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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley Series in Pure and Applied Mathematics
- Sprache: Englisch
Praise for the First Edition " ...beautiful and well worth the reading ... with many exercises and a good bibliography, this book will fascinate both students and teachers." Mathematics Teacher Fibonacci and Lucas Numbers with Applications, Volume I, Second Edition provides a user-friendly and historical approach to the many fascinating properties of Fibonacci and Lucas numbers, which have intrigued amateurs and professionals for centuries. Offering an in-depth study of the topic, this book includes exciting applications that provide many opportunities to explore and experiment. In addition, the book includes a historical survey of the development of Fibonacci and Lucas numbers, with biographical sketches of important figures in the field. Each chapter features a wealth of examples, as well as numeric and theoretical exercises that avoid using extensive and time-consuming proofs of theorems. The Second Edition offers new opportunities to illustrate and expand on various problem-solving skills and techniques. In addition, the book features: * A clear, comprehensive introduction to one of the most fascinating topics in mathematics, including links to graph theory, matrices, geometry, the stock market, and the Golden Ratio * Abundant examples, exercises, and properties throughout, with a wide range of difficulty and sophistication * Numeric puzzles based on Fibonacci numbers, as well as popular geometric paradoxes, and a glossary of symbols and fundamental properties from the theory of numbers * A wide range of applications in many disciplines, including architecture, biology, chemistry, electrical engineering, physics, physiology, and neurophysiology The Second Edition is appropriate for upper-undergraduate and graduate-level courses on the history of mathematics, combinatorics, and number theory. The book is also a valuable resource for undergraduate research courses, independent study projects, and senior/graduate theses, as well as a useful resource for computer scientists, physicists, biologists, and electrical engineers. Thomas Koshy, PhD, is Professor Emeritus of Mathematics at Framingham State University in Massachusetts and author of several books and numerous articles on mathematics. His work has been recognized by the Association of American Publishers, and he has received many awards, including the Distinguished Faculty of the Year. Dr. Koshy received his PhD in Algebraic Coding Theory from Boston University. "Anyone who loves mathematical puzzles, number theory, and Fibonacci numbers will treasure this book. Dr. Koshy has compiled Fibonacci lore from diverse sources into one understandable and intriguing volume, [interweaving] a historical flavor into an array of applications." Marjorie Bicknell-Johnson
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Veröffentlichungsjahr: 2017
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Table of Contents
COVER
TITLE PAGE
COPYRIGHT
DEDICATION
LIST OF SYMBOLS
PREFACE
CHAPTER 1: LEONARDO FIBONACCI
CHAPTER 2: FIBONACCI NUMBERS
2.1 FIBONACCI'S RABBITS
2.2 FIBONACCI NUMBERS
2.3 FIBONACCI AND LUCAS CURIOSITIES
CHAPTER 3: FIBONACCI NUMBERS IN NATURE
3.1 FIBONACCI, FLOWERS, AND TREES
3.2 FIBONACCI AND MALE BEES
3.3 FIBONACCI, LUCAS, AND SUBSETS
3.4 FIBONACCI AND SEWAGE TREATMENT
3.5 FIBONACCI AND ATOMS
3.6 FIBONACCI AND REFLECTIONS
3.7 PARAFFINS AND CYCLOPARAFFINS
3.8 FIBONACCI AND MUSIC
3.9 FIBONACCI AND POETRY
3.10 FIBONACCI AND NEUROPHYSIOLOGY
3.11 ELECTRICAL NETWORKS
CHAPTER 4: ADDITIONAL FIBONACCI AND LUCAS OCCURRENCES
4.1 FIBONACCI OCCURRENCES
4.2 FIBONACCI AND COMPOSITIONS
4.3 FIBONACCI AND PERMUTATIONS
4.4 FIBONACCI AND GENERATING SETS
4.5 FIBONACCI AND GRAPH THEORY
4.6 FIBONACCI WALKS
4.7 FIBONACCI TREES
4.9 FIBONACCI AND THE STOCK MARKET
CHAPTER 5: FIBONACCI AND LUCAS IDENTITIES
5.1 SPANNING TREE OF A CONNECTED GRAPH
5.2 BINET'S FORMULAS
5.3 CYCLIC PERMUTATIONS AND LUCAS NUMBERS
5.4 COMPOSITIONS REVISITED
5.5 NUMBER OF DIGITS IN AND
5.6 THEOREM 5.8 REVISITED
5.7 CATALAN'S IDENTITY
5.8 ADDITIONAL FIBONACCI AND LUCAS IDENTITIES
5.9 FERMAT AND FIBONACCI
5.10 FIBONACCI AND
CHAPTER 6: GEOMETRIC ILLUSTRATIONS AND PARADOXES
6.1 GEOMETRIC ILLUSTRATIONS
6.2 CANDIDO'S IDENTITY
6.3 FIBONACCI TESSELLATIONS
6.4 LUCAS TESSELLATIONS
6.5 GEOMETRIC PARADOXES
6.6 CASSINI-BASED PARADOXES
6.7 ADDITIONAL PARADOXES
CHAPTER 7: GIBONACCI NUMBERS
7.1 GIBONACCI NUMBERS
7.2 GERMAIN'S IDENTITY
CHAPTER 8: ADDITIONAL FIBONACCI AND LUCAS FORMULAS
8.1 NEW EXPLICIT FORMULAS
8.2 ADDITIONAL FORMULAS
CHAPTER 9: THE EUCLIDEAN ALGORITHM
9.1 THE EUCLIDEAN ALGORITHM
9.2 FORMULA (5.5) REVISITED
9.3 LAMé'S THEOREM
CHAPTER 10: DIVISIBILITY PROPERTIES
10.1 FIBONACCI DIVISIBILITY
10.2 LUCAS DIVISIBILITY
10.3 FIBONACCI AND LUCAS RATIOS
10.4 AN ALTERED FIBONACCI SEQUENCE
CHAPTER 11: PASCAL'S TRIANGLE
11.1 BINOMIAL COEFFICIENTS
11.2 PASCAL'S TRIANGLE
11.3 FIBONACCI NUMBERS AND PASCAL'S TRIANGLE
11.4 ANOTHER EXPLICIT FORMULA FOR
11.5 CATALAN'S FORMULA
11.6 ADDITIONAL IDENTITIES
11.7 FIBONACCI PATHS OF A ROOK ON A CHESSBOARD
CHAPTER 12: PASCAL-LIKE TRIANGLES
12.1 SUMS OF LIKE-POWERS
12.2 AN ALTERNATE FORMULA FOR
