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Summary: Magic Kingdom of the Golden Ratio
Review: "The Golden Ratio" is truly master piece by Mario Livio. I was impressed by Livio's first book: "The Accelerating Universe", not just by its style and contents, but also by the clear and engaging manner in which Livio had presented the material. In this book Livio has exceeded even the high standards of his earlier book in producing what I found to be a very fascinating and educative book on a seemingly ordinary number. Phi, the so called 'golden ratio', originated from a geometrical concept: if we divide a line into two segments such that the ratio of the length of whole line to that of the bigger segment is the same as the ratio of the length of the bigger segment to that of the smaller one, then this ratio is 'phi'. On the face of it, this looks like a mundane, dull and insignificant ratio. Then Livio leads us through innumerable examples --- examples as varied as the breeding patterns of rabbits to optics of light rays --- where the same golden ratio appears again and again. With Livio's magic touch, the seemingly dull number widens to a fascinating world of its own. We find the same ratio hidden in the delightful petal arrangements in a red rose, in Salvador Dali's famous painting "Sacrament of the last supper", in the spiral shells of mollusks, in the spiral patterns of face-on Galaxies, ..... the list goes on. The book also contains philosophical discussions on such topics as "Is God a Mathematician?", where Livio tries to pierce through the meaning of it all. This book is likely to remain a classic and true source book on the golden ratio for a long time. The book is full of information, and cleverly written. It makes for a very interesting reading: and in the process you will not only learn all about the fascinating world of the golden ratio but also about paintings, flowers, astronomy, and a lot more, in an effortless and enjoyable manner.....
Rating:
Summary: A 'Golden' Story of a Golden Number
Review: A captivating account of a remarkable ratio that permeates thorough nature and most of what we consider to be pure in form. Mario Livio takes the reader on a journey not only of the lives of the people associated with the discovery of this ratio but also paints for him or her a picture of how they explored the natural world, the world of art and beauty, of form and shape and of the philosophy of creation. He tells a story of how great men and women throughout our history have looked upon this number as a manifestation of divinity and speculates on how and why nature may have used this number as a guide to showcase its profound beauty. As the author may say, 'If there is a God, he may be a mathematician!'.
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Summary: Tremendous
Review: A number of disciplines are intertwined and addressed by this riveting work, and the author's depth and breadth of knowledge in each of them is formidable yet naturally conveyed. Artists, scientists, and generalists alike will find this book hard to put down. Turning onto each new page, one wonders with enthusiasm and amazement, "where will phi turn up next?"
Rating:
Summary: Excellent
Review: An easy to read, (as a math book can be), on this number as it came through antiquity and its significance to mankind and nature. Excellent!
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Summary: "The World's Most Astonishing Number"
Review: Are some numbers more important than others? Certainly numbers like the primes, pi and "e" have properties that make them interesting to mathematicians and physical scientists alike. Then there are numbers like 7, 13 and 666 that have other connotations for theologians, numerologists and the like. And yet, some numbers have not gotten their due in recent years. Phi--a number variously referred to as the golden ratio, golden section, and divine proportion among others--is one. But Mario Livio has written a book in an attempt to remedy this situation. Phi received its original definition from Euclid as an "extreme and mean ratio" when a straight line is cut so that the ratio of the entire line to the longer division of the segment is the same as the ratio of the longer division of the segment to the shorter. And yet, much like the better known geometrical example of pi, phi turns out to have many more applications beyond its simplest geometrical definition. Though measurable, phi is an irrational number with relationships to the Fibonacci sequence, fractals, the physical structure of things from plant growth and spiral shell development to the appearance of large-scale objects like galaxies, and more. And beyond this, phi has been used as a basis applications in numerology and aesthetics. Livio does a very good job of covering all this ground and more. He is especially good at giving us a historical overview of the development of our understanding of this important number as well as explaining the mathematics in a way that is complete but easy to understand. He is also very good at presenting the various mystical ways phi has been interpreted over the centuries, giving each a rigorous challenge--rejecting many but open-minded to the possibilities that any good Platonist would be. In fact, if there is a weakness in this book, it is that Livio spends a lot of time covering these more esoteric applications of phi. And yet, these applications are part of the history of the number and cannot be ignored whatever a reader might feel about the value of these applications. Phi may not quite live up to the hype as "the world's most astonishing number" but certainly any reader with an interest in mathematics will not want to miss this book.
