanda) 271p. 1971. Errico F., L’origine du stockage de l’information, Pour la Science, pp 6–9, October-Januari 2002. Fauvel J., Gerdes P., African Slave and Calculating Prodigy: Bicentenary of the Death of Thomas Fuller, Historia Mathematica 17, 141–151, 1990. Gannon G., Martelli M., The Farmer and the Goose, a Generalization, The Mathematics Teacher, Vol. 86, n° 3, maart 1993. Gardiner T., Once upon a Time, The Mathematical Gazette, Vol. 76 n 475, 1992. Gay J., Cole M., The New Mathematics and an Old Culture, A Study of Learning among the Kpelle of Liberia, New York: Holt, Rinehart and Winston, 1967. Gerdes P., Geometry from Africa, Mathematical and Educational Explorations, The Mathematical Association of America, 2002. Gerdes P., Vivendo a Matemática, Desenhos da Africa, Editora Scipione, 2001 (can be ordered for free at: Paulus Gerdes, Caixa Postal 915 Maputo, Republica Popular de Moçambique). Günther R., Musik In Rwanda, Ein Beitrag zur Musikethnologie Zentralafrikas, Annales du Musée Royal de l’ Afrique Centrale de Tervuren, Série IN-8, Sciences humaines, no 50, 1964. Hall R. de Z., Boa, Tanganika Notes No 34 (January 1953), pp. 57–61. Harris R., The Origin of Writing, La Salle, IL: Open Court, 1986. d’Hertefelt, M., Trouwborst A., Scherer J. H., Les anciens royaumes de la zone interlacustre méri- dionale: Rwanda, Burundi, Buha, Tervuren (Annales du Musée Royal de l’ Afrique Centrale, no 6) 248 p. (1962). d’Hertefelt, M. et Coupez, A., La Royauté sacré de l’ ancien Rwanda, Tervuren (Annales du Musée Royal de l’ Afrique Centrale, no 52) 520 p. (1954). Hurel E., Grammaire Kinyarwanda, Ruanda: Kabgaye 1951. Hurel E., La poésie chez les primitifs: contes fables, récits et proverbes du Rwanda, Bruxelles (1922). Huylebrouck D., Traditional Scientific Knowledge in Burundi and Rwanda, Mathematics in School, The Mathematical Association, Longman Group Ltd., November, 1995. References 223