12.3 DIFFERENCES OF LIKE-POWERS
12.4 CATALAN'S FORMULA REVISITED
12.5 A LUCAS TRIANGLE
12.6 POWERS OF LUCAS NUMBERS
12.7 VARIANTS OF PASCAL'S TRIANGLE
CHAPTER 13: RECURRENCES AND GENERATING FUNCTIONS
13.1 LHRWCCs
13.2 GENERATING FUNCTIONS
13.3 A GENERATING FUNCTION FOR
13.4 A GENERATING FUNCTION FOR
13.5 SUMMATION FORMULA (5.1) REVISITED
13.6 A LIST OF GENERATING FUNCTIONS
13.7 COMPOSITIONS REVISITED
13.8 EXPONENTIAL GENERATING FUNCTIONS
13.9 HYBRID IDENTITIES
13.10 IDENTITIES USING THE DIFFERENTIAL OPERATOR
CHAPTER 14: COMBINATORIAL MODELS I
14.1 A FIBONACCI TILING MODEL
14.2 A CIRCULAR TILING MODEL
14.3 PATH GRAPHS REVISITED
14.4 CYCLE GRAPHS REVISITED
14.5 TADPOLE GRAPHS
CHAPTER 15: HOSOYA'S TRIANGLE
15.1 RECURSIVE DEFINITION
15.2 A MAGIC RHOMBUS
CHAPTER 16: THE GOLDEN RATIO
16.1 RATIOS OF CONSECUTIVE FIBONACCI NUMBERS
16.2 THE GOLDEN RATIO
16.3 GOLDEN RATIO AS NESTED RADICALS
16.4 NEWTON'S APPROXIMATION METHOD
16.5 THE UBIQUITOUS GOLDEN RATIO
16.6 HUMAN BODY AND THE GOLDEN RATIO
16.7 VIOLIN AND THE GOLDEN RATIO
16.8 ANCIENT FLOOR MOSAICS AND THE GOLDEN RATIO
16.9 GOLDEN RATIO IN AN ELECTRICAL NETWORK
16.10 GOLDEN RATIO IN ELECTROSTATICS
16.11 GOLDEN RATIO BY ORIGAMI
16.12 DIFFERENTIAL EQUATIONS
16.13 GOLDEN RATIO IN ALGEBRA
16.14 GOLDEN RATIO IN GEOMETRY
CHAPTER 17: GOLDEN TRIANGLES AND RECTANGLES
17.1 GOLDEN TRIANGLE
17.2 GOLDEN RECTANGLES
17.3 THE PARTHENON
17.4 HUMAN BODY AND THE GOLDEN RECTANGLE
17.5 GOLDEN RECTANGLE AND THE CLOCK
17.6 STRAIGHTEDGE AND COMPASS CONSTRUCTION
17.7 RECIPROCAL OF A RECTANGLE
17.8 LOGARITHMIC SPIRAL
17.9 GOLDEN RECTANGLE REVISITED
17.10 SUPERGOLDEN RECTANGLE
CHAPTER 18: FIGEOMETRY
18.1 THE GOLDEN RATIO AND PLANE GEOMETRY
18.2 THE CROSS OF LORRAINE
18.3 FIBONACCI MEETS APOLLONIUS
18.4 A FIBONACCI SPIRAL
18.5 REGULAR PENTAGONS
18.6 TRIGONOMETRIC FORMULAS FOR
18.7 REGULAR DECAGON
18.8 FIFTH ROOTS OF UNITY
18.9 A PENTAGONAL ARCH
18.10 REGULAR ICOSAHEDRON AND DODECAHEDRON
18.11 GOLDEN ELLIPSE
18.12 GOLDEN HYPERBOLA
CHAPTER 19: CONTINUED FRACTIONS
19.1 FINITE CONTINUED FRACTIONS
19.2 CONVERGENTS OF A CONTINUED FRACTION
19.3 INFINITE CONTINUED FRACTIONS
19.4 A NONLINEAR DIOPHANTINE EQUATION
CHAPTER 20: FIBONACCI MATRICES
20.1 THE
Q
-MATRIX
20.2 EIGENVALUES OF
20.3 FIBONACCI AND LUCAS VECTORS
20.4 AN INTRIGUING FIBONACCI MATRIX
20.5 AN INFINITE-DIMENSIONAL LUCAS MATRIX
20.6 AN INFINITE-DIMENSIONAL GIBONACCI MATRIX
20.7 THE LAMBDA FUNCTION
CHAPTER 21: GRAPH-THEORETIC MODELS I
21.1 A GRAPH-THEORETIC MODEL FOR FIBONACCI NUMBERS
21.2 BYPRODUCTS OF THE COMBINATORIAL MODELS
21.3 SUMMATION FORMULAS
CHAPTER 22: FIBONACCI DETERMINANTS
22.1 AN APPLICATION TO GRAPH THEORY
22.2 THE SINGULARITY OF FIBONACCI MATRICES
22.3 FIBONACCI AND ANALYTIC GEOMETRY
CHAPTER 23: FIBONACCI AND LUCAS CONGRUENCES
23.1 FIBONACCI NUMBERS ENDING IN ZERO
23.2 LUCAS NUMBERS ENDING IN ZERO
23.3 ADDITIONAL CONGRUENCES
23.4 LUCAS SQUARES
23.5 FIBONACCI SQUARES
23.6 A GENERALIZED FIBONACCI CONGRUENCE
23.7 FIBONACCI AND LUCAS PERIODICITIES
23.8 LUCAS SQUARES REVISITED
23.9 PERIODICITIES MODULO
CHAPTER 24: FIBONACCI AND LUCAS SERIES
24.1 A FIBONACCI SERIES
24.2 A LUCAS SERIES
24.3 FIBONACCI AND LUCAS SERIES REVISITED
24.4 A FIBONACCI POWER SERIES
24.5 GIBONACCI SERIES
24.6 ADDITIONAL FIBONACCI SERIES
CHAPTER 25: WEIGHTED FIBONACCI AND LUCAS SUMS
25.1 WEIGHTED SUMS
25.2 GAUTHIER'S DIFFERENTIAL METHOD
CHAPTER 26: FIBONOMETRY I
26.1 GOLDEN RATIO AND INVERSE TRIGONOMETRIC FUNCTIONS
26.2 GOLDEN TRIANGLE REVISITED
26.3 GOLDEN WEAVES
26.4 ADDITIONAL FIBONOMETRIC BRIDGES
26.5 FIBONACCI AND LUCAS FACTORIZATIONS
CHAPTER 27: COMPLETENESS THEOREMS
27.1 COMPLETENESS THEOREM
27.2 EGYPTIAN ALGORITHM FOR MULTIPLICATION
CHAPTER 28: THE KNAPSACK PROBLEM
28.1 THE KNAPSACK PROBLEM
CHAPTER 29: FIBONACCI AND LUCAS SUBSCRIPTS
29.1 FIBONACCI AND LUCAS SUBSCRIPTS
29.2 GIBONACCI SUBSCRIPTS
29.3 A RECURSIVE DEFINITION OF
CHAPTER 30: FIBONACCI AND THE COMPLEX PLANE
30.1 GAUSSIAN NUMBERS
30.2 GAUSSIAN FIBONACCI AND LUCAS NUMBERS
30.3 ANALYTIC EXTENSIONS
APPENDIX 1: FUNDAMENTALS
SEQUENCES
SUMMATION AND PRODUCT NOTATIONS
INDEXED SUMMATION
THE PRODUCT NOTATION
THE FACTORIAL NOTATION
FLOOR AND CEILING FUNCTIONS
THE WELL-ORDERING PRINCIPLE (WOP)
MATHEMATICAL INDUCTION
SUMMATION FORMULAS
RECURSION
RECURSIVE DEFINITION OF A FUNCTION
THE DIVISION ALGORITHM
DIV AND MOD OPERATORS
DIVISIBILITY RELATION
DIVISIBILITY PROPERTIES
PIGEONHOLE PRINCIPLE
ADDITION PRINCIPLE
UNION AND INTERSECTION
GCD AND LCM
GREATEST COMMON DIVISOR
A SYMBOLIC DEFINITION OF GCD
RELATIVELY PRIME INTEGERS
GCD OF POSITIVE INTEGERS
FUNDAMENTAL THEOREM OF ARITHMETIC
CANONICAL DECOMPOSITION
LEAST COMMON MULTIPLE
A SYMBOLIC DEFINITION OF LCM
MATRICES AND DETERMINANTS
MATRICES
EQUALITY OF MATRICES
ZERO AND IDENTITY MATRICES
MATRIX OPERATIONS
MATRIX MULTIPLICATION
DETERMINANTS
MINORS AND COFACTORS
DETERMINANT OF A SQUARE MATRIX
CONGRUENCES
CONGRUENCE MODULO
APPENDIX 2: THE FIRST 100 FIBONACCI AND LUCAS NUMBERS
APPENDIX 3: THE FIRST 100 FIBONACCI NUMBERS AND THEIR PRIME FACTORIZATIONS
APPENDIX 4: THE FIRST 100 LUCAS NUMBERS AND THEIR PRIME FACTORIZATIONS
ABBREVIATIONS
REFERENCES
SOLUTIONS TO ODD-NUMBERED EXERCISES
INDEX
PURE AND APPLIED MATHEMATICS
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Begin Reading
List of Illustrations
CHAPTER 2: FIBONACCI NUMBERS
Figure 2.1 A Fibonacci tree.
