The Golden Ratio and Fibonacci Numbers

Author: R. A. Dunlap

Publisher: World Scientific

ISBN: 9789812386304

Category: Mathematics

Page: 162

View: 8578

In this invaluable book, the basic mathematical properties of the golden ratio and its occurrence in the dimensions of two- and three-dimensional figures with fivefold symmetry are discussed. In addition, the generation of the Fibonacci series and generalized Fibonacci series and their relationship to the golden ratio are presented. These concepts are applied to algorithms for searching and function minimization. The Fibonacci sequence is viewed as a one-dimensional aperiodic, lattice and these ideas are extended to two- and three-dimensional Penrose tilings and the concept of incommensurate projections. The structural properties of aperiodic crystals and the growth of certain biological organisms are described in terms of Fibonacci sequences. Contents: Basic Properties of the Golden Ratio; Geometric Problems in Two Dimensions; Geometric Problems in Three Dimensions; Fibonacci Numbers; Lucas Numbers and Generalized Fibonacci Numbers; Continued Fractions and Rational Approximants; Generalized Fibonacci Representation Theorems; Optimal Spacing and Search Algorithms; Commensurate and Incommensurate Projections; Penrose Tilings; Quasicrystallography; Biological Applications; Construction of the Regular Pentagon; The First 100 Fibonacci and Lucas Numbers; Relationships Involving the Golden Ratio and Generalized Fibonacci Numbers. Readership: Applied mathematicians.

The Golden Ratio & Fibonacci Sequence: Golden Keys to Your Genius, Health, Wealth & Excellence

Author: Matthew K. Cross,Robert D. Friedman M. D.

Publisher: Hoshin Media

ISBN: 9781939623003

Category: Self-Help

Page: 64

View: 1762

It's been called the Secret of the Universe, a Most Precious Jewel, Nature's Path of Least Resistance and Maximum Performance and The Golden Key to the Cosmos. It is the Grand Pattern of Life, Growth and Success. It has fascinated and appears in the works of the world's greatest geniuses, including Leonardo Da Vinci, Albert Einstein, Walt Disney and Steve Jobs. Now, the open secret of the Golden Ratio and Fibonacci Sequence is revealed for your enjoyment and application. In this book you'll journey through Golden Ratio appearances and applications throughout Nature and the Universe and in all fields of human existence and endeavor, including: art, architecture, astronomy, beauty, business, culture, design, finance, health and longevity, geometry, mathematics, medicine, relationships, science and spirituality. This book presents a delightful cornucopia of Golden Ratio facts and facets, offering a bright window into its universality and power. The information presented is laser-like, with the ability to activate your unique genius and transform your consciousness. Universal wisdom will be at your fingertips once your Golden Doors of Perception are opened to the sacred design code of the Universe: The Golden Ratio & Fibonacci Sequence.

Fibonacci and Lucas Numbers, and the Golden Section

Theory and Applications

Author: Steven Vajda

Publisher: Courier Corporation

ISBN: 0486462765

Category: Mathematics

Page: 189

View: 1916

This survey of the use of Fibonacci and Lucas numbers and the ancient principle of the Golden Section covers areas relevant to operational research, statistics, and computational mathematics. 1989 edition.

Topics in Mathematical Modeling

Author: K. K. Tung

Publisher: Princeton University Press

ISBN: 1400884055

Category: Mathematics

Page: 336

View: 5013

Topics in Mathematical Modeling is an introductory textbook on mathematical modeling. The book teaches how simple mathematics can help formulate and solve real problems of current research interest in a wide range of fields, including biology, ecology, computer science, geophysics, engineering, and the social sciences. Yet the prerequisites are minimal: calculus and elementary differential equations. Among the many topics addressed are HIV; plant phyllotaxis; global warming; the World Wide Web; plant and animal vascular networks; social networks; chaos and fractals; marriage and divorce; and El Niño. Traditional modeling topics such as predator-prey interaction, harvesting, and wars of attrition are also included. Most chapters begin with the history of a problem, follow with a demonstration of how it can be modeled using various mathematical tools, and close with a discussion of its remaining unsolved aspects. Designed for a one-semester course, the book progresses from problems that can be solved with relatively simple mathematics to ones that require more sophisticated methods. The math techniques are taught as needed to solve the problem being addressed, and each chapter is designed to be largely independent to give teachers flexibility. The book, which can be used as an overview and introduction to applied mathematics, is particularly suitable for sophomore, junior, and senior students in math, science, and engineering.

