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Two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Often symbolized as Phi (φ), is also expressed as one plus the square root of five, divided by two. The Golden Ratio has profoundly influenced both art and architecture, serving as a guiding principle for achieving balance and aesthetic beauty. Here we present that number printed, in his first two million digits, in a single volume.
This book contains the first one million digits of the square root of 2. Each page contains 45 rows, each divided into 5 columns of 10 digits, for a total of 446 pages! This is a great gift for friends or relatives who are passionate about math! Dimensions: 6.14 x 9.21 inches (15.6 x 23.39 cm). Other books in this series include: "One Million Digits of Pi", "One Million Digits of Zeta 3", "One Million Digits of e (Euler's Number)", "One Million Digits of the Golden Ratio (phi)".
A great gift for friends or relatives who are passionate about math, this book features: The first 1.000.000 digits of Pi, A total of 446 pages, Each page contains 42 rows, each divided into 5 columns of 10 digits, High quality printing, Standard format (6 x 9 in) Other books in this series include: "One Million Digits of e (Euler's Number)", "One Million Digits of Zeta 3", "One Million Digits of the Square Root of 2", "One Million Digits of the Golden Ratio (phi)".
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.
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.
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.
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.
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This book introduces readers to the art of doing mathematical proofs. Proofs are the glue that holds mathematics together. They make connections between math concepts and show why things work the way they do. This book teaches the art of proofs using familiar high-school concepts, such as numbers, polynomials, functions, and trigonometry. It retells math as a story, where the next chapter follows from the previous one. Readers will see how various mathematical concepts are tied and will see that mathematics is not a pile of formulas and facts; rather, it has an orderly and beautiful edifice. The author begins with basic rules of logic and then progresses through the topics already familiar to the students: numbers, inequalities, functions, polynomials, exponents, and trigonometric functions. There are also beautiful proofs for conic sections, sequences, and Fibonacci numbers. Each chapter has exercises for the reader. Reviewer Comments: I find the book very impressive. The choice and sequence of topics is excellent, and it is wonderful to have all of these things together in one volume. Theorems are clearly stated, and proofs are accurate. – Michael Comenetz The thoroughness of the narrative is one of the main strengths of the book. The book provides a perfect illustration of mathematical thinking. Each step of a given derivation is precise and clear. – Julie Gershunskaya Draganov’s book stands out from the many competing books. Draganov’s goal is to show that mathematics depends on the notion of proof. Unlike other transition books, he addresses mathematical topics at an accessible level, rather than topics studied later in the university curriculum. – Ken Rosen
Science 1001 provides clear and concise explanations of the most fundamental and fascinating scientific concepts. Distilled into 1001 bite-sized mini-essays arranged thematically, this unique reference book moves steadily from the basics through to the most advanced of ideas, making it the ideal guide for novices and science enthusiasts. Whether used as a handy reference, an informal self-study course or simply as a gratifying dip-in, this book offers--in one volume--a world of cutting-edge scientific knowledge for the general reader. Science 1001 is an incredibly comprehensive guide, spanning all of the key scientific disciplines including Physics, Chemistry, Biology, The Earth, Space, Health and Medicine, Social Science, Information Science, the Applied Sciences and Futurology. From Newton's elemental laws of motion and the physics of black holes, through the fundamental particles of matter, to the extraordinary Human Genome Project and the controversial possibilities of cloning and gene therapy, Dr. Paul Parsons demystifies the key concepts of science in the simplest language and answers its big questions: Will scientists find a cure for AIDS? How did the universe begin? And will we conquer space? Concluding with an exciting glimpse of what's to come for science--from the possibility of time travel to the specter of trans-humanism--this really is the only science book you'll ever need.