Figure 2.2 Tree diagram for recursive computing of .
Figure 2.3 The first five triangular numbers.
Figure 2.4 The Fibonacci chimney at the Turun Energia Power Plant.
Figure 2.5 Fibonacci numbers on a building in Sweden.
CHAPTER 3: FIBONACCI NUMBERS IN NATURE
Figure 3.1 Flowers.
Figure 3.2 (a) Cross section of an apple. (b) Starfish.
Figure 3.3 An assortment of flowers with five petals.
Figure 3.4 Elm, cherry, and pear limbs.
Figure 3.5 A sunflower.
Figure 3.6 The spiral pattern in a sunflower.
Figure 3.7 Pinecone.
Figure 3.8 Scale patterns (8 clockwise spirals, 13 counterclockwise spirals).
Figure 3.9 Artichokes.
Figure 3.10 Pineapple.
Figure 3.11 Hexagonal scale patterns.
Figure 3.12 The family tree of a male bee.
Figure 3.13
Figure 3.14
Figure 3.15
Figure 3.16
Figure 3.17
Figure 3.18
Figure 3.19
Figure 3.20
Figure 3.21
Figure 3.22
Figure 3.23
Figure 3.24
Figure 3.25 (a) Two separate glass plates. (b) The stack has four reflective faces, labeled 1–4.
Figure 3.26 Stacked glass plates with no reflections.
Figure 3.27 Stacked glass plates with one reflection.
Figure 3.28 Stacked glass plates with two reflections.
Figure 3.29 Stacked glass plates with three reflections.
Figure 3.30 Stacked glass plates with four reflections.
Figure 3.31
Figure 3.32
Figure 3.33 Ethane molecule .
Figure 3.34
Figure 3.35
Figure 3.36
Figure 3.37 Fibonacci numbers in the octave of a piano keyboard.
Figure 3.38 Fibonacci ratios in musical intervals.
Figure 3.39 A sample pore.
Figure 3.40 Tree of states of a pore with 5 cells.
Figure 3.41 A tree diagram of Figure 3.40.
Figure 3.42 Resistor network: ladder sections.
Figure 3.43
Figure 3.44
Figure 3.45
Figure 3.46
Figure 3.47
Figure 3.48
Figure 3.49
Figure 3.50
Figure 3.51
Figure 3.52
Figure 3.53
Figure 3.56
Figure 3.57
Figure 3.60
Figure 3.61 Resistor network: ladder sections.
Figure 3.62
Figure 3.63
CHAPTER 4: ADDITIONAL FIBONACCI AND LUCAS OCCURRENCES
Figure 4.1 Possible ways of painting the building.
Figure 4.2
Figure 4.3
Figure 4.4
Figure 4.5
Figure 4.6
Figure 4.7
Figure 4.8
Figure 4.9
Figure 4.10 Lattice paths.
Figure 4.11 Fibonacci walks of length .
Figure 4.12 The Bernoulli family tree.
Figure 4.13
Figure 4.14
Figure 4.15
Figure 4.16 Fibonacci trees.
Figure 4.17
Figure 4.18
Figure 4.19 The fundamental behavior of the Elliott Wave Principle.
Figure 4.20 Waves within waves.
Figure 4.21 Forming large waves.
Figure 4.22
Figure 4.23
CHAPTER 5: FIBONACCI AND LUCAS IDENTITIES
Figure 5.1
Figure 5.2
Figure 5.3 Fan graphs.
Figure 5.4 Spanning tree of .
Figure 5.5 Spanning trees of .
Figure 5.6 Spanning trees of .
Figure 5.7 Possible ways of having vertex 4 in a spanning tree of .
Figure 5.8
Figure 5.9
Figure 5.10 Graph of .
Figure 5.11
Figure 5.12
Figure 5.13
Figure 5.14
Figure 5.15
Figure 5.16
CHAPTER 6: GEOMETRIC ILLUSTRATIONS AND PARADOXES
Figure 6.1 .
Figure 6.2 .
Figure 6.3 .
Figure 6.4 .
Figure 6.5
Figure 6.6
Figure 6.7
Figure 6.8 .
Figure 6.11 .
Figure 6.12 .
Figure 6.13
Figure 6.14
Figure 6.15
Figure 6.16 A Fibonacci Tessellation.
Figure 6.17
Figure 6.18
Figure 6.19
Figure 6.20
Figure 6.21
Figure 6.22
Figure 6.23
Figure 6.24
Figure 6.25
Figure 6.26
Figure 6.27
Figure 6.28
Figure 6.29
Figure 6.30
Figure 6.31
Figure 6.32
Figure 6.33
Figure 6.34
CHAPTER 11: PASCAL'S TRIANGLE
Figure 11.1 Pascal's triangle.
Figure 11.2 Pascal's triangle.
Figure 11.3 Pascal's triangle.
