ISBN: 978- 1-4357-2629- 1 Copyright © 2007 by Paulus Gerdes 2 C JPYnghted matanal Paulu s Gerdes LUNDA GEOMETRY: Mirror Curves, Designs, Knots, Polyominoes, Patterns, Symmetries Universidade Pedag6gica Maputo, Mozambique 2007 3 C JPYrig eo maJ:lnal Title. LUNDA GEOMETRY: Designs, Knots Polyominoes, Patterns, Symmetrie . uthor: Paulus Gerde Research Center for Mathematics, Cultur,e and Education . P. 915 Maputo Mozambique pgerdes . irconn.com irstedition : niversidade Pedag6g.ica Maputo Mo ~ arohique 1996 ] 52 pp. (01362/FBM/96 Maputo Mozambique) Original title: Lunda Geometry: Designs, Polyominoes Patterns, Symmetries Second edition: Lulu Enterprises, MorrisviHe NC 27560 USA,2007 196 pp. Distribuition (print & download): http://stores.lulu.comlpg -rdes or search 'Paulus Gerdes ~ on the webpage: wwwJulu.cOJll opyright © 2007 by Paulus Gerdes 4 CONTENTS page Preface (First edition) 7 Preface (Second edition) 9 1 On ~' OfUl sand d.·awings, min·or curves and the 11 generation of Lunda-designs 1.1 A bout sOfia sa nd drawings 11 1.2 Towards a discovery 12 1.3 Some theorems about smooth and monolinear mirror lines 21 designs 1.4 Enumeration modulo 2 and Lunda-designs 33 2 On Lunda-designs and some ortheir symmetries 39 2.1 Mirror designs and mirror curves 39 2.2 The discovery of Lunda-designs 42 2.3 Examples 43 2.4 General symmetry properties 50 2.5 Special classes of Lunda-designs 51 2.6 Square Lunda-designs and fractals 66 2.7 Generalization of Lunda-designs 68 2.8 Hexagona l Lunda-designs 73 References 75 3 On mirror CUI"\les and Lunda-designs 77 3. 1 Mirror designs and mirror curves 77 3.2 Regular mirror curves 80 3.3 Non-regular mirror curves 82 3.4 Properties of Lunda-designs 85 3.5 Classes of Lunda-designs 86 1 6 Generalizations 89 Lunda-k-designs 90 Circular Lunda-designs 91 Polyominal Lunda-designs 92 Fractal Lunda-designs 97 References 98 5 C JPYng eo maJ:lnal 4 On Lunda-designs and Lunda-animals - Fibonacci 10 1 returns to Africa 4. 1 Introduction 101 4.2 Counting the number of possible paths of Lunda-animals 104 References 107 5 On Lunda-designs and polyominoes 109 5.1 Polyominoes in Lunda-designs 109 5.2 Types of Lunda-polyominoes 110 5.3 Approximation of the number of Lunda-n-ominoes 112 5.4 Symmetrical types 114 6 Symmetrical, closed Lunda-polyominoes 121 7 Examples of Lunda-spirals 133 8 137 8. lOne-dimensional Lunda-patterns (Lunda-strip-patterns) 137 8.2 Two-dimensional Lunda-patterns 146 Reference 160 9 On the geometry of Celtic knots and their Lunda­ 161 designs 10 Recent variations and generalizations 179 Appendix I: J89 Proofs of some theorems concerning the relations hip betvveen rectangle-filling mirror curves and Lunda- designs Books by the same author 197 6 C JPYnghted matanal PREFACE (First edition) Lunda-designs are a type of black-and-white design that I discovered while analyzing properties of a special class of sOlla sand drawings from eastern Angola and neighboring regions of Zambia and Congo / Zaire. The elements of this class are mirror-generated curves, and generate, in turn, Lunda-designs. The chapters of this book are comprised of papers on mirror curves, Lunda-designs and related concepts such as Lunda­ polyominoes and Lunda-patterns. Most chapters may be read independently of each other. Chapter I is a partial translation of Chapter 6 of the second volume of Sona Geometry, and analyses the mirror curve class of sona sand drawings and some of their basic properties. In this chapter, I also describe the discovery of Lunda-designs_ Chapters 2: and 3 present an introduction to Lunda-designs, their symmetries, and some of their generalizations such as Lunda-k-designs and hexagonal Lunda-designs, These papers were published in the international journals Visl/al Malhemalics (1999) and Compl/lers and Graphic.I' (1997) respectively_ In Chapter 4, I introduce the concepts of Lunda-polyominoes and Lunda-animals, and evaluate the number of possible paths of given lengths that may be traversed by Lunda-animals. The famous sequence of Fibonacci surprisingly appears in this context. Chapter 5 presents a first approximation for the number of Lunda-n-ominoes, In Chapters 6 and 7, I explore special classes of Lunda-polyominoes such as symmetrical closed Lunda-polyominoes and Lunda-spirals, Finally, in Chapter 8, I show that for all twenty­ four classes of one-color and two-color, one-dimensional patterns, it is possible to construct Lunda-strip-patterns, which belong to them_ Furthermore, I present examples of one-color and two-color, two­ dimensional Lunda-patterns. 