Product Description
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. ... Read more

Customer Reviews (88)

Nice and fast
This book came when it was advertised. Right out of the box, I noticed a few dust marks on the book so it wasn't a perfect pristine book, but it still looked new.

I'd give it six stars if I could
A thoroughly satisfying and objective examination of the subject, which has been a fascination to me since eighth-grade math class. Beginning with the speculative history of numbers and mathematics and weaving a rich narrative historically and scientifically through the present day, it takes time to scrutinize spurious legends of the golden ratio (many exist) while providing a wealth of information of it. It astonished me and then kept astonishing me. I recommend it to all numbers people.

The Golden Ratio

Not being inclined towards number mysticism, I found the book fuzzy and sluggish. Whether the golden ratio was or wasn't used in human construction or composition doesn't interest me, because I find no mystical meaning arising out of such use (much of the book is about how claims of its use are in fact wrong). The discovery of this ratio in nature, on the other hand, is interesting because the question arises of why it's occurring there. But the book isn't structured tightly and the author's meandering, even when ostensibly concerned with the golden ratio in mathematics or nature, frustrated my attention. It took me quite a while to finish it because it didn't hold my interest. Livio's final chapter on whether or not God is a mathematician doesn't bode well for his book on that subject.

A dry book best left for math nerds
I liked the idea of learning more about math and numbers, and the history of such things.This book seemed promising at first, but the further along I went, the drier it became.There is a lot of advanced math that leaves a non-mathematician feeling like they are missing the punchline.The number that represents the golden ratio certainly has interesting properties, but I really don't feel like my life was enriched by knowing more about phi.

Bottom line: too much "technical detail" for the casual reader that leaves one rather drowsy.

Decent
I bought this book from a designer's perspective and hoped to get some good insight on the use of Phi in art and design.The first chapter or two gave a good start but then it went into half a book of history of philosophy and mathematics that, while relevant (I guess), didn't really help me when it came to learning about the use of Phi.I ended up skimming through the rest of the book.Towards the end there's some more samples of how it may or may not have been used in the past, but by that time my mind was so glazed over I didn't really read any.
As a history of the number and a general historical mathematical book, it's not too bad.It's written well enough. But for a designer trying to get new ideas, it didn't really work. ... Read more

Product Description
In this text, 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 Finobacci 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, latice 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. ... Read more

Customer Reviews (2)

The definitive book on the subject
This book is not absolutely perfect, but it is so much better than any other one on the subject that it deserves a 5-star rating. The majority of books on Fibonacci numbers and the golden ratio fall into three categories: (1) Books for children, (2) Mystical mumbo-jumbo, and (3) Books claiming you can use Fibonacci numbers to win in the stock market (!). Even Mario Livio's book, which is better than most, uses so much of its space to _debunk_ some of the mystical mumbo-jumbo that it de-emphasizes the wonderful patterns that can be found in the Fibonacci sequence, which is beautiful enough in its mathematical properties that one doesn't need to make such digressions. Dunlap's book, by contrast, _does_ emphasize those patterns. And when it's not directly addressing the Fibonacci sequence's properties, it's discussing things like the golden ratio and Penrose tilings, which are completely within the category of mathematics related to the topic.

I really enjoyed the book, even though it has some typographical errors and minor omissions. The formulas collected in the third appendix alone justify buying the book. I could wish that there were more given, but this is the best collection I have seen.

It's the Beauty of Mathematics
Well, I have studied these numbers and other fascinating phenomena in mathematics, and I never found it enough. So, every book I read about this stuff I found a new set of new things, or at least a new view/review of the old things I knew. After my short study (7 years now) in the field of Number theory and Related topics, I dare say, that this book was another addition to my knowledge that -thanks GOD- I didn't waste. ... Read more

Product Description
An inspirational call for a return to the tenets of traditional architecture as a remedy for the dehumanizing standards of modern architecture

• Explains how modern architecture is emblematic of our current estrangement from the spiritual principles that shaped humanity’s greatest civilizations