Huylebrouck D., The bone that began the space odyssey, The Mathematical Intelligencer, 18-4, 56–60, 1996a. Huylebrouck D., Puzzles, patterns, drums: the dawn of mathematics in Rwanda and Burundi, Humanistic Mathematics Network, Journal #14, November 1996b. Huylebrouck D., Histoires célestes du Rwanda, Ciel et Terre, vol. 113 (2), 83–89, 1997. Huylebrouck D., A Stamp for World Mathematical Year, with F. Dumortier, European Mathematical Society Newsletter, September 2000 p. 27. Huylebrouck D., Le baton d’Ishango, l’objet le plus ancien en mathématiques, Kadath, Chroniques des Civilisations Disparues, n° 98, 2001. Ifrah G., From One to Zero : A Universal History of Number, New York: Viking, 1985. Irving G., Donckers J., Uiterwijk J., Solving Kalah, ICGA Journal, p. 139–147. September 2000. Jones G. G. M., Stolen Legacy, Africa World Press, Trenton, N.J., 1954. Joseph G. G., The Crest of the Peacock: Non-European Roots of Mathematics, Penguin Books, London, 1992. Kagame A., La poésie dynastique au Rwanda, Bruxelles, I.R.C.B., Brussel, 1951. Kagame A., La philosophie bantu-rwandaise de l’ Etre, Académie royale des Sciences coloniales, Bruxelles, 1956. Kagame A., La notion de génération appliquée à la généalogie dynastique et à l’ histoire du Rwanda des X - XII siècles à nos jours, Académie royale des Sciences coloniales, Bruxelles, 1959a. Kagame A., La Divine Pastorale, Editions du Marais, Bruxelles, 1952. Kagame A., La Naissance de l’ Univers, Deuxième Veillée de “La Divine Pastorale”, Editions du Marais, Bruxelles, 1955. Kagame A., La notion de génération appliquée à la généalogie dynastique et à l’ histoire du Rwanda des X° - XII° siècles à nos jours, Académie royale des Sciences coloniales, Bruxelles, 1959b. Kagame A., La langue du Rwanda et du Burundi expliquée aux autochtones, Institut pour la Recherche Scientifique en Afrique Centrale, Kabgayi, 1960. Kline M., Mathematics in Western Culture, New York, Oxford University Press (Reprinted by Penguin, London 1972), 1953. Kubik G., Mehrstimmigkeit und Tonsysteme in Zentral und Ostafrika, Vienna, 1968. Kwabena Nketia J. H., Die Musik Afrikas, Heinrichshofen’s Verlag, Wilhelmshaven (1979). Lagercrantz S., African Tally-Strings, Anthropos 63, p. 115, 1968 (1970). Lagercrantz S., Counting by Means of Tally Sticks or Cuts on the Body in Africa, Anthropos 68, p. 37, 1973. Lefkowitz M. R., Not Out of Africa, How Afrocentrism Became an Excuse to Teach Myth as History, BasicBooks, Hardcover, 1996. Lefkowitz M. R. en MacLean R. G. (eds), Black Athena Revisited, University of North Carolina Press 1996. Lerman S. (Ed.), Cultural Perspectives on the Mathematics Classroom, Mathematics Education Library, Kluwer Academic Publishers (1994). Loupias P., Tradition et légende des Batutsi sur la création du monde et leur établissement au Rwanda, Wien, Anthropos III 1, 1–13 pp. (1908). Maes J., Stryckmans M., Mazy M., Note sur les Populations Lalia et Yasayama du Territoire des Dzalia-Boyela, Congo vol. I-2, p. 172 – 179, 1934. Marshack A., Roots of Civilisation, The Cognitive beginnings of Man’s First Art, Symbol and Notation, McGraw-Hill Book Company, New York, 1972. Mathews, H. F., Duodecimal numeration in Northern Nigeria, The Nigerian Field, and Notes on the Nungu tribe, Nassawara Province, Northern Nigeria, and the neighboring tribes which use theduodecimal system of numeration, Harvard African Studies, 1917, Vol. 1, 83–93. McGee W. J., Primitive Numbers, Bureau of American Ethnology, 1900. Meeussen A. E., Essai de grammaire rundi, Tervuren Musée Royal du Congo Belge, 236 p. 1959. Menninger K., Number words and Number Symbols, Cambridge, Mass.: M. I. T. Press, 1969. Merriam A. P., African Music Re-examined in the light of New materials from Belgian Congo and Ruanda Urundi, African Music Society Newsletter, I/6 (1953a), 57–64. 224 References