Figure 11.4
Figure 11.5
Figure 11.6
Figure 11.7
CHAPTER 12: PASCAL-LIKE TRIANGLES
Figure 12.1
Figure 12.2
Figure 12.3
Figure 12.4
Figure 12.5
Figure 12.6 Array
F
.
Figure 12.7
Figure 12.8
CHAPTER 13: RECURRENCES AND GENERATING FUNCTIONS
Figure 13.1
Figure 13.2 Paths from (0, 0) to (
n
, 0), where .
CHAPTER 14: COMBINATORIAL MODELS I
Figure 14.1 Tilings of a board.
Figure 14.2 5-Tilings.
Figure 14.3 5-Tilings with one or more dominoes.
Figure 14.4 Tiling unbreakable at cell
k
.
Figure 14.5 Tilings breakable at cell
k
.
Figure 14.6 5-Tilings with .
Figure 14.7 5-Tilings with .
Figure 14.8 5-Tilings with .
Figure 14.9 5-Tilings.
Figure 14.10 Tilings of a circular board.
Figure 14.11
Figure 14.12
Figure 14.13
Figure 14.14
Figure 14.15
Figure 14.16
Figure 14.17
Figure 14.18
Figure 14.19
Figure 14.20
Figure 14.21
Figure 14.22
Figure 14.23
Figure 14.24
Figure 14.25
Figure 14.26
Figure 14.27
Figure 14.28
Figure 14.29
Figure 14.30
Figure 14.31 Tadpole graph .
Figure 14.32 Tadpole graph .
Figure 14.33 Tadpole graph .
Figure 14.34 Tadpole triangle.
CHAPTER 15: HOSOYA'S TRIANGLE
Figure 15.1 Hosoya's triangle
H
.
Figure 15.2 A magic rhombus.
Figure 15.3
Figure 15.4
Figure 15.5
Figure 15.6
Figure 15.7
Figure 15.8
Figure 15.9
Figure 15.10
Figure 15.11
Figure 15.12
Figure 15.13
Figure 15.15
Figure 15.16
Figure 15.17
CHAPTER 16: THE GOLDEN RATIO
Figure 16.1 (a) The pyramids at Giza.
*
(b) Diagram of pyramid showing the golden section.
Figure 16.2
Figure 16.3
Figure 16.4
Figure 16.5 Graph of , where .
Figure 16.6 Proportions of the human body.
*
Figure 16.7
Figure 16.8 The point
B
on the violin, where the two lines drawn through the centers of the
f
holes intersect, divides the instrument in the golden ratio; the body and the neck are likewise in the golden ratio.
*
Figure 16.9 Calibrations on a ruler used in ancient mosaics.
*
Figure 16.10 An infinite network consisting of resistors.
Figure 16.11
Figure 16.12
Figure 16.13
Figure 16.14
Figure 16.15
Figure 16.16
Figure 16.17
Figure 16.18
Figure 16.19
Figure 16.20
Figure 16.21
Figure 16.22
Figure 16.23
Figure 16.24
Figure 16.25
Figure 16.26
Figure 16.27
Figure 16.28
CHAPTER 17: GOLDEN TRIANGLES AND RECTANGLES
Figure 17.1
Figure 17.2
Figure 17.3
Figure 17.4
Figure 17.5
Figure 17.6
Figure 17.7
Figure 17.8
Figure 17.9
Figure 17.10
Figure 17.11 The lighthouse divides the picture in a way that creates a golden rectangle.
*
Figure 17.12
Figure 17.13 (a)
St. Jerome
by da Vinci fits into a golden rectangle.
*
(b) Michelangelo's
David
also illustrates a golden rectangle.
*
Figure 17.14
Figure 17.15 (a) A view of the Parthenon in Athens. (b) This magnificent building fits into a golden rectangle.
Figure 17.16 The Parthenon in Nashville, Tennessee.
*
Figure 17.17 The
modulator
, a scale of proportions developed by Le Corbusier.
*
Figure 17.18 The doorway of the
Cathedral of Chartres
.
#
Figure 17.19 The Tower of Saint Jacques.
*
Figure 17.20 The golden proportions in a human head and face.
#
Figure 17.21 The golden proportions in a human hand.
*
Figure 17.22 A personal golden rectangle formed by a pointer finger.
*
Figure 17.23 An ancient graveyard cross.
*
Figure 17.24 A modern cross.
*
Figure 17.25 A Prostate Cancer Awareness stamp.
Figure 17.26 A Greek urn.
*
Figure 17.27 A wristwatch with hands set at 10:09.
Figure 17.28 The setting on a watch is related to the golden ratio.
Figure 17.29
Figure 17.30
Figure 17.31
Figure 17.32
Figure 17.33
Figure 17.34 Shells displaying the logarithmic spiral.
Figure 17.35 AWM logo.
Figure 17.36 Indigo hotel.
*
Figure 17.37
Figure 17.38
Figure 17.39 Supergolden rectangle.
Figure 17.40 Equal areas.
Figure 17.41 Graph of the equation .
Figure 17.42 Graph of the equation .
Figure 17.43 Geometric properties of the supergolden rectangle.
Figure 17.44
CHAPTER 18: FIGEOMETRY
Figure 18.1
Figure 18.2
Figure 18.3
Figure 18.4
Figure 18.5
Figure 18.6
Figure 18.7
Figure 18.8
Figure 18.9
Figure 18.10
Figure 18.11 The cross of Lorraine.
Figure 18.12
Figure 18.13 A Fibonacci spiral.
Figure 18.14 (a) A starfish. (b) The former Chrysler logo.*
Figure 18.15
Figure 18.16
Figure 18.17
Figure 18.18
Figure 18.19
Figure 18.20
Figure 18.21
Figure 18.22 Fifth roots of unity.
Figure 18.23
Figure 18.24
Figure 18.25 Pentagonal arch.
Figure 18.26 A regular icosahedron.
*
Figure 18.27 Three golden rectangles.*
Figure 18.28 The corners of three golden rectangles meet at those of a regular icosahedron.*
Figure 18.29 The corners of the same rectangles coincide with the centers of the sides of a regular dodecahedron.
*
Figure 18.30 The golden ellipse.
Figure 18.31 The golden hyperbola.
Figure 18.32
Figure 18.33
Figure 18.34
Figure 18.35
CHAPTER 19: CONTINUED FRACTIONS
Figure 19.1
Figure 19.2
Figure 19.3
CHAPTER 20: FIBONACCI MATRICES
Figure 20.1
CHAPTER 21: GRAPH-THEORETIC MODELS I
Figure 21.1
CHAPTER 22: FIBONACCI DETERMINANTS
Figure 22.1
Figure 22.2 Wheel graphs.
Figure 22.3 Spanning trees.
CHAPTER 26: FIBONOMETRY I
Figure 26.1 The golden ratio and trigonometric functions.
Figure 26.2
Figure 26.3
Figure 26.4 The development of a golden weave on a square loom of unit side for ; (a) after 3 reflections; (b) after 5 reflections; (c) after 15 reflections.
Figure 26.5 The weaving patterns for the first 15 reflections.