• Reveals how the ancient laws of sacred proportion and harmony can be restored

The ugly buildings that characterize the modern landscape are inferior not only to the great cathedrals of medieval Europe and the temples of ancient Egypt and Greece, but even to lesser buildings of the more recent past. The great masterworks of our ancestors spoke to humanity’s higher nature. Architect Herbert Bangs reveals how today’s dysfunctional buildings bring out the worst in humanity, reinforcing that which is most base within us. He shows how, through the ancient laws of proportion and number, architecture once expressed the harmonious relationship between man and the cosmos. In early times, the architect worked within a sacred and esoteric tradition of creating structures through which human beings could gain insight into the nature of the divine reality. Today, that tradition has been abandoned in favor of narrowly defined utilitarian principles of efficiency and economy.

In The Return of Sacred Architecture, Bangs provides the key to freeing architecture from the crude functionality of the twentieth century: the architects of the modern human landscape must find the deep-felt connection to the cosmos that guided the inner lives of those who built the temples of the past. The form of their buildings will then reflect the sacred patterns of geometry and proportion and bring forth greater harmony in the world.
... Read more

Customer Reviews (4)

Good Therapy?
The author does a very good job of describing the pathetic state of contemporary architecture to include all the isms.He starts with the school of the Bauhaus and continues with the concepts of machine age architecture or what was termed the international style and then classifies those associated with the modern movements as "materialists".The author covers interesting flaws associated with celebrated architectural works by famous architects such as Mies van der Rohe, Le Corbusier and Frank Gehry.A sampling of his humor is his reference to Frank Gehry's Guggenheim Museum in Bilbao, Spain of which he terms this building "hidiotic", a combination of hideous and idiotic.The intent is to drive home just how inhuman machine architecture "is". I especially enjoyed the spiritual references to Zen and Gnosticism as well as references to Carl Jung's archetypes.The author touches on four archetypes of great architecture such as the cave, the clearing, the garden and water and the associated balance and harmony needed to successfully experience these archetypes in great architecture.Mathematical symbolism is discussed along with the Golden Ratio and two small modern examples are analyzed.Of interest to note is that this book does not contain any architectural gibberish.While I can't say I agree with everything in it, it is easy to comprehend.Andrea Palladio's proportion concepts are covered as architecture as frozen music.I admire the author for having the courage to create such a work.The author has convinced me that architecture has taken a detour towards the "ego" or mind (logic) and has ignored the architecture of the heart/spirit/intuition.Great architecture combines the mind and the heart and thus becomes wholehearted.After living in the contemporary world of "ego architecture", it is refreshing and therapeutic to learn there is another like me in this world.

We Create Our Own Reality
Do you wonder why some buildings can change the way you feel?Well wonder no more.

There are certain mathematical ratios that occur repeatedly in the universe, and have been known to the most advanced civilizations preceding us.Most of the currently built structures in the US do not reflect these geometric ratios, and architects do not learn any of this in school.Nowadays it is all about costs and efficiencies.Project managers have more to say about the construction than the architects.Buildings are merely boxes in which we work or things are sold.

As the author reflects, our buildings reflect our current nature.Looks like we're not as advanced as we perceive ourselves to be.

Enlightening & Comprehensive Overview of Art & History of Sacred Geometry in Architecture
Herbert Bangs brings extraordinary insights into his scholarly yet impassioned exploration of sacred architecture, which is a relatively new and vitally important subject. Bangs presents illustrations and historical accounts of a wide variety of architectural structures, letting the stories of each example demonstrate the points he makes regarding archetypes of shelter (cave, clearing, garden, water, elements), archetypes of design (duality, hierarchy, materials), and resolution of form. Bangs enthusiastically praises those architectural masterpieces which prove themselves to be aesthetically pleasing and livable, and denounces designs which may have been lauded by others, yet actually are tolerable by few. There are reasons why some spaces can lead people to feel relaxed and refreshed... or confused and stressed... and Bangs brilliantly describes the architectural principles and qualities that create these results. THE RETURN OF SACRED ARCHITECTURE is a clarion call for adopting a more intuitive approach to design and a better appreciation for the sources of divine inspiration. I give this book my highest recommendation to everyone, since all of us are at some point involved in designing, buying, visiting, and living in architectural structures.

