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: 5125

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 Glorious Golden Ratio

Author: Alfred S. Posamentier,Ingmar Lehmann

Publisher: Prometheus Books

ISBN: 1616144246

Category: Mathematics

Page: 363

View: 2013

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.

Is God a Mathematician?

Author: Mario Livio

Publisher: Simon and Schuster

ISBN: 9781416594437

Category: Mathematics

Page: 320

View: 4679

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.

The Golden Ratio

The Divine Beauty of Mathematics

Author: Gary B. Meisner

Publisher: Race Point Publishing

ISBN: 076036026X

Category: Mathematics

Page: 224

View: 4777

The Golden Ratio examines the presence of this divine number in art and architecture throughout history, as well as its ubiquity among plants, animals, and even the cosmos. This gorgeous book—with layflat dimensions that closely approximate the golden ratio—features clear, enlightening, and entertaining commentary alongside stunning full-color illustrations by Venezuelan artist and architect Rafael Araujo. From the pyramids of Giza, to quasicrystals, to the proportions of the human face, the golden ratio has an infinite capacity to generate shapes with exquisite properties. This book invites you to take a new look at this timeless topic, with a compilation of research and information worthy of a text book, accompanied by over 200 beautiful color illustrations that transform this into the ultimate coffee table book. Author Gary Meisner shares the results of his twenty-year investigation and collaboration with thousands of people across the globe in dozens of professions and walks of life. The evidence will close the gaps of understanding related to many claims of the golden ratio’s appearances and applications, and present new findings to take our knowledge further yet. Whoever you are, and whatever you may know about this topic, you’ll find something new, interesting, and informative in this book, and may find yourself challenged to see, apply, and share this unique number of mathematics and science in new ways.

The Divine Proportion

Author: H. E. Huntley

Publisher: Courier Corporation

ISBN: 0486131874

Category: Mathematics

Page: 208

View: 3733

Discussion ranges from theories of biological growth to intervals and tones in music, Pythagorean numerology, conic sections, Pascal's triangle, the Fibonnacci series, and much more. Excellent bridge between science and art. Features 58 figures.

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: 3955

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.

Fibonacci Numbers

Author: Nikolai Nikolaevich Vorob'ev

Publisher: Courier Corporation

ISBN: 048629885X

Category: Mathematics

Page: 80

View: 2635

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.

The Golden Ratio and Fibonacci Numbers

Author: Richard A Dunlap

Publisher: World Scientific

ISBN: 9814496944

Category: Mathematics

Page: 172

View: 6756

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 RatioGeometric Problems in Two DimensionsGeometric Problems in Three DimensionsFibonacci NumbersLucas Numbers and Generalized Fibonacci NumbersContinued Fractions and Rational ApproximantsGeneralized Fibonacci Representation TheoremsOptimal Spacing and Search AlgorithmsCommensurate and Incommensurate ProjectionsPenrose TilingsQuasicrystallographyBiological ApplicationsConstruction of the Regular PentagonThe First 100 Fibonacci and Lucas NumbersRelationships Involving the Golden Ratio and Generalized Fibonacci Numbers Readership: Applied mathematicians. keywords:Golden Ratio;Fibonacci Numbers;Number Theory;Quasicrystallography;Tilings;Geometry;Optimization Algorithms;Numerical Sequences;Lucas Numbers;Continued Fractions;Aperiodicity;Incommensurate Structures "This attractive and carefully written book addresses the general reader with interest in mathematics and its application to the physical and biological sciences ... this one will make a valuable addition to academic and personal libraries." The Fibonacci Quarterly

The Golden Ratio

The Facts and the Myths

Author: Francis D. Hauser, Ph.d.

Publisher: CreateSpace

ISBN: 9781517518776


Page: 56

View: 4346

Euclid's masterpiece textbook, The Elements, was written twenty-three hundred years ago. It is primarily about geometry and contains dozens of figures. Five of these are constructed using a line that "is cut in extreme and mean ratio." Today this is called the golden ratio and is often referred to by the symbol Φ. Many myths have grown up around this ratio. This book was written to learn about them. They arise from the pyramids, the Pythagorean Brotherhood, the platonic solids, the Fibonacci numbers, sea shells, and others. There is a common thread among these myths. Φ is an irrational number (a number whose digits after the decimal point go on forever and never form a repeating pattern). Φ can be used to draw pleasing figures. But its numerical value cannot be written down using integers and fractions, which were the only numbers used in Euclid's time. Mathematicians before Euclid knew that irrational numbers existed. But to many people, a number that can't be written down was absurd. For centuries, many scientists and engineers believed that Φ was godlike. This book discusses the myths from an engineering viewpoint. The last chapter of the book shows how Euclid handled irrational numbers; how Euclid did algebra using geometry; and a simple visual proof of why there are only five platonic solids.

Divine Proportion

Phi in Art, Nature, and Science

Author: Priya Hemenway

Publisher: Sterling Publishing Company, Inc.

ISBN: 9781402735226

Category: Science

Page: 203

View: 6343

The number Phi, simply defined, is one plus the square root of five, all divided by two. But its myriad occurrences in art, nature, and science have been a source of speculation and wonder for thousands of years. Divine Proportion draws upon both religion and science to tell the story of Phi and to explore its manifestations in such diverse places as the structure of the inner ear, the spiral of a hurricane, the majesty of the Parthenon, and the elusive perfection of the Mona Lisa. A universal key to harmony, regeneration, and balance, Phi is at the heart of a tantalizing story begun on clay tablets in ancient Babylon, and which will continue to be written for centuries to come.

