



Mystery of golden ratio explained
December 21, 2009
DURHAM, N.C.  The Egyptians supposedly used it to guide the construction the Pyramids. The architecture of ancient Athens is thought to have been based on it. Fictional Harvard symbologist Robert Langdon tried to unravel its mysteries in the novel The Da Vinci Code. "It" is the golden ratio, a geometric proportion that has been theorized to be the most aesthetically pleasing to the eye and has been the root of countless mysteries over the centuries. Now, a Duke University engineer has found it to be a compelling springboard to unify vision, thought and movement under a single law of nature's design. Also know the divine proportion, the golden ratio describes a rectangle with a length roughly one and a half times its width. Many artists and architects have fashioned their works around this proportion. For example, the Parthenon in Athens and Leonardo da Vinci's painting Mona Lisa are commonly cited examples of the ratio. Adrian Bejan, professor of mechanical engineering at Duke's Pratt School of Engineering, thinks he knows why the golden ratio pops up everywhere: the eyes scan an image the fastest when it is shaped as a goldenratio rectangle. The natural design that connects vision and cognition is a theory that flowing systems  from airways in the lungs to the formation of river deltas  evolve in time so that they flow more and more easily. Bejan termed this the constructal law in 1996, and its latest application appears early online in the International Journal of Design & Nature and Ecodynamics. "When you look atwhat so many people have been drawing and building, you see these proportions everywhere," Bejan said. "It is well known that the eyes take in information more efficiently when they scan sidetoside, as opposed to up and down." Bejan argues that the world  whether it is a human looking at a painting or a gazelle on the open plain scanning the horizon  is basically oriented on the horizontal. For the gazelle, danger primarily comes from the sides or from behind, not from above or below, so their scope of vision evolved to go sidetoside. As vision developed, he argues, the animals got "smarter" by seeing better and moving faster and more safely. "As animals developed organs for vision, they minimized the danger from ahead and the sides," Bejan said. "This has made the overall flow of animals on earth safer and more efficient. The flow of animal mass develops for itself flow channels that are efficient and conducive to survival  straighter, with fewer obstacles and predators." For Bejan, vision and cognition evolved together and are one and the same design as locomotion.The increased efficiency of information flowing from the world through the eyes to the brain corresponds with the transmission of this information through the branching architecture of nerves and the brain. "Cognition is the name of the constructal evolution of the brain's architecture, every minute and every moment," Bejan said. "This is the phenomenon of thinking, knowing, and then thinking again more efficiently. Getting smarter is the constructal law in action." While the golden ratio provided a conceptual entryway into this view of nature's design, Bejan sees something even broader. "It is the oneness of vision, cognition and locomotion as the design of the movement of all animals on earth," he said. "The phenomenon of the golden ratio contributes to this understanding the idea that pattern and diversity coexist as integral and necessary features of the evolutionary design of nature." In numerous papers and books over past decade, Bejan has demonstrated that the constructal law (www.constructal.org) predicts a wide range of flow system designs seen in nature, from biology and geophysics to social dynamics and technology evolution. Duke University

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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...


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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.


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One of the delights of life is the discovery and rediscovery of patterns of order and beauty in nature—designs revealed by slicing through a head of cabbage or an orange, the forms of shells and butterfly wings. These images are awesome not just for their beauty alone, but because they suggest an order underlying their growth, a harmony existing in nature. What does it mean that such an order exists; how far does it extend?
The Power of Limits was inspired by those simple discoveries of harmony. The author went on to investigate and measure hundreds of patterns—ancient and modern, minute and vast. His discovery, vividly illustrated here, is that certain proportions occur over and over again in all these forms. Patterns are also repeated in how things grow and are made—by...


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by Public Domain Books
This book was converted from its physical edition to the digital format by a community of volunteers. You may find it for free on the web. Purchase of the Kindle edition includes wireless delivery.


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The power to activate your genetic potential for maximum quality and quantity of life is in your hands. The Golden Ratio Lifestyle Diet is Nature's universal code for robust health and longevity. The Diet is based on a fascinating formulathe Golden Ratiothat when applied to anything, creates greater beauty, unity, efficiency, value and successa greater whole exceeding the sum of its parts. This formula guides the structure of matter and movement of energy throughout the Universe. The benefits are staggering and transformational: the manifestation of your full potential for robust health and longevity. Through the Golden Ratio Lifestyle Diet, you'll learn how to upgrade your body's structure, movement and energy to higher levels. Countless geniuses, including Da Vinci and...


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Quadrivium is the first volume to bring together these four subjects in many hundreds of years. Composed of six successful titles in the Wooden Books series—Sacred Geometry, Sacred Number, Harmonograph, The Elements of Music, Platonic & Archimedean Solids, and A Little Book of Coincidence—it makes ancient wisdom and its astonishing...


The Divine Proportion: A Study in Mathematical Beauty (Dover Books on Mathematics)
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Using simple mathematical formulas, most as basic as Pythagoras's theorem and requiring only a very limited knowledge of mathematics, Professor Huntley explores the fascinating relationship between geometry and aesthetics. Poetry, patterns like Pascal's triangle, philosophy, psychology, music, and dozens of simple mathematical figures are enlisted to show that the "divine proportion" or "golden ratio" is a feature of geometry and analysis which awakes answering echoes in the human psyche. When we judge a work of art aesthetically satisfying, according to his formulation, we are making it conform to a pattern whose outline is laid down in simple geometrical figures; and it is the analysis of these figures which forms the core of Professor Huntley's book. For the philosopher, scientist,...


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by Abrazol Publishing
Mathematics is full of mystery and beauty. Few mathematicians would disagree with that. Most people would agree with the mystery part but not the beauty. There is however one mathematical object that has often been associated with beauty and aesthetics, at least by artists. That object is an irrational number called the golden ratio. It is believed by some that a piece of art that exhibits this ratio in some way is more aesthetically pleasing.
In a painting this could be as simple as making the ratio of the width to height equal to the golden ratio. In architecture it could mean making the dimensions of a room agree with the golden ratio. Whether these beliefs are true or not, is not for us to say.
What we have found, and what this book is about, is the fact that...