Especially recommended for any college-level architectural library.
Both architecture and spirituality collections will relish Return of Sacred Architecture: The Golden Ratio and the End of Modernism: it contrasts and modern religious architectural structure with the grand monuments of the past, revealing how modern dysfunctional buildings bring out the worst in humanity and how ancestral masterworks focus on the best. Architecture once was closer to expressing the sacred between man and cosmos: THE RETURN OF SACRED ARCHITECTURE focuses on the elements of this expression and is especially recommended for any college-level architectural library. ... Read more

Product Description
What do the Parthenon in Greece, crosses in graveyards, the ratio of the height of the navel to a woman's total height, sunflowers and Leonardo da Vinci's "Mona Lisa" have in common? The answer is that these disparate elements share a geometrical proportion of ancient origin, commonly known as the 'Golden Ratio', or expressed numerically as phi. In a captivating journey through art and architecture, botany and biology, taking in fractals and DNA on the way, Livio takes us into the heart of this extraordinary number, and gives an account of the phi-obsessed individuals who devoted their lives to discovering its secrets - from the Pythagoreans who believed that the study of the Golden Ratio would reveal the hand of God, to the great astronomer Johannes Kepler who felt that phi was the greatest treasure of geometry, to the modern day scientists who are discovering ever more remarkable ramifications of this strange, almost mystical, number. It is a journey into the heart of beauty itself. ... Read more

Product Description
As a master baker, painter, and woodworker, Stephen Lanzalotta has used the mathematical principles of The Golden Ratio (an integral plot element in The Da Vinci Code) for more than 30 years. His realization that this seemingly magic formula, once used by Da Vinci, held the secret to optimal health and weight loss led him to apply it to his menu at his popular bakery/caf. Thus was born THE DIET CODE, a revolutionary Mediterranean-style program that uses the ancient mathematical sequence to balance carbs, proteins, and fats in order to boost metabolism and spark weight loss. The plan reveals Renaissance foods that promote weight control and includes unique tips for diet success, including: * Eat breadbut not without fat orprotein * Cook pasta al dente to boost fat loss * Balance your plate by The Golden Ratio; 52% carbs, 20% protein and 28% fat. Combining menu plans and recipes, as well as Renaissance lore and Italian tips on healthy eating, this is a unique diet plan from the ages for the ages. ... Read more

Customer Reviews (18)

An interesting concept
If you look beyond the obvious Da Vinci Code association, this is actually a very well written book with a well thought out diet. There is no starvation, no exclusion of entire food groups, just good sensible eating based on nutrient rich foods in balance and proportion to one another. The diet is very simple to follow. It doesn't require you to buy a bunch of exotic ingredients. It is also very easy for those with a family, whether they choose to join you on the diet or not.

For those looking to lose a pound a day, don't bother reading this. Go stick your head in another miracle cure book or buy yourself another bottle of can't miss pills. For those looking for a sensible guide to not only weight loss, but a healthier lifestyle as well, then this book is well worth reading.

And for those who complained because Da Vinci may have been a vegetarian, you weren't reading the book carefully. It wasn't about what Leonardo may or may not have personally eaten. It was about what was available to be eaten in Leonardo's era.

good
I liked the rpesentation of this book. The format, the graphics, the cover all were excellent. the content was also good which was refreshing because sometimes these types of books are found lacking.

is this book going to change your life as some reviews claim? well, if they say so... it did not change my life but I enjoyed it nonetheless. Recommended.

