The Golden Ratio and Fibonacci Numbers

Author: R. A. Dunlap

Publisher: World Scientific

ISBN: 9789812386304

Category: Mathematics

Page: 162

View: 1955


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.

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: 5329


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.

MATLAB Guide to Fibonacci Numbers and the Golden Ratio

A Simplified Approach

Author: Peter I. Kattan

Publisher: CreateSpace

ISBN: 9781466229877


Page: 172

View: 7999


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: 6545


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: 9879


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.

The Interesting Golden Ratio

A Simple Mathematical Approach

Author: Vincent Siu

Publisher: Partridge Publishing Singapore

ISBN: 1482855437

Category: Mathematics

Page: 158

View: 7285


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 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: 9455


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,Ingmar Lehmann

Publisher: N.A


Category: Mathematics

Page: 385

View: 6981


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 Glorious Golden Ratio

Author: Alfred S. Posamentier,Ingmar Lehmann

Publisher: Prometheus Books

ISBN: 1616144246

Category: Mathematics

Page: 363

View: 2822


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.