CHAPTER 28: THE KNAPSACK PROBLEM
Figure 28.1
Figure 28.2
List of Tables
CHAPTER 2: FIBONACCI NUMBERS
Table 2.1 Growth of the Rabbit Population
CHAPTER 3: FIBONACCI NUMBERS IN NATURE
Table 3.1 An Assortment of Flowers
Table 3.2 Phyllotactic Ratios
Table 3.3 Number of Bees per Generation
Table 3.4 Number of Paths
Table 3.5
Table 3.6 Atomic Number and Fibonacci Number
Table 3.7
Table 3.8 Topological Indices of Paraffins
Table 3.9 Topological Indices of Cycloparaffins
CHAPTER 4: ADDITIONAL FIBONACCI AND LUCAS OCCURRENCES
Table 4.1
Table 4.1
Table 4.2
Table 4.2
Table 4.3
Table 4.4
Table 4.5
Table 4.6
Table 4.7
Table 4.8
Table 4.9 Number of Permutations of
Table 4.10
Table 4.11
Table 4.12 Number of Fibonacci Walks
Table 4.13
Table 4.14
CHAPTER 5: FIBONACCI AND LUCAS IDENTITIES
Table 5.1 Cyclic
n
-bit Words With No Consecutive 1s
Table 5.2
CHAPTER 6: GEOMETRIC ILLUSTRATIONS AND PARADOXES
Table 6.1
CHAPTER 7: GIBONACCI NUMBERS
Table 7.1 Gibonacci Growth of Bees
Table 7.2
CHAPTER 10: DIVISIBILITY PROPERTIES
Table 10.1
CHAPTER 12: PASCAL-LIKE TRIANGLES
Table 12.1 Array
A
Table 12.2 Array
B
Table 12.3 Array
C
: a Lucas Triangle
Table 12.4 Array
D
: a Reflection of the Lucas Triangle
Table 12.5 Array
E
CHAPTER 13: RECURRENCES AND GENERATING FUNCTIONS
Table 13.1
CHAPTER 14: COMBINATORIAL MODELS I
Table 14.1
Table 14.2 Numbers of Independent Subsets of
CHAPTER 16: THE GOLDEN RATIO
Table 16.1 Ratios of Consecutive Fibonacci and Lucas Numbers
CHAPTER 18: FIGEOMETRY
Table 18.1 Exact Trigonometric Values
CHAPTER 19: CONTINUED FRACTIONS
Table 19.1
CHAPTER 20: FIBONACCI MATRICES
Table 20.1
Table 20.2
Table 20.3
CHAPTER 21: GRAPH-THEORETIC MODELS I
Table 21.1 Paths of Length 4
Table 21.2 Closed Paths of Length 6 Starting at
Table 21.3 Closed Paths of Length 5 from to Itself
Table 21.4 Closed Paths of Length 5 Originating at
Table 21.5 Closed Paths of Length 5 Originating at
Table 21.6 Closed Paths of Length 5 Originating at
Table 21.7 Closed Paths of Length 5 from to Itself with at Least One d-edge
CHAPTER 23: FIBONACCI AND LUCAS CONGRUENCES
Table 23.1 Lucas Numbers Modulo 8
CHAPTER 27: COMPLETENESS THEOREMS
Table 27.1
Table 27.2
Table 27.3
PURE AND APPLIED MATHEMATICS
A Wiley Series of Texts, Monographs, and Tracts
Founded by RICHARD COURANT
Editors Emeriti: MYRON B. ALLEN III, PETER HILTON, HARRY
HOCHSTADT, ERWIN KREYSZIG, PETER LAX, JOHN TOLAND
A complete list of the titles in this series appears at the end of this volume.
FIBONACCI AND LUCAS NUMBERS WITH APPLICATIONS
Volume One
Second Edition
THOMAS KOSHY
Framingham State University
Copyright © 2018 by John Wiley & Sons, Inc. All rights reserved
Published by John Wiley & Sons, Inc., Hoboken, New Jersey
Published simultaneously in Canada
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Library of Congress Cataloging-in-Publication Data:
Names: Koshy, Thomas.
Title: Fibonacci and Lucas numbers with applications / Thomas Koshy, Framingham State University.
Description: Second edition. | Hoboken, New Jersey : John Wiley & Sons, Inc., [2017]- | Series: Pure and applied mathematics: a Wiley series of texts, monographs, and tracts | Includes bibliographical references and index.
Identifiers: LCCN 2016018243 | ISBN 9781118742129 (cloth : v. 1)
Subjects: LCSH: Fibonacci numbers. | Lucas numbers.
Classification: LCC QA246.5 .K67 2017 | DDC 512.7/2-dc23 LC record available at https://lccn.loc.gov/2016018243
Dedicated to Suresh, Sheeba, Neethu, and Vinod
LIST OF SYMBOLS
Symbol
Meaning
set of positive integers 1, 2, 3, 4,
set of whole numbers 0, 1, 2, 3,
set of integers
set of real numbers
sequence with general term
sum of the values of
as
runs
over the values in
sum of the values of
, where
satisfies properties
(
factorial)
, where
absolute value of
(the floor of
)
the greatest integer
(the ceiling of
)
the least integer
PMI
principle of mathematical induction
div
quotient when
is divided by
mod
remainder when
is divided by
is a factor of
is not a factor of
set consisting of the elements
PREFACE
Man has the faculty of becoming completely absorbed in one subject,
no matter how trivial and no subject is so trivial that it will not assume
infinite proportions if one's entire attention is devoted to it.
–Tolstoy, War and Peace
THE TWIN SHINING STARS
The Fibonacci sequence and the Lucas sequence are two very bright shining stars in the vast array of integer sequences. They have fascinated amateurs, and professional architects, artists, biologists, musicians, painters, photographers, and mathematicians for centuries; and they continue to charm and enlighten us with their beauty, their abundant applications, and their ubiquitous habit of occurring in totally surprising and unrelated places. They continue to be a fertile ground for creative amateurs and mathematicians alike, and speak volumes about the vitality of this growing field.
This book originally grew out of my fascination with the intriguing beauty and rich applications of the twin sequences. It has been my long-cherished dream to study and to assemble the myriad properties of both sequences, developed over the centuries, and to catalog their applications to various disciplines in an orderly and enjoyable fashion. As the cryptanalyst Sophie Neveu in Dan Brown's bestseller The Da Vinci Code claims, “the [Fibonacci] sequence … happens to be one of the most famous mathematical progressions in history.”
An enormous wealth of information is available in the mathematical literature on Fibonacci and Lucas numbers; but, unfortunately, most of it continues to be widely scattered in numerous journals, so it is not easily accessible to many, especially to non-professionals. The first edition was the end-product of materials collected and presented from a wide range of sources over the years; and to the best of my knowledge, it was the largest comprehensive study of this beautiful area of human endeavor.