Merriam A. P., African Music Re-examined in the light of New materials from Belgian Congo and Ruanda Urundi, The Anthropology of Music (Evanston, 1964). Merriam A. P., The game of kubuguza among the Abatutsi of North-East Ruanda, Man 53, 169–172, no 262, 1953b. Michiels F., (1984) Ingoma, Socio-historische en Musicologische Studie van de trommel uit Rwanda, dissertation KU Leuven, July ’84. Moiso B., Ngandi L., Numération Cardinale dans les Langues Bantu du Haut-Zaïre, Annales Aequatoria 6, 189–196, 1985. Moiso B., Etude comparée du système de numérotation de 1 à 10 dans quelques Langues non- Bantu du Haut-Zaïre, Annales Aequatoria 12, 475–479, 1991. Mubumbila V. M., Sur le sentier mystérieux des Nombres Noirs, Editions Harmattan Paris, 1988. Mubumbila V. M., Sciences et Traditions Africaines, Les Messages du Grand Zimbabwe, Editions Harmattan Paris, 1992. Nicolas F.-J., La mine d’or de Poura et le nombre quatre-vingt en Afrique occidentale, Notes Africaines, Notes Africaines, n° 161, p. 19–21, 1979. Nelson D., Joseph G. G., Williams J., Multicultural Mathematics, Teaching Mathematics from a Global Perspective, Oxford University Press, 1993. Njock G. E., Document on Co-operation Presented at the First Pan African Congress of Mathematics Held in Rabat, Morocco in 1976, A.A.U. Bull. XI, 1976, vol. 3, no 1, p. 135–147. Njock G. E., Langues africaines et non africaines dans l’enseignement des mathématiques en Afrique. Séminaire Interafricain sur l’Enseignement des Mathématiques à l’Ecole Primaire et Secondaire en Afrique, Accra (Ghana), avril 1979. Nketia J. H. K., Die Musik Afrikas, Heinrichshofen’s Verlag, Wilhelmshaven, 1979. Nkongori L., Kamanzi T., Proverbes du Rwanda, Tervuren 75 p. (1957). Noguera A., How African Was Egypt? A Comparative Study of Ancient Egyptian and Black African Cultures, Vantage Press, New York, 1976. Nsimbi M.B., Omweso, A Game People Play in Uganda, Occasional Paper, no 6, African Studies Center, University of California at Los Angeles, 1968. Pagès A., Visions, Oracles et Prophéties, Grands Lacs 2I 11–12 517/522, 1935. Pagès A., La vie Intellectuelle …, Congo XV, t. I, 3, 357/389; 4, 481/503; 5, 657/671, 1934. Pagès A., Au Rwanda, sur les bords du lac Kivu, Congo Belge. Un royaume hamite au centre de l’ Afrique, Institut royal colonial belge, Bruxelles, 1933. Pauwels M., Les métiers et les objets en usage au Rwanda, Rome Annali Lateranensi, XIX pp. 185–294. (1955a). Pauwels M., Jeux et divertissement au Rwanda, Rome, Annali Lateranensi, XXIII pp. 185–294. (1955b). Pauwels M., L’ agriculture au Rwanda et ce qui s’ y rapporte, Annali del Pont. Museo Miss. Etn., Vol. XXXIII 31/123, 1969. Pauwels M., Les couleurs et les dessins au Ruanda, Anthropos no 47, p. 474, 1952. Perugia Del P., Les Derniers Rois Mages, Editions Phébus, Paris (1978). Pletser V., Huylebrouck D., Does the Ishango bone indicate knowledge of the base 12? An interpretation of a prehistoric discovery, the first mathematical tool of humankind,http://etopia. sintlucas.be/3.14/, 1999a. Pletser V., Huylebrouck D., The Ishango artifact: the missing base 12 link, Proc. Katachi U Symmetry Congress KUS2, paper C11, Tsukuba Univ., Japan, 18 Nov. 1999; Forma 14-4, 339–346, 1999b. Pletser V., Huylebrouck D., Research and promotion about the first mathematical artefact: the Ishango bone, Abstr. Proc. PACOM 2000 Meeting on Ethnomathematics and History of Mathematics in Africa, Cape Town, South Africa, November 1999c. Ponsaerts S., Counting systems, a general historical introduction (Telsystemen, een algemene historische inleiding), masters’ dissertation Department Language Studies KU Leuven, Belgium, 2002. Rodegem F. M., Sagesse kirundi, Tervuren, Annales du Musée Royal du Congo Belge, vol. 34 416 p. (1961a). References 225