MATLAB Guide to Fibonacci Numbers and the Golden Ratio

A Simplified Approach

Author: Peter I. Kattan

Publisher: CreateSpace

ISBN: 9781466229877


Page: 172

View: 1675

This book is written for people who love MATLAB and Fibonacci numbers. One of the objectives of writing this book is to introduce MATLAB and Fibonacci numbers to students in high schools. The material presented is very easy and simple to understand -- written in a very simplified manner. The book is divided into two parts - A review of MATLAB is presented in the first part. This is followed by a presentation of Fibonacci numbers and the Golden Ratio in the second part. The topics covered in the book include symbolic computing with MATLAB, solving equations with MATLAB, MATLAB functions, graphs in MATLAB, Fibonacci numbers, The Golden Ratio, Properties of the Golden Ratio, Lucas numbers, Generalizations of Fibonacci numbers, and Random Fibonacci numbers. The book includes numerous M-files written specifically to compute different kinds of Fibonacci sequences and the Golden Ratio. The author has been using MATLAB for the past seventeen years and is the author of the best selling books "MATLAB for Beginners" and "MATLAB Guide to Finite Elements".

Fibonacci’s Liber Abaci

A Translation into Modern English of Leonardo Pisano’s Book of Calculation

Author: Laurence Sigler

Publisher: Springer Science & Business Media

ISBN: 1461300797

Category: Mathematics

Page: 638

View: 5133

First published in 1202, Fibonacci’s Liber Abaci was one of the most important books on mathematics in the Middle Ages, introducing Arabic numerals and methods throughout Europe. This is the first translation into a modern European language, of interest not only to historians of science but also to all mathematicians and mathematics teachers interested in the origins of their methods.

Fibonacci Numbers

Author: Nikolai Nikolaevich Vorob'ev

Publisher: Courier Corporation

ISBN: 048629885X

Category: Mathematics

Page: 80

View: 8105

An engaging treatment of an 800-year-old problem explores the occurrence of Fibonacci numbers in number theory, continued fractions, and geometry. Its entertaining style will appeal to recreational readers and students alike.

Solving the Riddle of Phyllotaxis

Why the Fibonacci Numbers and the Golden Ratio Occur on Plants

Author: Irving Adler

Publisher: World Scientific Publishing Company Incorporated

ISBN: 9789814407625

Category: Mathematics

Page: 205

View: 7010

Foreword by Stephen L Adler (Institute for Advanced Study, USA)Illustrations by Peggy Adler The term Phyllotaxis refers to the patterns on plants formed by the arrangement of repeated biological units. In nearly all cases, the Fibonacci Numbers and the Golden Ratio occur in these arrangements. This topic has long fascinated scientists. Over a period of more than two decades, Irving Adler wrote a number of papers that construct a rigorously derived mathematical model for Phyllotaxis, which are major and enduring contributions to the field. These papers are collected in this reprint volume to enable their access to a wider readership.

Math for Mystics

From the Fibonacci sequence to Luna's Labyrinth to the Golden Section and Other Secrets of Sacred Geometry

Author: Renna Shesso

Publisher: Weiser Books

ISBN: 9781609252748

Category: Body, Mind & Spirit

Page: 208

View: 7139

Much of math history comes to us from early astrologers who needed to be able to describe and record what they saw in the night sky. Whether you were the king’s court astrologer or a farmer marking the best time for planting, timekeeping and numbers really mattered. Mistake a numerical pattern of petals and you could be poisoned. Lose the rhythm of a sacred dance or the meter of a ritually told story and the intricately woven threads that hold life together were spoiled. Ignore the celestial clock of equinoxes and solstices, and you’d risk being caught short of food for the winter. Shesso’s friendly tone and clear grasp of the information make the math “go down easy” in this marvelous book.