So why this new edition? Since the publication of the original volume, I have had the advantage and fortune of hearing from a number of Fibonacci enthusiasts from around the globe, including students. Their enthusiasm, support, and encouragement were really overwhelming. Some opened my eyes to new sources and some to new charming properties; and some even pointed out some inexcusable typos, which eluded my own eyes. The second edition is the byproduct of their ardent enthusiasm, coupled with my own.
Many Fibonacci enthusiasts and amateurs know the basics of Fibonacci and Lucas numbers. But there are a multitude of beautiful properties and applications that may be less familiar. Fibonacci and Lucas numbers are a source of great fun; teachers and professors often use them to generate excitement among students, who find them stimulating their intellectual curiosity and sharpening their mathematical skills, such as pattern recognition, conjecturing, proof techniques, and problem-solving. In the process, they invariably appreciate and enjoy the beauty, power, and ubiquity of the Fibonacci family.
AUDIENCE
As can be predicted, this book is intended for a wide audience, not necessarily of professional mathematicians. College undergraduate and graduate students often opt to study Fibonacci and Lucas numbers because they find them challenging and exciting. Often many students propose new and interesting problems in periodicals. It is certainly delightful and rewarding that they often pursue Fibonacci and Lucas numbers for their senior and master's thesis. In short, it is well-suited for projects, seminars, group discussions, proposing and solving problems, and extending known results.
High School students have enjoyed exploring this material for a number of years. Using Fibonacci and Lucas numbers, students at Framingham High School in Massachusetts, for example, have published many of their discoveries in Mathematics Teacher.
As in the first edition, I have included a large array of advanced material and exercises to challenge mathematically sophisticated enthusiasts and professionals in such diverse fields as art, biology, chemistry, electrical engineering, neurophysiology, physics, music, and the stock market. It is my sincere hope that this edition will also serve them as a valuable resource in exploring new applications and discoveries, and advance the frontiers of mathematical knowledge, experiencing a lot of satisfaction and joy in the process.
MAJOR CHANGES
In the interest of brevity and aestheticism, I have consolidated several closely-related chapters, resulting in fewer chapters in the new edition. I also have rearranged some chapters for a better flow of the development of topics. A number of new and charming properties, exercises, and applications have been added; so are a number of direct references to Fibonacci numbers, the golden ratio, and the pentagram to D. Brown's The Da Vinci Code. The chapters on Combinatorial Models I (Chapter 14) and Graph-theoretic Models I (Chapter 21) present spectacular opportunities to interpret Fibonacci and Lucas numbers combinatorially; so does the section on Fibonacci Walks (Section 4.6). I also have added a new way of looking at and studying them geometrically (Chapter 6).
Again, in the interest of brevity, the chapters on Fibonacci, Lucas, Jacobsthal, and Morgan-Voyce polynomials have been dropped from this edition; but they will be treated extensively in the forthcoming Volume Two. The chapters on tribonacci numbers and polynomials also will appear in the new volume.
ORGANIZATION
In the interest of manageability, the book is divided into 30 chapters. Nearly all of them are well within reach of many users. Most chapters conclude with a substantial number of interesting and challenging exercises for Fibonacci enthusiasts to explore, conjecture, and confirm. I hope, the numerous examples and exercises are as exciting for readers as they are for me. Where the omission can be made without sacrificing the essence of development or focus, I have omitted some of the long, tedious proofs of theorems. Abbreviated solutions to all odd-numbered exercises are given in the back of the book.
SALIENT FEATURES
Salient features of this edition remain the same as its predecessor: a user-friendly, historical approach; a non-intimidating style; a wealth of applications, exercises, and identities of varying degrees of difficulty and sophistication; links to combinatorics, graph theory, matrices, geometry, and trigonometry; the stock market; and relationships to everyday life. For example, works of art are discussed vis-à-vis the golden ratio (phi), one of the most intriguing irrational numbers. It is no wonder that Langdon in The Da Vinci Code claims that “PHI is the most beautiful number in the universe.”
APPLICATIONS
This volume contains numerous and fascinating applications to a wide spectrum of disciplines and endeavors. They include art, architecture, biology, chemistry, chess, electrical engineering, geometry, graph theory, music, origami, poetry, physics, physiology, neurophysiology, sewage/water treatment, snow plowing, stock market trading, and trigonometry. Most of the applications are well within the reach of mathematically sophisticated amateurs, although they vary in difficulty and sophistication.
HISTORICAL PERSPECTIVE
Throughout, I have tried to present historical background for the material, and to humanize the discourse by giving the name and affiliation of every contributor to the field, as well as the year of contribution. I have included photographs of a number of mathematicians, who have contributed significantly to this exciting field. My apologies to any discoverers whose names or affiliations are still missing; I would be pleased to hear of any such inadvertent omissions.
NUMERIC AND GEOMETRIC PUZZLES
This volume contains several numeric and geometric puzzles based on Fibonacci and Lucas numbers. They are certainly a source of fun, excitement, and surprise for every one. They also provide opportunities for further exploration.
LIST OF SYMBOLS
An updated List of symbols appears between the Contents and Preface. Although they are all standard symbols, they will come in handy for those not familiar with them.
APPENDIX
The Appendix contains a short list of the fundamental properties from the theory of numbers and the theory of matrices. It is a good idea to review them as needed. Those who are curious about their proofs will find them in Elementary Number Theory with Applications by the author.
The Appendix also contains a list of the first 100 Fibonacci and Lucas numbers, and their prime factorizations. They all should come in handy for computations.
A WORK IN PROGRESS
A polynomial approach to Fibonacci and Lucas numbers creates new opportunities for optimism, creativity, and elegance. It acts like a thread unifying Fibonacci, Lucas, Pell, Pell-Lucas, Chebyshev, and Vieta polynomials. Such polynomials, and their combinatorial and graph-theoretic models, among other topics, will be studied in detail in a successor volume.
ACKNOWLEDGMENTS
It is my great pleasure and joy to express my sincere gratitude to a number of people who have helped me to improve the manuscript of both editions with their constructive suggestions, comments, and support, and to those who sent in the inexcusable typos in the first edition. To begin with, I am deeply grateful to the following reviewers of the first or second edition for their boundless enthusiasm and input:
Napoleon Gauthier
The Royal Military College, Kingston, Ontario, Canada
Henry W. Gould
West Virginia University, Morgantown, West Virginia
Marjorie Bicknell-Johnson
Santa Clara, California
Thomas E. Moore
Bridgewater State University, Bridgewater, Massachusetts
Carl Pomerance
University of Georgia, Athens, Georgia
M.N.S. Swamy
Concordia University, Montreal, Quebec, Canada
Monte J. Zerger
Adams State University, Alamosa, Colorado
Gerald Alexanderson
Santa Clara University, Santa Clara, California
Anonymous Reviewer
(2nd edition)
Thanks to (the late) Angelo DiDomenico of Framingham High School, who read an early version of the first edition and offered valuable suggestions; to Margarite Landry for her superb editorial assistance; to Kevin Jackson-Mead for preparing the canonical prime factorizations of the Lucas numbers through , and for proofreading the entire work of the first edition; to (the late) Thomas Moore for the graphs in Figures 5.10, 17.41, and 17.42; to Marjorie Bicknell-Johnson and Krishnaswami Alladi for their quotes; and to the staff at Wiley, especially, Susanne Steitz-Filler, Allison McGinniss, and Kathleen Pagliaro for their enthusiasm, cooperation, support, and confidence in this huge endeavor.