Rodegem F. M., Le rire et l’ humour au Burundi, Ngozi (Burundi) Patrimoine culturel rundi, 11 52–104 p. (1961b). Rodegem F. M., Précis de grammaire rundi, Story-Scientia, Brussel-Gent, 1967. Romein J. W., Bal H. E., Awari is solved, Notes, ICGA Journal, p. 162–165. September 2002. Rowe W. F., School Daze: A Critical Review of the ‘African-American Baseline Essays’ for Science and Mathematics, The Skeptical Inquirer, Sept.-Oct, 1995. Seidenberg A., Ritual Origin of Counting, in Archive for the History of Exact Sciences, Berlin: Springer Verlag, 1962, pp. 1–40. Seidenberg A., On the Eastern Bantu Root for Six, African Studies, 18, (1959), p. 28–34. Smith P., La forge de l’ intelligence, Paris L’ homme X, 2. pp. 5–21, 1970. Struik D. J., Concise History of Mathematics, New York: Dover, 1987. Thomas N. W., Duodecimal Base of Numeration, Man (London, Royal Anthropological Institute), p. 25–29 Nos. 13–14 February 1920. Tjon Sie Fat F. E., Representing kinship simple models of elementary structures, dissertation, University Leiden, november 1990. Van Sertima I., ed. Blacks in Science, Ancient and Modern, New Brunswick, NJ: Transaction Books, 1984. Van Sertima I., ed. Black Athena Egypt Revisited, London and New Brunswick, Transactions Publications, 2nd Ed. 1995. Vansina J., L’ évolution du royaume rwanda des origines à 1900, Académie royale des sciences d’ outre-mer, 1962. Williamson J., Dabida Numerals, African Studies, (Johannesburg, 1943), n° 4, p. 215–216. Wirth N., Algorithms + data structures = programs, Prentice-Hall Englewood Cliffs N. Y., 1976. Zaslavsky C., Africa Counts, Lawrence Hill Books, New York, 1973. Zaslavsky C., Black African Traditional Mathematics, The Mathematics Teacher, Vol. LXIII, Number 4, April 1970, p. 345–354. Index

A Dogon, 14, 87 Africa museum, 33, 39, 41, 46 Doubling, 37–39, 58, 78, 114–118, 122, 142, Algebra, 78, 87, 123, 213 143, 145, 147, 148, 150 Arithmetic, 35, 49–51, 78–79, 113, 143, 147, Dozen, 48 148, 166, 204 Duodecimal, 48–53, 87, 150, 153 Ascher, M., 58, 59, 204 Ashanti, 8, 66, 69, 205 Associativity, 213 E Egypt, 12, 51, 114, 115, 139, 153, 154, 157–163, 166, 173, 174, 197 B Errico, F., 169, 170 Baali, 46 Bach, J.S., 29 Bal, H.E., 212 F Beaumont, P., 169 Fractals, 72–76 Bell, R.C., 87 Fulani, 44, 163 Brandel, R., 30, 32 Broline, D.M., 211, 212 Brooks, A.S., 137, 140, 172 G Buekenhout, F., 169, 170, 208 Gamma function, 212 Burundi, vi, viii, 5, 6, 10, 16, 24, 26, 29, 39, Gauss, C.F., 87, 212 46, 47, 79–81, 127 Gay, J., 3, 4 Gerdes, P., 59, 62, 63, 76, 79, 176 Golden section, 63, 64, 74, 117, 191, 194 C Greece, 31, 32, 114, 156–159, 161–163 Celis, G., 68, 70 Group theory, 71 Circle, 203 Günther, R., 32 Cole, M., 3, 4 Congo, v, viii, 38, 44, 45, 47, 56, 71, 72, 90, 127, 128, 134, 164, 206 H Harroy, J.-P., 29 Harvard, 148 D Hauzeur, 138 de Heinzelin, J., v, 128, 134, 141, 154, 167 Hemiola, 29, 30 Demaine, 179 Hindu, 31

Hogben, L., 143 Ptolemy, 12, 14, 191, 192 Hypergeometric function, 212 Pythagoras, 12, 143, 157