The Fibonacci Resonance and Other New Golden Ratio Discoveries

Maths, Music, Archaeology, Architecture, Art, Science: Quasicrystals and Metamaterials, Lucas Numbers, Ori32

Author: Clive Menhinick

Publisher: N.A

ISBN: 9780993216602


Page: 618

View: 3354

A new & definitive reference for the Golden Ratio & Fibonacci numbers. With Mondrian, Seurat, Toulouse-Lautrec, the Great Pyramid, Tiwanaku, Le Corbusier, Kepler, Penrose, quasicrystals, Pendry, green energy & the latest light technologies, this recreational maths & science book is written to be enjoyed. Explore bohemian Paris 'capital of phi' in fresh analyses of art, architecture & music. Lavishly illustrated, this book includes a history of Fibonacci & Lucas numbers spirals, sunflowers, pine cones, megaliths, and ornamental tilings. The Fibonacci Resonance discovery is revealed step-by-step from its unique beginnings to abacus bead visualizations. Comparisons are made with Silver Ratio Pell numbers and ties are shown to perfect numbers & record-breaking Mersenne primes.

The Glorious Golden Ratio

Author: Alfred S. Posamentier,Ingmar Lehmann

Publisher: Prometheus Books

ISBN: 1616144246

Category: Mathematics

Page: 363

View: 5665

What exactly is the Golden Ratio? How was it discovered? Where is it found? These questions and more are thoroughly explained in this engaging tour of one of mathematics' most interesting phenomena. The authors trace the appearance of the Golden Ratio throughout history, demonstrate a variety of ingenious techniques used to construct it, and illustrate the many surprising geometric figures in which the Golden Ratio is embedded. Requiring no more than an elementary knowledge of geometry and algebra, the authors give readers a new appreciation of the indispensable qualities and inherent beauty of mathematics. From the Hardcover edition.

The Golden Ratio

The Story of PHI, the World's Most Astonishing Number

Author: Mario Livio

Publisher: Broadway Books

ISBN: 9780307485526

Category: Mathematics

Page: 304

View: 1122

Throughout history, thinkers from mathematicians to theologians have pondered the mysterious relationship between numbers and the nature of reality. In this fascinating book, Mario Livio tells the tale of a number at the heart of that mystery: phi, or 1.6180339887...This curious mathematical relationship, widely known as "The Golden Ratio," was discovered by Euclid more than two thousand years ago because of its crucial role in the construction of the pentagram, to which magical properties had been attributed. Since then it has shown a propensity to appear in the most astonishing variety of places, from mollusk shells, sunflower florets, and rose petals to the shape of the galaxy. Psychological studies have investigated whether the Golden Ratio is the most aesthetically pleasing proportion extant, and it has been asserted that the creators of the Pyramids and the Parthenon employed it. It is believed to feature in works of art from Leonardo da Vinci's Mona Lisa to Salvador Dali's The Sacrament of the Last Supper, and poets and composers have used it in their works. It has even been found to be connected to the behavior of the stock market! The Golden Ratio is a captivating journey through art and architecture, botany and biology, physics and mathematics. It tells the human story of numerous phi-fixated individuals, including the followers of Pythagoras who believed that this proportion revealed the hand of God; astronomer Johannes Kepler, who saw phi as the greatest treasure of geometry; such Renaissance thinkers as mathematician Leonardo Fibonacci of Pisa; and such masters of the modern world as Goethe, Cezanne, Bartok, and physicist Roger Penrose. Wherever his quest for the meaning of phi takes him, Mario Livio reveals the world as a place where order, beauty, and eternal mystery will always coexist. From the Hardcover edition.

