The Golden Ratio and Fibonacci Numbers

Author: R. A. Dunlap

Publisher: World Scientific

ISBN: 9789812386304

Category: Mathematics

Page: 162

View: 3400

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.

Divina Proportione

Author: F.L. Pacioli

Publisher: Рипол Классик

ISBN: 5879564436

Category: History

Page: N.A

View: 5287


MATLAB Guide to Fibonacci Numbers and the Golden Ratio

A Simplified Approach

Author: Peter I. Kattan

Publisher: CreateSpace

ISBN: 9781466229877


Page: 172

View: 7341

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 and Lucas Numbers, and the Golden Section

Theory and Applications

Author: Steven Vajda

Publisher: Courier Corporation

ISBN: 0486462765

Category: Mathematics

Page: 189

View: 5272

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.

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: 349

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.

Geometry of Design

Author: Kimberly Elam

Publisher: Princeton Architectural Press

ISBN: 9781568985848

Category: Architecture

Page: 107

View: 9792

Kimberly Elam fA1/4hrt den Leser auf eine geometrische Reise und gibt Einsicht in den Designprozess, indem sie visuelle Beziehungen untersucht, die sowohl auf mathematischen Prinzipien als auch auf grundlegenden Eigenschaften des Lebens basieren. Elams ErklArungen machen die mathematischen Beziehungen, die sich hinter diesen Designs verbergen, offensichtlich, und ihre EinfA1/4hrung in die Technik der visuellen Analyse macht dieses Buch zu einer unerlAsslichen Hilfe fA1/4r jeden, der grafisch arbeitet. Die Autorin behandelt dabei nicht nur die klassischen Proportionierungssysteme wie den Goldenen Schnitt und wurzelbasierte Rechtecke, sondern auch weniger bekannte Systeme wie z.B. die Fibonaccifolge.

Symmetrien der Natur

Ein Handbuch zur Natur- und Wissenschaftsphilosophie

Author: Klaus Mainzer

Publisher: Walter de Gruyter

ISBN: 3110853655

Category: Philosophy

Page: 750

View: 5802


Pascal, Fermat und die Berechnung des Glücks

eine Reise in die Geschichte der Mathematik

Author: Keith J. Devlin

Publisher: C.H.Beck

ISBN: 9783406590993

Category: Wahrscheinlichkeitstheorie - Geschichte

Page: 204

View: 9979

Der Autor stellt die Entstehung der Wahrscheinlichkeitsrechnung und den damit verbundenen Wandel des menschlichen Alltagslebens dar.

The Interesting Golden Ratio

A Simple Mathematical Approach

Author: Vincent Siu

Publisher: Partridge Publishing Singapore

ISBN: 1482855437

Category: Mathematics

Page: 158

View: 6317

This book uses simple geometry, trigonometry and algebra to explain how to construct and calculate the golden ratio. Starting from Euclids propositions in The Elements, the golden ratio and its related geometry such as the pentagon, pentagram and Vesica Piscis are constructed and determined graphically. Then the value of the ratio is solved by quadratic equations, and depending on the initial assumptions, two values are found. It is proposed that the ratio can be obtained easily by applying the Pythagoras theorem. Common terms like golden triangle, golden rhombus, golden spiral and golden angle are deduced and explained. The connections between the golden ratio and Fibonacci numbers, continued fractions, fractals, chaos and tiling are also introduced. The conclusion is that it is mathematics and not the golden ratio that is fascinating.

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: 3291

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 Fabulous Fibonacci Numbers

Author: Alfred S. Posamentier,Ingmar Lehmann

Publisher: N.A


Category: Mathematics

Page: 385

View: 2973

Explores the complexity and wide-ranging applications of the Fibonacci sequence, which appears in nature, art, economics, and the "golden ratio," which is derived from this simple pattern of numbers.

The Golden Relationship: Universal patterns

Author: Martha Boles,Rochelle Newman

Publisher: N.A


Category: Art

Page: 266

View: 6739

In the series, The Golden Relationship: Art, Math & Nature, the common denominator is space. "Nature is relationships...,geometry describes relationships...,art creates relationships in space." Geometry, beyond theorems & proofs, is a powerful tool for approaching design problems, while art allows for the physical realization of geometric ideas. With their union, mathematics is given a forum for displaying its beauty & art for expressing its logic. For readers with limited experience with mathematics & art, the authors write to demystify the disciplines while showing nature as the source of both. The texts are elegantly & accurately illustrated with equal emphasis on visual & written information. Chapters contain problems to reinforce mathematical concepts, & project suggestions offer experiences for structuring artworks. Diagrams augment the glossary & appendices provide extended content. Book 1, UNIVERSAL PATTERNS, examines certain triangles, rectangles, spirals, & number sequences that relate to recurring patterns in nature. Book 2, THE SURFACEPLANE, plays out these patterns on the two dimensional surface, focusing on grids, symmetry, circles, tilings, fractals, & pliability. In both volumes, the concept of the Golden Ratio is the spatial thread that weaves the ideas into a unified fabric.

New Visual Perspectives on Fibonacci Numbers

Author: Krassimir T. Atanassov

Publisher: World Scientific

ISBN: 9789812381347

Category: Mathematics

Page: 313

View: 6337

Discusses new ground on Fibonacci sequences and the well-known Fibonacci numbers. There is a continuing emphasis on diagrams, both geometric and combinatorial, which helps to tie disparate topics together, weaving around the unifying themes of the golden mean and various generalizations of the Fibonacci recurrence relation.

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: 4594

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.

Über Wachstum und Form

Author: D'Arcy Wentworth Thompson,Anita Albus

Publisher: N.A

ISBN: 9783821845685


Page: 477

View: 3249


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: 508

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.

Untersuchungen über höhere Arithmetik

Author: Carl Friedrich Gauss

Publisher: American Mathematical Soc.

ISBN: 0821842137

Category: Mathematics

Page: 695

View: 4623

In this volume are included all of Gauss's number-theoretic works: his masterpiece, Disquisitiones Arithmeticae, published when Gauss was only 25 years old; several papers published during the ensuing 31 years; and papers taken from material found in Gauss's handwriting after his death. These papers include a fourth, fifth, and sixth proof of the Quadratic Reciprocity Law, researches on biquadratic residues, quadratic forms, and other topics. This reprint of the German translation from Latin of the second edition published in 1889 includes an extensive appendix and concludes with a commentary on the papers (with references, where appropriate, to the relevant pages of the Disquisitiones).