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Summary: A Difficult Mathematical Concept Revealed
Review: As a non-mathematician I appreciate any help I can get in understanding the more esoteric parts of math. The Golden Ratio is just such a concept. Fortunately, Mario Livio has shown much light on this remarkable corner of geometry in his book "The Golden Ratio." It is little wonder that such numbers as the Golden Ratio were considered magical. The never ending, never repeating number that cannot ever be expressed as a fraction has an uncanny tendency to show up in the oddest places, not only galactic structure and nautilus shells, but in plant parts and composition of paintings and music. Unfortunately magical numerology can lead to far-fetched relationships, as to the so-called number of the beast (666), and to academicism in art. Just because the Golden Ratio results in a pleasing relationship in a composition we are not tied to always measure art on how well it fits that ratio! Livio has illuminated the history of the Golden Ratio in such a way that much of the associated themes can be understood by the reasonably educated laymen. While some of the book can be tough sledding for most of us non-mathematicians, the gist is available to all with some effort. Read this book to learn about the history of interpretation and misinterpretation of mathematical concepts.
Rating:
Summary: A Difficult Mathematical Concept Revealed
Review: As a non-mathematician I appreciate any help I can get in understanding the more esoteric parts of math. The Golden Ratio is just such a concept. Fortunately, Mario Livio has shown much light on this remarkable corner of geometry in his book "The Golden Ratio." It is little wonder that such numbers as the Golden Ratio were considered magical. The never ending, never repeating number that cannot ever be expressed as a fraction has an uncanny tendency to show up in the oddest places, not only galactic structure and nautilus shells, but in plant parts and composition of paintings and music. Unfortunately magical numerology can lead to far-fetched relationships, as to the so-called number of the beast (666), and to academicism in art. Just because the Golden Ratio results in a pleasing relationship in a composition we are not tied to always measure art on how well it fits that ratio! Livio has illuminated the history of the Golden Ratio in such a way that much of the associated themes can be understood by the reasonably educated laymen. While some of the book can be tough sledding for most of us non-mathematicians, the gist is available to all with some effort. Read this book to learn about the history of interpretation and misinterpretation of mathematical concepts.
Rating:
Summary: Remarkable Catalyst for Thought
Review: Author Mario Livio has added another wonderful book that bridges the gap between writings meant for the academic and for the layperson. "The Golden Ratio", was not quite as accessible as some other books that I have read, but it is certainly worth the extra effort even if many of the proofs found at the book's end remain a mystery. The book spent a great deal of time disproving the myths of application that have been attributed to the remarkable relationships of the number PHI. Like the more familiar PI it is a decimal that has yet to be proven to ever repeat itself, and it demonstrates its presence both widely and in fascinating manners. Whether or not Mozart used the number or Leonardo Da'Vinci did, or even if a building long considered to be predicated upon the number truly is or is not, does not detract from the wonders of this numbers appearance. The writer will take you through the commonality in the structure of a Nautilus shell, the arrangement of leaves around the stem of a plant, and even how by tossing a coin can prove truth or fraud in the accounting practices of business. He shares an example of having two groups, one tosses a coin 200 times and records each result. The second group does nothing with the coin, they just manufacture the results. The two groups can be detected, the why is a wonderful find. Certain shapes whether they are the outline of a room or the dimensions of a painting will generally be found the most pleasing by the majority of people. And lest you think these rules are confined to the shell on the beach or an image on your wall, they extend to those galaxies of which we are a part, why planets move in the orbits they do, and what would happen if the slightest changes were made. As I said, this particular work requires more effort than others I have read and commented upon recently, and the enjoyment or aversion you have to math will play a role in how much pleasure this read will bring. However, even if the formulas leave you cold, the spirals upon rectangles that form, "The Eye Of God", are guaranteed to fascinate.
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Summary: Not quite what you might expect
Review: Bought this after reading Da Vinci Code, and I was looking for more of the same. it is a little dry. It is a quick read though, and how much do you want to know about 1.618?
Rating:
Summary: Mathematically Profound
Review: Broad streams of literary, historical, aethsetic and religious thought are pooled together in a concise and well-illustrated review of this powerful proportion, which recurs in the natural world in surprising places both large and small. Clearly presented mathematical proofs give the book a solid backbone. Mathematical ideas are expressed in the book through a combination of prose, appendix proofs, and plentiful illustrations & diagrams. This allows readers of varying mathematical ability and learning styles to appreciate the beautiful ideas that Livio gracefully presents. A must for serious lovers of proportion & geometry, architects, mystics, painters, graphic designers and mathematicians.
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