The Book that changed my life!
First let me say that Jimmy Moore does NOT know what he is talking about! He has it wrong right from the start because the author is NOT a baker from Portland, Oregon.He is a baker from Portland, Maine. Now you can make your own judgements on the rest of his review.It is true the beginning can make your head spin with all the unnecessary information, but if you perservere and read it through to the actual eating plan and recipes, you will find it makes alot of sense.Don't worry if you don't understand what he is talking about with all the formulas and numbers. It doesn't matter. I started eating according to this book and I have lost 17 lbs in 2 1/2 months. No added exercise!I am NEVER hungry and I feel great. I had stomach trouble and decided to follow this just for the sake of cutting out processed foods. Believe me, it IS as easy as it sounds. I love the food, never feel deprived and I can't believe how well it works.
If you have tried everything else, you might as well try this. You won't be sorry and it definitely cannot hurt you! I started losing weight the first week and it continues. I can't say enough about this book, and NO, I don't know the author. I had never heard of him until I saw him on PBS. Its a great book, great recipes, great food.

nutrition that tastes good.
What i liked about the author and the book is he loves food and loves to eat, This not a book about what to cut out as it is a about eating lots of wonderful tasting food that just happens to be good for you. This a positive response to all the food police. No one will stick a food regiment that doesn'taste good. You look forward to his meals as opposed to feeling you are stuck with the meal if you are to stick to the diet. I would have rated it a 5 if they had just incuded color photos. 2 thumbs up.

Food for Thought
Well written and helpful.Gives you something to chew on both figuratively and actually. ... Read more

Product Description
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. ... Read more

Product Description
Purchase includes free access to book updates online and a free trial membership in the publisher's book club where you can select from more than a million books without charge. Chapters: Unary Numeral System, Golden Ratio Base, Fibonacci Coding, Quater-Imaginary Base, Negative Base, Factorial Number System, Bijective Numeration, Non-Integer Representation, Balanced Ternary, Mixed Radix, Redundant Binary Representation, Non-Standard Positional Numeral Systems, Pentimal System, Negafibonacci Coding, Signed-Digit Representation, Bubble Babble, Non-Adjacent Form, Complex Base Systems. Excerpt:Numeral systems by culture Balanced ternary is a non-standard positional numeral system (a balanced form ), useful for comparison logic. It is a ternary system, but unlike the standard (unbalanced) ternary system, the digits have the values 1, 0, and 1. This combination is especially valuable for ordinal relationships between two values, where the three possible relationships are less-than, equals, and greater-than. Balanced ternary can represent all integers without resorting to a separate minus sign . Balanced ternary is counted as follows. (In this example, the symbol 1 denotes the digit 1, but alternatively for easier parsing " " may be used to denote 1 and "+" to denote +1.) Decimal: 6: 5: 4: 3: 2: 1: 0: 1: 2: 3: 4: 5: 6 Unbalanced ternary can be converted to balanced ternary notation by adding 1111.. with carry, then subtracting 1111... without borrow (the string of 1s must be the same length, so if the result of the addition has more digits, subtract nothing from these extra digits). For example, 0213 + 1113 = 2023, 2023 1113 = 1113(bal) = 710. Balanced ternary is easily represented as electronic signals, as potential can either be negative, neutral, or positive. Utilizing a third state encompasses more data per digit; linearly approximately 1.5849 (log23) bits per trit. Applications Balanced ternary has other applications besides computing. For example, a classical "2... ... Read more

Product Description
Chapters: Golden Ratio Base, List of Works Designed With the Golden Ratio, Pentagram, Proportion, Golden Section Search, Golden Spiral, Golden Rectangle, Jay Hambidge, Golden Angle, Golden Triangle, Golomb Sequence, Lute of Pythagoras, Mark Barr. Source: Wikipedia. Pages: 103. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to (=) the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.6180339887. Other names frequently used for the golden ratio are the golden section (Latin: sectio aurea) and golden mean. Other terms encountered include extreme and mean ratio, medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut, golden number, and mean of Phidias. The golden ratio is often denoted by the Greek letter phi, usually lower case (). The figure on the right illustrates the geometric relationship that defines this constant. Expressed algebraically: This equation has as its unique positive solution the algebraic irrational number At least since the Renaissance, many artists and architects have proportioned their works to approximate the golden ratioespecially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratiobelieving this proportion to be aesthetically pleasing. Mathematicians have studied the golden ratio because of its unique and interesting properties. Construction of a golden rectangle:1. Construct a unit square (red).2. Draw a line from the midpoint of one side to an opposite corner.3. Use that line as the radius to draw an arc that defines the long dimension of the rectangl...More: http://booksllc.net/?id=12386 ... Read more

