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Author: Alfred S. Posamentier Publisher: Prometheus Books ISBN: 1616144246 Category : Mathematics Languages : en Pages : 312
Book Description
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.
Author: Alfred S. Posamentier Publisher: Prometheus Books ISBN: 1616144246 Category : Mathematics Languages : en Pages : 312
Book Description
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.
Author: Gary B. Meisner Publisher: Quarto Publishing Group USA ISBN: 076036026X Category : Mathematics Languages : en Pages : 227
Book Description
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.
Author: Alfred S. Posamentier Publisher: Prometheus Books ISBN: 1616147482 Category : Mathematics Languages : en Pages : 281
Book Description
Two veteran math educators demonstrate how some "magnificent mistakes" had profound consequences for our understanding of mathematics' key concepts. In the nineteenth century, English mathematician William Shanks spent fifteen years calculating the value of pi, setting a record for the number of decimal places. Later, his calculation was reproduced using large wooden numerals to decorate the cupola of a hall in the Palais de la Découverte in Paris. However, in 1946, with the aid of a mechanical desk calculator that ran for seventy hours, it was discovered that there was a mistake in the 528th decimal place. Today, supercomputers have determined the value of pi to trillions of decimal places. This is just one of the amusing and intriguing stories about mistakes in mathematics in this layperson's guide to mathematical principles. In another example, the authors show that when we "prove" that every triangle is isosceles, we are violating a concept not even known to Euclid - that of "betweenness." And if we disregard the time-honored Pythagorean theorem, this is a misuse of the concept of infinity. Even using correct procedures can sometimes lead to absurd - but enlightening - results. Requiring no more than high-school-level math competency, this playful excursion through the nuances of math will give you a better grasp of this fundamental, all-important science.
Author: Mark C. Chu-Carroll Publisher: Pragmatic Bookshelf ISBN: 168050360X Category : Computers Languages : en Pages : 269
Book Description
Mathematics is beautiful--and it can be fun and exciting as well as practical. Good Math is your guide to some of the most intriguing topics from two thousand years of mathematics: from Egyptian fractions to Turing machines; from the real meaning of numbers to proof trees, group symmetry, and mechanical computation. If you've ever wondered what lay beyond the proofs you struggled to complete in high school geometry, or what limits the capabilities of computer on your desk, this is the book for you. Why do Roman numerals persist? How do we know that some infinities are larger than others? And how can we know for certain a program will ever finish? In this fast-paced tour of modern and not-so-modern math, computer scientist Mark Chu-Carroll explores some of the greatest breakthroughs and disappointments of more than two thousand years of mathematical thought. There is joy and beauty in mathematics, and in more than two dozen essays drawn from his popular "Good Math" blog, you'll find concepts, proofs, and examples that are often surprising, counterintuitive, or just plain weird. Mark begins his journey with the basics of numbers, with an entertaining trip through the integers and the natural, rational, irrational, and transcendental numbers. The voyage continues with a look at some of the oddest numbers in mathematics, including zero, the golden ratio, imaginary numbers, Roman numerals, and Egyptian and continuing fractions. After a deep dive into modern logic, including an introduction to linear logic and the logic-savvy Prolog language, the trip concludes with a tour of modern set theory and the advances and paradoxes of modern mechanical computing. If your high school or college math courses left you grasping for the inner meaning behind the numbers, Mark's book will both entertain and enlighten you.
Author: Larry Witham Publisher: Open Road Media ISBN: 1480447668 Category : Biography & Autobiography Languages : en Pages : 497
Book Description
An innovative painter in the early generation of Renaissance artists, Piero dell Francesca was also an expert on religious topics and a mathematician who wanted to use perspective and geometry to make painting a “true science.” Although only sixteen of Piero’s works survive, few art historians doubt his importance in the Renaissance. A 1992 conference of international experts meeting at the National Gallery of Art deemed Piero “one of the most highly regarded painters of the early Renaissance, and one of the most respected artists of all time.” In recent years, the quest for Piero has continued among intrepid scholars, and Piero’s Light uncovers the life of this remarkable artistic revolutionary and enduring legacy of the Italian Renaissance.