7 c Appendix I presents the proof of the theorem that every mirror design generates a Lunda-design, and, inversely, for every (finite) Lunda-design a mirror design that generates it may be constructed. Lunda-designs present a concrete example of how ethnomathematical research can lead to both fruitful and interesting ideas of se ri ous mathematical reflection. I hope the book LUNDA Geumelly wi ll stimulate further research on these aesthetically attractive figures. Acknowledgements My thanks go to my colleagues at the Unil'ersidade Pedagogica (UP, Mozambique) for their permanent interest and encouragement of my research; to Jill Gerrish (Department of English, UP) for the linguistic revision. I am very grateful for the interest that colleagues in various parts of the world have shown in the preparation of this book. My thanks go, in particular, 10 Slavik Jablan (The Mathematical Institute, Belgrade, Serbia), Maurice Bazin (Exploratorium, San Francisco, USA), Erhard Scholz (Bergische Universitat, Wuppertal, Germany), Clifford Pickover (TB M Thomas Watson Research Center, New York, USA), and Arthur Powell (Rutgers University, Newark, USA) for their stimulating reactions on the chapters they have read. Above all, I thank my wife Marcela Libombo and my daughter Lesira for their encouragement. Maputo, June 25, [996 Paulus Gerdes , C JPYnghted matanal PREFACE (Second edition) The new edi tion of the book Lllnda Geometry contains two additional chapters. Chapter 9, entitled 011 the Geometry of Celtic knots and their Lllnda..(/esigns, was published in the British journal Mathematics in School (\999). Chapter lOis the last section of an invited paper presented al the Wenner-Gren International Symposium "Symmetry 2000" (Stockholm, September 13-16, 2000) and published in the proceedings. The new edition does not include the review of Solomon Golomb's book Po/yolllinoes, published in Archives infernariollales d 'Hisloire des Sciences (1998,48. No. 140, 174- 176). Chapter 4 of my book GeOIl/CllY from Africa: Malhclllaticaf and Educational Explorations, published by the MAA ( 1999), presents an introduction to Lunda-designs and includes a section on Lunda­ polyhedral-designs and a board game, The relationship between Lunda-designs and magic squares is explored in a paper published in 71le College Mathematics Journal (2000). The study of Lunda-designs led to the discovery of Liki-designs (2002a, 2002b) and of various types of matrices, including cycle matrices (see e.g. the papers published in the electronic journal Visual Mathematics). The book Adventures in the World a/Matrices (2007d) presents an introduction to cycle matrices. Further books on the beautiful geometry and linear algebra of Lunda-designs are forthcoming, The paper Llinda Symmetry where Geomelry meeli)' Al'l (2 005) explores some relationships between Lunda Geometry and art. Mathematician and artist John Sims of the Ringling School of Art and Design (Sarasota, Florida) organized a math-art exhibition including some of colorful Lunda-designs I had prepared. A book on Lunda­ designs and art is in preparation. Maputo, October 2007 Paulus Gerdes c Referellce .~ Gerdes. Paul us (1999), Geometry from Africa: Mathematical and Educational Explorations. The Mathemat ical Association of America. Washington DC __ (2000). On Lunda-designs and the con struction of associated magic squares of order 4p. The College Ma/hemalics Journal, Washington DC, 3 1(3). 182- 188 __ (2002a), New designs from Africa, Pills Magazine. Ox ford, March __ (2002b). Prom Liki-designs to cycle matrices: The discovery of atlractive new symmetries, ViSlial Ma/hemarics, Belgrade. 4(1 ) __ (2002c), m-Canonic mirror curves, Visual Malhematics. 4( I ) __ (2002d). Variazio ni sui disegni Lunda, in: Mi chele Emmer (Ed.), Malemalica e Gllllllm 2002, Springer. Milan, 135- 146 __ (2002e), tlciix matrices. Visual Marh eflla/ics, 4(2) __ (2002f), Cylinder matrices, Vimal Marlll'lIIa/ic:;, 4(2) __ (2003). forom African 'sona' drawings 10 the discovery of new symmetri es and matrices, in: Nouzha £1 Yw.:oubi e/ al. (Eds.), ProcN'diligs of rill' 13" Pall African M athemaric~' Olympiad, Min isterio da Educ a ~iio . Maputo, 5 1-64 __ (2004a), On mirror-curves, alternating kn ots. rope mats and plaited polyhedra: A Foreword, in: Ni ls Kr. Rossing & Christoph Kirfcl, Maremarisk beskrivdse av rawnarrer, NTNU. Trondheim . 9-14 __ (2005). Lunda Symmetry where Geometl)' meets Art, in : Michele Emmer (Ed.), The Visual Mind 1/, MIT Press, Boston, 335-34 8 __ (2006a), Symmetries o f alternating cycle matrices, Visual Malhemalics. 8(2) __ (2006b), On the representation and multiplication of basic altern at i ng cycle matrices. Viwal MathematiCS, 8(2) __ (2006c). Sona Geometryfrom Angola: Mathematics of an Afncall Traditioll, Polimctrica International Science Publishers, Mo nza, 232 pp. __ (2007 11), MWllni colour inversion, symmetry lind cycle matrices, Visllal MathematiCS, Belgrade, 9(3) __ (2007c). From th e African sona tradition to new types of desi gns and matrices, in: Konatc, Diall a (Ed.). Proceedings oltlle International Workshop on Mmhematical Modellillg, Siml/lmion, Visllalizmion and e­ Learning, Springer, New York (in press). __ (2007d), Advelllllres ill rlie World of Matrias, Nova Science Publishers, New York __ (2007c), The Beautifu! Geometry Wid Lineur Algebra of Lunda-Desiglls, 204 pp.