NRC Publications Archive Archives des publications du CNRC Fibonacci, his numbers and his rabbits Drozdyuk, Andriy; Drozdyuk, Denys NRC Publications Record / Notice d'Archives des publications de CNRC: https://nrc-publications.canada.ca/eng/view/object/?id=fbfdddf4-d0e9-45fe-bef6-fe6d3cbecfa0 https://publications-cnrc.canada.ca/fra/voir/objet/?id=fbfdddf4-d0e9-45fe-bef6-fe6d3cbecfa0 Access and use of this website and the material on it are subject to the Terms and Conditions set forth at https://nrc-publications.canada.ca/eng/copyright READ THESE TERMS AND CONDITIONS CAREFULLY BEFORE USING THIS WEBSITE. L’accès à ce site Web et l’utilisation de son contenu sont assujettis aux conditions présentées dans le site https://publications-cnrc.canada.ca/fra/droits LISEZ CES CONDITIONS ATTENTIVEMENT AVANT D’UTILISER CE SITE WEB. Questions? Contact the NRC Publications Archive team at [email protected]. If you wish to email the authors directly, please see the first page of the publication for their contact information. Vous avez des questions? Nous pouvons vous aider. Pour communiquer directement avec un auteur, consultez la première page de la revue dans laquelle son article a été publié afin de trouver ses coordonnées. Si vous n’arrivez pas à les repérer, communiquez avec nous à [email protected]. ANDRIY DROZDYUK DENYS DROZDYUK FIBONACCI, HIS NUMBERS AND HIS RABBITS Choven Publishing Corp. TORONTO – 2010 Andriy Drozdyuk, Denys Drozdyuk Fibonacci, his numbers and his rabbits. — Toronto : Choven, 2010. — [xii], 130, [2] p. ISBN 978-0-9866300-1-9 This book contains what is known today about Fibonacci. The basic concepts about the Fibonacci numbers are given. The role of Fibonacci in the development of mathematics is considered and the value of Fibonacci’s work “Liber Abaci ” in the spread of Arabic numerals in Europe is assessed. The book is intended for a wide range of readers. It may be useful to students, teachers and scientists as it contains ex- cerpts and quotes, taken directly from the original works (primary sources) and presented both in the original language and in the translation. LCC: QA29.F5 D7 2010 DDC or MDS: 510.92 Copyright © 2010 • All rights reserved www. choven.ca © A. Drozdyuk, 2010 © D. Drozdyuk, 2010 to our mom and dad v vi C O N T E N T S CONTENTS........................................................... v Copyrights .......................................................... viii Foreword ...................................................................ix Acknowledgement.........................................................xi Chapter I. FIBONACCI............................................... 1 FIBONACCI’S BIOGRAPHY: CONJECTURES ONLY...................... 1 PISAN DOCUMENT ....................................................... 2 AUTOBIOGRAPHY ....................................................... 8 The second paragraph in Grimm’s interpretation ......................8 Grimm’s translation....................................................9 scriba ..................................................................10 duana ..................................................................12 Period of study ........................................................13 bugee ..................................................................13 NAMES IN THE TIMES OF FIBONACCI ............................... 13 About Leonardo’s names..............................................14 bigolli, bigollus, Bigollo ..............................................15 Fibonacci..............................................................16 LEONARDO’S FATHER ................................................. 19 Chapter II. FIBONACCI’S WORKS ............................. 21 FREDERICK II AND PROBLEMS FOR FIBONACCI .................... 23 Flos ...................................................................24 LUCAS’ PROPOSITION ................................................. 33 Chapter III. LIBER ABACI ........................................ 34 Extant manuscripts....................................................36 Liber Abaci or Liber Abbaci?.........................................37 Abbaco culture ........................................................37 Translation of the manuscript’s title ..................................38 The Rabbit Problem ..................................................39 vii Chapter IV. FIBONACCI NUMBERS ........................... 41 Commentary by Albert Girard ....................................... 48 Generalizations of Fibonacci numbers .............................. 49 Examples of generalizations ......................................... 52 Tribonacci, Tetranacci ............................................... 54 Binet’s formula and the curve M .................................... 55 One of the properties of Fibonacci sequence......................... 56 Why in the limit only................................................. 57 Chapter V. GOLDEN SECTION .................................. 58 Division of a line segment............................................ 60 Kepler on the two treasures of geometry............................. 62 PRO ET CONTRA (THE PROS AND CONS)............................. 65 The term “golden section” is an ancient ............................. 66 The Great Pyramid of Giza and φ .................................... 67 The Parthenon temple and φ ......................................... 68 The most aesthetically pleasing rectangle............................ 69 Artists and φ .......................................................... 70 Human body and φ ................................................... 71 Nautilus pompilius ................................................... 71 Fibonacci numbers in nature ......................................... 72 SPIRAL ................................................................. 74 Fibonacci spiral....................................................... 75 What shall we measure?.............................................. 78 STAGES OF RESEARCH ................................................ 81 Fechner’s experiment with the rectangles............................ 81 Ratios of the sides of the paintings................................... 84 OUR EXPERIMENTS ................................................... 86 Stage one ............................................................. 86 Stage two............................................................. 86 The choice............................................................ 87 The perception........................................................ 88 Stage three............................................................ 89 Aesthetic upbringing ................................................. 90 Our advice............................................................ 91 THE NORMAL PROCESS OF STUDYING ............................... 92 Chapter VI. ABOUT FIBONACCI AND HIS WORKS....... 93 Liber Abaci........................................................... 94 Practica Geometriae.................................................. 95 Liber Quadratorum................................................... 95 Flos, Epistola, Liber Quadratorum................................... 96 Aspects that are touched upon........................................ 96 Fibonacci’s qualification ............................................. 97 viii “Arabic” numbers and Fibonacci .....................................97 NOT THE ONLY BOOK ................................................. 99 Arithmetic of Al-Khwarizmi..........................................99 Poem by Alexander de Villa Dei.................................... 101 Sacrobosco’s “Algorismus ”......................................... 101 ON THE INFLUENCE OF LIBER ABACI .............................. 102 BORROWINGS......................................................... 104 Borrowings from the works of Fibonacci ........................... 105 Translations and copies of Fibonacci’s works ...................... 107 Pioneers of “Fibonacci numbers” ................................... 107 Chapter VII. FIBONACCI’S CONTRIBUTION .............. 111 400 years later....................................................... 111 600 years later....................................................... 113 Gabriel Lamé........................................................ 113 Édouard Lucas...................................................... 114 Leonard Eugene Dickson............................................ 118 Chapter VIII. WHY? ............................................... 122 Conclusion ............................................................. 124 REFERENCES..................................................... 125 ix Copyrights 1. All material is the sole property of the authors Andriy Drozdyuk and Denys Drozdyuk. 2. You are allowed to use up to 20% of the volume of this work in quoting without a special permission from the authors. 3. Free for non-commercial use. 4. The illustrations (two dimensional images), contained in the given work which are not the property of authors of the given work, are in the public domain because their copyrights have ex- pired — due to the dates of death of their authors, or due to their dates of publications. 5. The quotations from the primary sources, protected by the copyrights, are used under the terms of “fair dealings” or “fair use” — without special permissions from the copyrights owners because of the insignificant amounts used.