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Keith Flynn's fourth and most innovative collection of poems to date, The Golden Ratio, is filled with themes of music and balance, creating a literary world in opposition to our discordant contemporary times, where the human population is at odds with one another and the Natural World, out of sync and unfocused, spiritually and physically adrift. The book's six sections deal separately with the state of our government, our art forms, and our relationship and responsibility to the society that surrounds us. This book is a tour de force by one of our most distinctive poetic voices, seeking to redefine what it means to be human and creative in the face of overwhelming opposition. ... Read more

Customer Reviews (2)

Flynn Spins The World Into A Beautiful Balance
Keith Flynn's latest book, The Golden Ratio, is one of the most compelling and inventive collections of poetry that I have ever read. From the exquisite imagery in "Mysterium," the book's intro poem, through the historical masterpiece, "Chang and Eng," and the political theater of "Hitler's Yacht in America," the poems coalesce to create a multi-faceted picture of the contemporary world. In the last section, Flynn shows us how music is the real moral law in the brilliant rhythmic jazz profiles of giants such as Duke Ellington, Chet Baker, Herbie Hancock, Ellis Marsalis and others, by using surreal juxtapositions, intricate wordplay, and thought-provoking descriptions to bring these lives and the music they play into fuller focus. Flynn is one of our best contemporary poets and this book shows his incredible talents in full flower. The poems about divorce in the book's mid-section will break your heart as the poet struggles to find a proper balance in his relationship with the new landscape alone, and the ways to reconcile his artistic growth with the personal process of aging and loving again. This might be the first real Baby Boomer coming-of-age collection that defines this era in clear, unflinching terms. This book creates an entirely new world between its covers and builds sentences that are unforgettable. Don't miss it...

When the Mountain Boy becomes the Intellectual
Keith Flynn is a mountain boy at heart, but there is an urbanite rolling around under the surface. His poems reflect the duality of his nature: the hillbilly and the intellectual.

In this collection there are poems that honor nature, jazz, and variety.The set of poems from the voices of Chang and Eng (the legendary Siamese twins who retired to NC) are fascinating as are the poems that delve into the world of jazz, hoping to honor the narrator's heroes of that world as well.

My favorite was Welsh Fantasia with lines such as "Only mystery enables us to live."

This is a long collection of over 150 pages and many of the poems are lengthy with long lines as well so you will certainly get your money's worth.My only complaint with this collection is that at times the author's lust for language overpowers me as so many complex words are strung together. Or maybe, I'm just not as smart as Mr. Flynn :)

I'd defintiely suggest this collection for Southern Literature fans and for anyone out there who has a love of the world of jazz. ... Read more

Product Description
Chapters: Mathematical Constants, Physical Constants, Dimensionless Quantity, Avogadro Constant, Pi, Golden Ratio, Gas Constant, Stefan-boltzmann Constant, Faraday Constant, Brun's Constant, Catalan's Constant, Feigenbaum Constants, de Bruijn-Newman Constant, Euler-mascheroni Constant, Parts-Per Notation, Bohr Radius, Planck Mass, Planck Time, Planck Temperature, Square Root of 2, Time Constant, Square Root of 5, Silver Ratio, Particle Data Group, Schnirelmann Density, Astronomical Constant, Elementary Charge, Apéry's Constant, Degree, Electron Rest Mass, Zeta Constant, Madelung Constant, Euler Product, Gaussian Gravitational Constant, Khinchin's Constant, G-Factor, Particular Values of the Gamma Function, Electron Magnetic Dipole Moment, Rydberg Constant, Loschmidt Constant, Representations of E, Viswanath's Constant, Champernowne Constant, Plastic Number, Classical Electron Radius, Mrb Constant, Period, Bohr Magneton, Mills' Constant, Molar Mass Constant, 6174, Square Root of 3, Gelfond's Constant, Golomb-dickman Constant, Twelfth Root of Two, Gelfond-schneider Constant, Parabolic Constant, Astronomical Almanac, Fransén-robinson Constant, Cahen's Constant, Reciprocal Fibonacci Constant, Komornik-loreti Constant, Stieltjes Constants, Omega Constant, Gauss's Constant, Somos' Quadratic Recurrence Constant, Prouhet-thue-morse Constant, Glaisher-kinkelin Constant, Feller's Coin-Tossing Constants, Conic Constant, Bernstein's Constant, Curie Constant, Niven's Constant, Hartree, Meissel-mertens Constant, Hilbert Number, Hafner-sarnak-mccurley Constant, Landau's Constants, Embree-trefethen Constant, Erdős-borwein Constant, Laplace Limit, Legendre's Constant, Ramanujan-soldner Constant, Atomic Mass Constant, Heath-Brown-moroz Constant, Backhouse's Constant, Lieb's Square Ice Constant, Landau-ramanujan Constant, Lévy's Constant, Seshadri Constant, Sierpiński's Constant, Kepler-bouwkamp Constant, Oort Constants, Favard Constant, Lebesg...More: http://booksllc.net/?id=12386 ... Read more