The Golden Ratio

And Other Mathematical Patterns Inspired by Nature and Art

Author: Michael O'Mara Books,Steve Richards

Publisher: Lark Books (NC)

ISBN: 9781454710226

Category: Art

Page: 96

View: 6285

The elegance of mathematics meets the creativity of art in this unique coloring book. The golden ratio is the mathematical equation that describes what many consider the "divine proportion," and it's found throughout nature, as well as in art and architecture. This stunning coloring book showcases the beauty of Fibonacci's most famous formula, exploring the many ways numbers and beauty combine. Logical and creative minds alike will enjoy filling in the spirals and designs in these remarkable images.

Brilliant Blunders

From Darwin to Einstein - Colossal Mistakes by Great Scientists That Changed Our Understanding of Life and the Universe

Author: Mario Livio

Publisher: Simon and Schuster

ISBN: 1439192375

Category: Biography & Autobiography

Page: 352

View: 3545

We all make mistakes. Nobody is perfect. And that includes five of the greatest scientists in history -- Charles Darwin, William Thomson (Lord Kelvin), Linus Pauling, Fred Hoyle, Albert Einstein. But the mistakes that these great scientists made helped science to advance. Indeed, as Mario Livio explains in this fascinating book, science thrives on error; it advances when erroneous ideas are disproven. All five scientists were great geniuses and fascinating human beings. Their blunders were part of their genius and part of the scientific process. Livio brilliantly analyses their errors to show where they were wrong and right, but what makes his book so enjoyable to read is Livio's analysis of the psychology of these towering figures. Along the way the reader learns an enormous amount about the evolution of life on earth and in the universe, but from an unusual vantage point -- the mistakes of great scientists rather than the achievements that made them famous.

The Golden Section

Nature's Greatest Secret

Author: Scott Olson

Publisher: Bloomsbury Publishing USA

ISBN: 0802715397

Category: Mathematics

Page: 58

View: 8831

Tells the story of the golden section, a line segment divided into two parts such that the ratio of the short portion to the longer portion is equal to the ratio of the longer portion to the whole, and its impact on civilization and the natural world.

New Geometries for New Materials

Author: Eric A. Lord,Alan L. Mackay,S. Ranganathan

Publisher: Cambridge University Press

ISBN: 0521861047

Category: Mathematics

Page: 238

View: 8815

Advances in materials science have given rise to novel materials with unique properties, through the manipulation of structure at the atomic level. Elucidating the shape and form of matter at this scale requires the application of mathematical concepts. This 2006 book presents the geometrical ideas that are being developed and integrated into materials science to provide descriptors and enable visualisation of the atomic arrangements in three-dimensional space. Emphasis is placed on the intuitive understanding of geometrical principles, presented through numerous illustrations. Mathematical complexity is kept to a minimum and only a superficial knowledge of vectors and matrices is required, making this an accessible introduction to the area. With a comprehensive reference list, this book will appeal to those working in crystallography, solid state and materials science.

Beautiful, Simple, Exact, Crazy

Mathematics in the Real World

Author: Apoorva Khare,Anna Lachowska

Publisher: Yale University Press

ISBN: 0300190891

Category: Mathematics

Page: 480

View: 2393

In this vibrant work, which is ideal for both teaching and learning, Apoorva Khare and Anna Lachowska explain the mathematics essential for understanding and appreciating our quantitative world. They show with examples that mathematics is a key tool in the creation and appreciation of art, music, and literature, not just science and technology. The book covers basic mathematical topics from logarithms to statistics, but the authors eschew mundane finance and probability problems. Instead, they explain how modular arithmetic helps keep our online transactions safe, how logarithms justify the twelve-tone scale commonly used in music, and how transmissions by deep space probes are similar to knights serving as messengers for their traveling prince. Ideal for coursework in introductory mathematics and requiring no knowledge of calculus, Khare and Lachowska s enlightening mathematics tour will appeal to a wide audience."

The Equation that Couldn't Be Solved

How Mathematical Genius Discovered the Language of Symmetry

Author: Mario Livio

Publisher: Simon and Schuster

ISBN: 9780743274623

Category: Mathematics

Page: 368

View: 466

What do Bach's compositions, Rubik's Cube, the way we choose our mates, and the physics of subatomic particles have in common? All are governed by the laws of symmetry, which elegantly unify scientific and artistic principles. Yet the mathematical language of symmetry-known as group theory-did not emerge from the study of symmetry at all, but from an equation that couldn't be solved. For thousands of years mathematicians solved progressively more difficult algebraic equations, until they encountered the quintic equation, which resisted solution for three centuries. Working independently, two great prodigies ultimately proved that the quintic cannot be solved by a simple formula. These geniuses, a Norwegian named Niels Henrik Abel and a romantic Frenchman named Évariste Galois, both died tragically young. Their incredible labor, however, produced the origins of group theory. The first extensive, popular account of the mathematics of symmetry and order, The Equation That Couldn't Be Solved is told not through abstract formulas but in a beautifully written and dramatic account of the lives and work of some of the greatest and most intriguing mathematicians in history.

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: 6653

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.