Author: Richard A Dunlap Publisher: World Scientific ISBN: 9814496944 Category : Mathematics Languages : en Pages : 172
Book Description
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.
Author: Priya Hemenway Publisher: Sterling Publishing Company Incorporated ISBN: 9781402735226 Category : Science Languages : en Pages : 203
Book Description
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.
Author: Francis D. Hauser, Ph.d. Publisher: CreateSpace ISBN: 9781517518776 Category : Languages : en Pages : 56
Book Description
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.
Author: Andrew Hayduke Publisher: ISBN: 9781735918020 Category : Languages : en Pages : 294
Book Description
Why are only certain people considered physically beautiful? What is it - specifically - within their anatomy that these lucky few individuals possess that most of us simply do not? Can we somehow objectively test our own anatomy for beauty and then use our test results as a guide to change our physical appearance? Can we join this highly elite group of physically beautiful people? Dr. Andrew Hayduke's groundbreaking book is the first of its kind to explore a potential scientific explanation for why only certain faces and breasts appear physically beautiful. After decades of experience as a plastic surgeon in Beverly Hills and Palm Springs, California, Dr. Hayduke shares his unique perspective on this fascinating subject. The Golden Ratio Within the Human Face and Breast unveils the mysterious underlying secret blueprint for human physical beauty that exists hidden deep within the anatomic layout of attractive (i.e., beautiful) faces and breasts. The Golden Ratio Within the Human Face and Breast will:1) Convince you to look beyond the expression "Beauty is in the eye of the beholder" and consider something far more intriguing as the true explanation for physical beauty. 2) Provide evidence that beauty is an objective quality rather than a mere subjective opinion. 3) Provide an all-access backstage pass to the secret clinical analysis methodologies used by a Beverly Hills plastic surgeon. 4) Illustrate how the faces and breasts of average-looking people can be transformed to a more beautiful appearance with the help of Dr. Hayduke's state-of-the-art facial and breast beauty tests. 5) Disclose a detailed description of the once-secret inner mechanism of both the Hayduke Facial Analysis test and the Hayduke Breast Analysis test-including the exact specifications of each anatomic test grid used within Dr. Hayduke's patented beauty analysis methodologies. 6)Explain the basics of the golden ratio (also known as the divine proportion, golden section, golden mean, golden proportion, divine section and golden number). 7) Illustrate how the golden ratio relationship of 1.618 to 1.0 is secretly hidden within many beautiful world-famous works of art, architecture, logos, and even nature.
Author: Alfred S. Posamentier Publisher: Rowman & Littlefield ISBN: 1633887618 Category : Mathematics Languages : en Pages : 249
Book Description
We see numbers on automobile license plates, addresses, weather reports, and, of course, on our smartphones. Yet we look at these numbers for their role as descriptors, not as an entity in and unto themselves. Each number has its own history of meaning, usage, and connotation in the larger world. The Secret Lives of Numbers takes readers on a journey through integers, considering their numerological assignments as well as their significance beyond mathematics and in the realm of popular culture. Of course we all know that the number 13 carries a certain value of unluckiness with it. The phobia of the number is called Triskaidekaphobia; Franklin Delano Roosevelt was known to invite and disinvite guests to parties to avoid having 13 people in attendance; high-rise buildings often skip the 13th floor out of superstition. There are many explanations as to how the number 13 received this negative honor, but from a mathematical point of view, the number 13 is also the smallest prime number that when its digits are reversed is also a prime number. It is honored with a place among the Fibonacci numbers and integral Pythagorean triples, as well as many other interesting and lesser-known occurrences. In The Secret Lives of Numbers, popular mathematician Alfred S. Posamentier provides short and engaging mini-biographies of more than 100 numbers, starting with 1 and featuring some especially interesting numbers –like 6,174, a number with most unusual properties –to provide readers with a more comprehensive picture of the lives of numbers both mathematically and socially.
Author: Alfred S. Posamentier
Author: Alfred S. Posamentier
Author: Gary B. Meisner
Author: Alfred S. Posamentier
Author: Mark C. Chu-Carroll
Author: Larry Witham
Author: Richard A Dunlap
Author: Priya Hemenway
Author: Francis D. Hauser, Ph.d.
Author: Andrew Hayduke
Author: Alfred S. Posamentier