Finally, I would be delighted to hear from Fibonacci enthusiasts about any possible elusive errors. If you should have any questions, or should come across or discover any additional properties or applications, I would be delighted to hear about them.
Thomas Koshy [[email protected]] Framingham, Massachusetts August, 2017
If I have been able to see farther, it was only because I stood on the shoulders of giants
– Sir Isaac Newton (1643–1727)
The author has provided a lucid and comprehensive treatment of Fibonacci and Lucas numbers. He has emphasized the beauty of the identities they satisfy, indicated the settings in mathematics and in nature where they occur, and discussed several applications. The book is easily readable and will be useful to experts and non-experts alike.
–Krishnaswami Alladi, University of Florida
LEONARDO FIBONACCI
Leonardo Fibonacci, also called Leonardo Pisano or Leonard of Pisa, was the most outstanding mathematician of the European Middle Ages. Little is known about his life except for the few facts he gives in his mathematical writings. Ironically, none of his contemporaries mention him in any document that survives.
Fibonacci was born around 1170 into the Bonacci family of Pisa, a prosperous mercantile center. (“Fibonacci” is a contraction of “Filius Bonacci,” son of Bonacci.) His father Guglielmo (William) was a successful merchant, who wanted his son to follow his trade.
Around 1190 when Guglielmo was appointed collector of customs in the Algerian city of Bugia (now called Bougie), he brought Leonardo there to learn the art of computation. In Bougie, Fibonacci received his early education from a Muslim schoolmaster, who introduced him to the Indian numeration system and Indian computational techniques. He also introduced Fibonacci to a book on algebra, Hisâb al-jabr w'almuqabâlah, written by the Persian mathematician al-Khowarizmi (ca. 825). (The word algebra is derived from the title of this book.)
As an adult, Fibonacci made frequent business trips to Egypt, Syria, Greece, France, and Constantinople, where he studied the various systems of arithmetic then in use, and exchanged views with native scholars. He also lived for a time at the court of the Roman Emperor, Frederick II (1194–1250), and engaged in scientific debates with the Emperor and his philosophers.
Fibonacci
*
Around 1200, at the age of 30, Fibonacci returned home to Pisa. He was convinced of the elegance and practical superiority of the Indian numeration system over the Roman system then in use in Italy. In 1202 Fibonacci published his pioneering work, Liber Abaci (The Book of the Abacus). (The word abaci here does not refer to the hand calculator called an abacus, but to computation in general.) Liber Abaci was devoted to arithmetic and elementary algebra; it introduced the Indian numeration system and arithmetic algorithms to Europe. In fact, Fibonacci demonstrated in his book the power of the Indian numeration system more vigorously than in any mathematical work up to that time. Liber Abaci's 15 chapters explain the major contributions to algebra by al-Khowarizmi and Abu Kamil (ca. 900), another Persian mathematician. Six years later, Fibonacci revised Liber Abaciand dedicated the second edition to Michael Scott, the most famous philosopher and astrologer at the court of Frederick II.
After Liber Abaci, Fibonacci wrote three other influential books. Practica Geometriae (Practice of Geometry), published in 1220, is divided into eight chapters and is dedicated to Master Domonique, about whom little is known. This book skillfully presents geometry and trigonometry with Euclidean rigor and some originality. Fibonacci employs algebra to solve geometric problems and geometry to solve algebraic problems, a radical approach for the Europe of his day.
The next two books, the Flos (Blossom or Flower) and the Liber Quadratorum (The Book of Square Numbers) were published in 1225. Although both deal with number theory, Liber Quadratorum earned Fibonacci his modern reputation as a major number theorist, ranked with the Greek mathematician Diophantus (ca. 250 A.D.) and the French mathematician Pierre de Fermat (1601–1665). Both Flos and Liber Quadratorum exemplify Fibonacci's brilliance and originality of thought, which outshine the abilities of most scholars of his time.
In 1225, Frederick II wanted to test Fibonacci's talents, so he invited Fibonacci to his court for a mathematical tournament. The contest consisted of three problems, prepared by Johannes of Palumbo, who was on the Emperor's staff. The first was to find a rational number such that both and are squares of rational numbers†. Fibonacci gave the correct answer 41/12: and .
The second problem was to find a solution to the cubic equation . Fibonacci showed geometrically that it has no solutions of the form , but gave an approximate solution, 1.3688081075, which is correct to nine decimal places. This answer appears in the Flos without any explanation.
The third problem, also recorded in Flos, was to solve the following:
Three people share 1/2, 1/3, and 1/6 of a pile of money. Each takes some money from the pile until nothing is left. The first person then returns one-half of what he took, the second one-third, and the third one-sixth. When the total thus returned is divided among them equally, each possesses his correct share. How much money was in the original pile? How much did each person take from the pile?
Fibonacci established that the problem is indeterminate and gave 47 as the smallest answer. None of Fibonacci's competitors in the contest could solve any of these problems.
The Emperor recognized Fibonacci's contributions to the city of Pisa, both as a teacher and as a citizen. Today, a statue of Fibonacci stands in the Camposanto Monumentale at Piazza dei Miracoli, near the Cathedral and the Leaning Tower of Pisa. Until 1990, it had been at a garden across the Arno River for some years.
Not long after Fibonacci's death in 1240,* Italian merchants# began to appreciate the beauty and power of the Indian numeration system, and gradually adopted it for business transactions. By the end of the sixteenth century, most of Europe had accepted it. Liber Abaci remained the European standard for more than two centuries, and played a significant role in displacing the unwieldy Roman numeration system, thereby spreading the more efficient Indian number system to the rest of world.
Figure source
: David Eugene Smith Collection, Rare Book & Manuscript Library, Columbia University in the City of New York. Reproduced with permission of Columbia University.
†
A solution to the problem appears in
The Mathematics Teacher
, Vol. 45 (1952), 605–606. R.A. Laird of New Orleans, Louisiana, reproduced it in
The Fibonacci Quarterly
3 (1965), pp. 121–122. The general solution to the problem that both
and
be rational squares appears in O. Ore,
Number Theory and its History
, McGraw-Hill, New York, 1948, pp. 188–193.
*
Figure source
:
www.epsilones.com/paginas/artes/artes-027-fibonacci-estatua.html
. Reproduced with permission of Alberto Rodríquez Santos.
#
Figure source
: Reproduced with permission of Marjorie Bicknell-Johnson.
FIBONACCI NUMBERS
It may be hard to define mathematical beauty,
but that is true of beauty of any kind.