The Fabulous Fibonacci Numbers

Author: Alfred S. Posamentier

Publisher: Prometheus Books

ISBN: 1615920226

Category: Mathematics

Page: N.A

View: 8874

The most ubiquitous, and perhaps the most intriguing, number pattern in mathematics is the Fibonacci sequence. In this simple pattern beginning with two ones, each succeeding number is the sum of the two numbers immediately preceding it (1, 1, 2, 3, 5, 8, 13, 21, ad infinitum). Far from being just a curiosity, this sequence recurs in structures found throughout nature-from the arrangement of whorls on a pinecone to the branches of certain plant stems. All of which is astounding evidence for the deep mathematical basis of the natural world.With admirable clarity, math educators Alfred Posamentier and Ingmar Lehmann take us on a fascinating tour of the many ramifications of the Fibonacci numbers. The authors begin with a brief history of their distinguished Italian discoverer, who, among other accomplishments, was responsible for popularizing the use of Arabic numerals in the West. Turning to botany, the authors demonstrate, through illustrative diagrams, the unbelievable connections between Fibonacci numbers and natural forms (pineapples, sunflowers, and daisies are just a few examples). In art, architecture, the stock market, and other areas of society and culture, they point out numerous examples of the Fibonacci sequence as well as its derivative, the golden ratio. And of course in mathematics, as the authors amply demonstrate, there are almost boundless applications in probability, number theory, geometry, algebra, and Pascal's triangle, to name a few.Accessible and appealing to even the most math-phobic individual, this fun and enlightening book allows the reader to appreciate the elegance of mathematics and its amazing applications in both natural and cultural settings.Alfred S. Posamentier (New York, NY) is dean of the School of Education and professor of mathematics education at The City College of the City University of New York. He has published over 40 books in the area of mathematics and mathematics education, including Pi: A Biography of the World's Most Mysterious Number and Math Charmers: Tantalizing Tidbits for the Mind.Ingmar Lehmann (Berlin, Germany) is on the mathematics faculty at Humboldt University in Berlin and the coauthor of Pi: A Biography of the World's Most Mysterious Number.

Leonardo Pisano (Fibonacci)

The Book of Squares

Author: L. E. Sigler

Publisher: Elsevier

ISBN: 0080886507

Category: Mathematics

Page: 144

View: 4000

The Book of Squares by Fibonacci is a gem in the mathematical literature and one of the most important mathematical treatises written in the Middle Ages. It is a collection of theorems on indeterminate analysis and equations of second degree which yield, among other results, a solution to a problem proposed by Master John of Palermo to Leonardo at the Court of Frederick II. The book was dedicated and presented to the Emperor at Pisa in 1225. Dating back to the 13th century the book exhibits the early and continued fascination of men with our number system and the relationship among numbers with special properties such as prime numbers, squares, and odd numbers. The faithful translation into modern English and the commentary by the translator make this book accessible to professional mathematicians and amateurs who have always been intrigued by the lure of our number system.

Fibonacci Trading, Chapter 1 - Fibonacci Numbers and the Golden Ratio

Author: Carolyn Boroden

Publisher: McGraw Hill Professional

ISBN: 0071716173

Category: Business & Economics

Page: 6

View: 458

This chapter comes from Fibonacci Trading, which shows you how to identify numeric clusters and measure timing signals, in order to achieve the highest rate of profitable trades. Filled with real-world trading situations, the book combines classic trading concepts and Fibonacci analysis to enable you to take advantage of the naturally occurring and recurring patterns within the market.