Product Description
Chapters: E, Pi, Golden Ratio, Brun's Constant, Catalan's Constant, Feigenbaum Constants, de Bruijn-Newman Constant, Euler-mascheroni Constant, Mathematical Constant, Square Root of 2, Square Root of 5, Silver Ratio, Schnirelmann Density, Apéry's Constant, Degree, Zeta Constant, Madelung Constant, Euler Product, Khinchin's Constant, Particular Values of the Gamma Function, Representations of E, Viswanath's Constant, Champernowne Constant, Plastic Number, Mrb Constant, Period, Mills' Constant, 6174, Square Root of 3, Gelfond's Constant, Golomb-dickman Constant, Twelfth Root of Two, Gelfond-schneider Constant, Parabolic Constant, Fransén-robinson Constant, Cahen's Constant, Reciprocal Fibonacci Constant, Komornik-loreti Constant, Stieltjes Constants, Omega Constant, Gauss's Constant, Somos' Quadratic Recurrence Constant, Prouhet-thue-morse Constant, Glaisher-kinkelin Constant, Feller's Coin-Tossing Constants, Conic Constant, Bernstein's Constant, Niven's Constant, Meissel-mertens Constant, Hilbert Number, Hafner-sarnak-mccurley Constant, Landau's Constants, Embree-trefethen Constant, Erdős-borwein Constant, Laplace Limit, Legendre's Constant, Ramanujan-soldner Constant, Heath-Brown-moroz Constant, Backhouse's Constant, Lieb's Square Ice Constant, Landau-ramanujan Constant, Lévy's Constant, Seshadri Constant, Sierpiński's Constant, Kepler-bouwkamp Constant, Favard Constant, Lebesgue Constant, Linnik's Constant. Source: Wikipedia. Pages: 277. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to (=) the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.6180339887. Other names frequently ...More: http://booksllc.net/?id=12386 ... Read more

Product Description
Chapters: Golden Ratio. Source: Wikipedia. Pages: 103. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to (=) the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.6180339887. Other names frequently used for the golden ratio are the golden section (Latin: sectio aurea) and golden mean. Other terms encountered include extreme and mean ratio, medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut, golden number, and mean of Phidias. The golden ratio is often denoted by the Greek letter phi, usually lower case (). The figure on the right illustrates the geometric relationship that defines this constant. Expressed algebraically: This equation has as its unique positive solution the algebraic irrational number At least since the Renaissance, many artists and architects have proportioned their works to approximate the golden ratioespecially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratiobelieving this proportion to be aesthetically pleasing. Mathematicians have studied the golden ratio because of its unique and interesting properties. Construction of a golden rectangle:1. Construct a unit square (red).2. Draw a line from the midpoint of one side to an opposite corner.3. Use that line as the radius to draw an arc that defines the long dimension of the rectangle. Two quantities a and b are said to be in the golden ratio if: This equation unambiguously defines . The right equation shows that a = b, which can be substituted in the left part, giving Dividing out b yields Multiplying...More: http://booksllc.net/?id=12386 ... Read more