— G.H. Hardy (1877–1947), A Mathematician's Apology
2.1 FIBONACCI'S RABBITS
Fibonacci's Classic work, Liber Abaci, contains many elementary problems, including the following famous problem about rabbits:
Suppose there are two newborn rabbits, one male and the other female. Find the number of rabbits produced in a year if:
Each pair takes one month to become mature;
Each pair produces a mixed pair every month, beginning with the second month; and
Rabbits are immortal.
Suppose, for convenience, that the original pair of rabbits was born on January 1. They take a month to become mature, so there is still only one pair on February 1. On March 1, they are two months old and produce a new mixed pair, a total of two pairs. Continuing like this, there will be three pairs on April 1, five pairs on May 1, and so on; see the last row of Table 2.1.
Table 2.1 Growth of the Rabbit Population
Number of Pairs
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Adults
0
1
1
2
3
5
8
13
Babies
1
0
1
1
2
3
5
8
Total
1
1
2
3
5
8
13
21
2.2 FIBONACCI NUMBERS
The numbers in the bottom row are called Fibonacci numbers, and the sequence is the Fibonacci sequence. Table A.2 in the Appendix lists the first 100 Fibonacci numbers.
The sequence was given its name in May, 1876, by the outstanding French mathematician François Édouard Anatole Lucas, who had originally called it “the series of Lamé,” after the French mathematician Gabriel Lamé (1795–1870). It is a bit ironic that despite Fibonacci's numerous mathematical contributions, he is primarily remembered for the sequence that bears his name.
François Édouard Anatole Lucas* was born in Amiens, France, in 1842. After completing his studies at the École Normale in Amiens, he worked as an assistant at the Paris Observatory. He served as an artillery officer in the Franco-Prussian war and then became professor of mathematics at the Lycée Saint-Louis and Lycée Charlemagne, both in Paris. He was a gifted and entertaining teacher.
Lucas died of a freak accident at a banquet; his cheek was gashed by a shard from a plate that was accidentally dropped; he died from the infection within a few days, on October 3, 1891. Lucas loved computing and developed plans for a computer, but it never materialized. Besides his contributions to number theory, he is known for his four-volume classic on recreational mathematics, Récréations mathématiques (1891–1894). Best known among the problems he developed is the Tower of Brahma (or Tower of Hanoi).
The Fibonacci sequence is one of the most intriguing number sequences. It continues to provide ample opportunities for professional mathematicians and amateurs to make conjectures and expand their mathematical horizon.
The sequence is so important and beautiful that The Fibonacci Association, an organization of mathematicians, has been formed for the study of Fibonacci and related integer sequences. The association was co-founded in 1963 by Verner E. Hoggatt, Jr. of San Jose State College (now San Jose State University), California, Brother Alfred Brousseau of St. Mary's College in California, and I. Dale Ruggles of San Jose State College. The association publishes The Fibonacci Quarterly, devoted to articles related to integer sequence.
Verner Emil Hoggatt, Jr.* (1921–1980) received his Ph.D. in 1955 from Oregon State University. His “life was marked by dedication to the study of the Fibonacci sequence. His production and creativity were [truly] astounding” [44], according to Marjorie Bicknell-Johnson, who has written extensively in The Fibonacci Quarterly.
Hoggatt was the founding editor of the Quarterly. He authored or co-authored more than 150 research articles. In addition, he wrote the book Fibonacci Numbers in 1969, and edited three other books. In short, Hoggatt had a brilliant and productive professional life.
Brother Alfred Brousseau* (1907–1988) began teaching at St. Mary's College, Moraga, California, in 1930. While there, he continued his studies in physics and earned his Ph.D. in 1937 from the University of California. Four years later, he became Principal of Sacred Heart High School. In 1937, Br. Alfred returned to St. Mary's College.
The April 4, 1969 issue of Time [170] featured Hoggatt and Br. Alfred in the article “The Fibonacci Numbers.”
Br. Alfred Brousseau later became Br. U. Alfred, when the Catholic brotherhood changed the way brothers were named; see Chapter 5.
The Fibonacci sequence has a fascinating property: every Fibonacci number, except the first two, is the sum of the two immediately preceding Fibonacci numbers. (At the given rate, there will be 144 pairs of rabbits on December 1. This can be confirmed by extending Table 2.1 through December.)
RECURSIVE DEFINITION
This observation yields the following recursive definition of the th Fibonacci number :
where . We will formally confirm the validity of this recurrence shortly.
It is not known whether Fibonacci knew of the recurrence. If he did, we have no record to that effect. The first written confirmation of the recurrence appeared three centuries later, when the great German astronomer and mathematician Johannes Kepler (1571–1630) wrote that Fibonacci must have surely noticed this recursive relationship. In any case, it was first noted in the west by the Dutch mathematician Albert Girard (1595–1632).
Numerous scholars cite the Fibonacci sequence in Sanskrit. Susantha Goonatilake attributes its discovery to the Indian writer Pingala (200 B.C.?). Parmanand Singh of Raj Narain College, Hajipur, Bihar, India, writes that what we call Fibonacci numbers and the recursive formulation were known in India several centuries before Fibonacci proposed the problem; they were given by Virahanka (between 600 and 800 A.D.), Gopala (prior to 1135 A.D.), and the Jain scholar Acharya Hemachandra (about 1150 A.D.). Fibonacci numbers occur as a special case of a formula established by Narayana Pandita (1156 A.D.).
The growth of the rabbit population can be displayed nicely in a tree diagram, as Figure 2.1 shows. Each new branch of the “tree” becomes an adult branch in one month and each adult branch, including the trunk, produces a new branch every month.
Figure 2.1 A Fibonacci tree.
Table 2.1 shows several interesting relationships among the numbers of adult pairs, baby pairs, and total pairs. To see them, let denote the number of adult pairs and the number of baby pairs in month , where . Clearly, , and .
Suppose . Since each adult pair produces a mixed baby pair in month , the number of baby pairs in month equals that of adult pairs in the preceding month; that is, . Then
Thus satisfies the Fibonacci recurrence, where . Consequently, , where .
Notice that
That is, , where . This establishes the Fibonacci recurrence observed earlier.
Since , it follows that the th element in row 1 is , where . Likewise, since , the th element in row 2 is , where .
The tree diagram in Figure 2.2 illustrates the recursive computing of , where each dot represents an addition.
Figure 2.2 Tree diagram for recursive computing of .
Using Fibonacci recurrence, we can assign a meaningful value to . Since , it follows that . This will come in handy in our pursuit of Fibonacci properties later.
Fibonacci recurrence has an immediate consequence to geometry. To see this, consider a nontrivial triangle. By the triangle inequality, the sum of the lengths of any two sides is greater than the length of the third side. Consequently, it follows by the Fibonacci recurrence that no three consecutive Fibonacci numbers can be the lengths of the sides of a nontrivial triangle.
Next we introduce another integer family.
LUCAS NUMBERS
The Fibonacci recurrence, coupled with different initial conditions, can be used to construct new number sequences. For instance, let be the th term of a sequence with and , where . The resulting sequence is the Lucas sequence