I R Igisoro, 89, 93–96, 99–105, 117, 212 Romein, J.W., 212 Ishango rod, v, 8, 135, 140, 141, 167, 173, Royal Belgian Institute for Natural Sciences, 175, 176, 191, 198 v, 128 Ishango site, v, 133, 136, 172 Rutanzige, 5, 127, 136, 164, 172 Isometry, 73 Rwanda, viii, 5, 14, 19, 22, 23, 26–28, Ivory Coast, 39, 66 30, 32, 33, 40, 43, 46, 47, 50, 67–71, 89, 94, 95, 98, 104, 105, 193, 199, 201 J Joseph, G.G., 24, 35, 156, 158, 173, 176 S Semi-group, 214, 215 K Semliki, v, 127, 128, 134, 154, 172 Kagame, A., 21, 47, 104 Shallit, J., 140 Kakira ka Kimenyi, 70 Square, 158 Kpelle, 4, 11 Star, 19, 23, 24, 26, 201 Struik, D.J., 176 Sun, 20, 22, 23, 26, 48, 56, 130, 199 L Symmetry group, 204 Loeb, D.E., 211, 212

T M Tanzania, 40, 89 Mancala, 87–92, 94, 95, 104, 200, 211–213 Timbuktu, 107, 108 Marshack, A., 16, 18, 145, 146, 148, 151, Triangle, 154 167, 172 Mathematics teacher, 66 Michiels, F., 33 U Middle East, 31, 48, 114, 153, 156 Uganda, 20, 89, 127, 161 Moon, 13, 16, 18, 20–21, 24, 26, 50, 51, 56, 81, 145, 147, 151 Multiplication, 63, 77, 78, 111–114, 117, 120, V 122, 145, 150, 158, 172, 213 van Straelen, V., 128 Venus, 24, 26, 173 von Neumann regular, 214 N Namibia, 169, 172 Navajo, 8 W Njock, G.E., 66 Walegga, 154, 195 Nyali, ix, 46, 150 Wirth, N., 122 Nzohabonayo, P., 26

Y P Yasayama, 45 Pattern, 16, 31, 57–59, 61, 66, 70–73, 76, 149, Yuhi, 27, 28 176, 205 Pi, 211 Pletser, V., 148, 149, 166, 167 Z Polyhedral, 169 Zaslavsky, C., 71, 176 Prime number, 50, 142, 143, 145, 147, 148, 151 Zulu, 5 Africa + Mathematics: Summary

Are the sources of mathematics to be found at the sources of the Nile? In 1950, Belgian archaeologist Jean de Heinzelin found a 22,000-year-old bone with intrigu- ing carvings in Ishango, a village in the volcanic region on the border of the Congo and Uganda, not far from Rwanda. The author defends the hypothesis of Belgium’s space scientist Vladimir Pletser that the bone was neither an arithmetical game with prime numbers nor an astronomical calendar, but a simple tally stick of a people counting in a different, non-Western way. The mathematics of the rod is an example of ethnomathematics, an emerging subfield in mathematics. Oddly enough, examples from the Congo, Rwanda, and Burundi are scarce, though Belgium has historical ties with this region. The author spent about 12 years teaching in the region, and through this experience and exam- ining valuable collections in Belgian Africa museums, he hopes to show the mathematics from Central Africa represent a particular challenge, even for contemporary mathematicians. Still, critical considerations are not neglected and some subse- quently turn out to be correct, as de Heinzelin revealed a second Ishango rod on his death bed. The Ishango rods waited half a century in a dusty drawer before their rediscov- ery – on one hand at the mercy of Moliere’s colonial bourgeois-gentilhommes, who did not want to admit that Africans did mathematics, and on the other hand the Afrocentrists who saw in the rods the beginning of all mathematics. Perhaps the third scientific dog will finally run away with the mathematical bones.