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The Golden Ratio (Phi) – original contributions & new discoveries

Author: Andrei-Lucian Drăgoi

Publisher: Dr. Andrei-Lucian Drăgoi

Published: 2024-01-14

Total Pages: 32

ISBN-13:

The Golden Ratio (PHI) (aka “golden mean”, “golden section” or “divine section”: standardly denoted with the capital Greek letter PHI) is the only positive (irrational) quadratic solution of the (minimal polynomial) equation x^2-x-1=0, so that PHI=[sqrt(5)+1]/2. The purpose of this paper is to demonstrate some aspects recently discovered by the author of this paper: (1) a recursive geometrical method for constructing PHI (inspired by the Taoist religious iconography) which is more simple and elegant than Odom’s method; (2) an occurrence of PHI in the water molecule geometry; (3) a quasi-Pythagorean property of the (PHI, e, PI) triad; (4) some other interesting quasi-exact relations between the elements of the (PHI, e, PI) triad; (5) the tendency of the human systolic/diastolic blood pressure different (combinational) ratios to stabilize (with age) to values close to PHI;

The Golden Ratio

Author: Gary B. Meisner

Publisher: Quarto Publishing Group USA

Published: 2018-10-23

Total Pages: 227

ISBN-13: 076036026X

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This enlightening and gorgeously illustrated book explores the beauty and mystery of the divine proportion in art, architecture, nature, and beyond. 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. Author Gary Meisner has spent decades researching the subject, investigating and collaborating with people across the globe in dozens of professions and walks of life. In The Golden Ratio, he shares his enlightening journey. Exploring the long history of this fascinating number, as well as new insights into its power and potential applications, The Golden Ratio invites you to take a new look at this timeless topic.

The Mathematics of Harmony

Author: Alexey Stakhov

Publisher: World Scientific

Published: 2009

Total Pages: 745

ISBN-13: 9812775838

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Assisted by Scott Olsen ( Central Florida Community College, USA ). This volume is a result of the author's four decades of research in the field of Fibonacci numbers and the Golden Section and their applications. It provides a broad introduction to the fascinating and beautiful subject of the OC Mathematics of Harmony, OCO a new interdisciplinary direction of modern science. This direction has its origins in OC The ElementsOCO of Euclid and has many unexpected applications in contemporary mathematics (a new approach to a history of mathematics, the generalized Fibonacci numbers and the generalized golden proportions, the OC goldenOCO algebraic equations, the generalized Binet formulas, Fibonacci and OC goldenOCO matrices), theoretical physics (new hyperbolic models of Nature) and computer science (algorithmic measurement theory, number systems with irrational radices, Fibonacci computers, ternary mirror-symmetrical arithmetic, a new theory of coding and cryptography based on the Fibonacci and OC goldenOCO matrices). The book is intended for a wide audience including mathematics teachers of high schools, students of colleges and universities and scientists in the field of mathematics, theoretical physics and computer science. The book may be used as an advanced textbook by graduate students and even ambitious undergraduates in mathematics and computer science. Sample Chapter(s). Introduction (503k). Chapter 1: The Golden Section (2,459k). Contents: Classical Golden Mean, Fibonacci Numbers, and Platonic Solids: The Golden Section; Fibonacci and Lucas Numbers; Regular Polyhedrons; Mathematics of Harmony: Generalizations of Fibonacci Numbers and the Golden Mean; Hyperbolic Fibonacci and Lucas Functions; Fibonacci and Golden Matrices; Application in Computer Science: Algorithmic Measurement Theory; Fibonacci Computers; Codes of the Golden Proportion; Ternary Mirror-Symmetrical Arithmetic; A New Coding Theory Based on a Matrix Approach. Readership: Researchers, teachers and students in mathematics (especially those interested in the Golden Section and Fibonacci numbers), theoretical physics and computer science."

A Mathematical History of the Golden Number

Author: Roger Herz-Fischler

Publisher: Courier Corporation

Published: 2013-12-31

Total Pages: 228

ISBN-13: 0486152324

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This comprehensive study traces the historic development of division in extreme and mean ratio ("the golden number") from its first appearance in Euclid's Elements through the 18th century. Features numerous illustrations.

Is God a Mathematician?

Author: Mario Livio

Publisher: Simon and Schuster

Published: 2011-02-22

Total Pages: 320

ISBN-13: 1416594434

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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.

Fibonacci’s Liber Abaci

Author: Laurence Sigler

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 736

ISBN-13: 1461300797

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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.

A Mathematical History of Division in Extreme and Mean Ratio

Author: Roger Herz-Fischler

Publisher: Wilfrid Laurier Univ. Press

Published: 1987

Total Pages: 208

ISBN-13: 0889201528

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The Golden Ratio And Fibonacci Numbers

Author: Richard A Dunlap

Publisher: World Scientific

Published: 1997-12-16

Total Pages: 170

ISBN-13: 9814496944

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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.

The Divine Proportion

Author: H. E. Huntley

Publisher: Courier Corporation

Published: 2012-06-08

Total Pages: 208

ISBN-13: 0486131874

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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.

Brilliant Blunders

Author: Mario Livio

Publisher: Simon and Schuster

Published: 2014-05-27

Total Pages: 352

ISBN-13: 1439192375

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"Drawing on the lives of five great scientists -- Charles Darwin, William Thomson (Lord Kelvin), Linus Pauling, Fred Hoyle and Albert Einstein -- scientist/author Mario Livio shows how even the greatest scientists made major mistakes and how science built on these errors to achieve breakthroughs, especially into the evolution of life and the universe"--