Fibonacci and Lucas Numbers with Applications

Author: Thomas Koshy

Publisher: John Wiley & Sons

ISBN: 1118742176

Category: Mathematics

Page: 704

View: 6596

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

The Divine Proportion

A Study in Mathematical Beauty

Author: H. E. Huntley

Publisher: Courier Corporation

ISBN: 0486222543

Category: Mathematics

Page: 186

View: 5575

Explores the aesthetic, emotional, artistic and philosophical significance of geometric figures, scientific patterns and mathematical formulas

Divina Proportione; Die Lehre Vom Goldenen Schnitt

Author: Luca Pacioli

Publisher: Theclassics.Us

ISBN: 9781230311012


Page: 120

View: 5623

Dieses historische Buch kann zahlreiche Tippfehler und fehlende Textpassagen aufweisen. Kaufer konnen in der Regel eine kostenlose eingescannte Kopie des originalen Buches vom Verleger herunterladen (ohne Tippfehler). Ohne Indizes. Nicht dargestellt. 1889 edition. Auszug: ...uber den Tiber bei seinem beruhmten erlangten Siege regelrecht anordnete. Auch kam durch keine andern Mittel unser sehr scharfsinniger Scotus zu den grossen Speculationen der heiligen Theologie, als durch die Kenntniss der mathematischen Disciplinen, wie aus allen seinen heiligen Werken erhellt. Besonders wenn man wohl beachtet die Untersuchung seines zweiten Buches uber die Meinungen, als er forschend fragt, ob der Engel seinen eigenen und bestimmten Aufenthaltsort fur seine Existenz habe, Worin er wohl zeigt, dass er das ganze herrliche Buch unseres scharfsinnigsten Megarenser Philosophen Euklid-) verstanden habe. Durch nichts Anderes zeigen sich gleichfalls alle Texte des Fursten deren, welche Physik, hohere Metaphvsik verstehen und auch die ubrigen schwer, als durch die Unkenntniss der schon genannten Disciplinen. Durch nichts Anderes ist Mangel an guten Astronomen als in Folge des Mangels an Arithmetik, Geometrie, Proportionen und Proportionalitat. Und von zehn richten sich neun in ihren Urtheilen nach Tafeln, Taschen 1) Verbum zu erzeugen. 5) Durchgehends falschlich anstatt des in Aegypten lebenden Mathe mathikers gleichen Namens angegeben. buchern und anderen Dingen, die von Ptolomaus. Albumansar, Ali al Fragano, Gebe, Alfonso, Biancho. Prodocino und Anderen berechnet sind, Folge der wenigen Umsicht ihrer Schreiber mangelhaft oder verschlechtert sein konnen. Und wenn sie sich in Folge dessen auf jene verlassen, verfallen sie in die grossten und evidentesten Irrthumer, zu nicht geringem Schaden und Yorurtheil derer, welche sich auf sie verlassen. Die ausserste Scharte aller Municipalgesetze...

Is God a Mathematician?

Author: Mario Livio

Publisher: Simon and Schuster

ISBN: 9781416594437

Category: Mathematics

Page: 320

View: 2338

Bestselling author and astrophysicist Mario Livio examines the lives and theories of history’s greatest mathematicians to ask how—if mathematics is an abstract construction of the human mind—it can so perfectly explain the physical world. Nobel Laureate Eugene Wigner once wondered about “the unreasonable effectiveness of mathematics” in the formulation of the laws of nature. Is God a Mathematician? investigates why mathematics is as powerful as it is. From ancient times to the present, scientists and philosophers have marveled at how such a seemingly abstract discipline could so perfectly explain the natural world. More than that—mathematics has often made predictions, for example, about subatomic particles or cosmic phenomena that were unknown at the time, but later were proven to be true. Is mathematics ultimately invented or discovered? If, as Einstein insisted, mathematics is “a product of human thought that is independent of experience,” how can it so accurately describe and even predict the world around us? Physicist and author Mario Livio brilliantly explores mathematical ideas from Pythagoras to the present day as he shows us how intriguing questions and ingenious answers have led to ever deeper insights into our world. This fascinating book will interest anyone curious about the human mind, the scientific world, and the relationship between them.