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The Golden Ratio: The Story of PHI, the World's Most Astonishing Number

The Golden Ratio: The Story of PHI, the World's Most Astonishing Number [Kindle Edition]

Mario Livio
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From Publishers Weekly

Most readers will have at least dim memories from geometry class of the irrational number pi. Theoretical astrophysicist Livio gives pi's overlooked cousin phi its due with this lively account, the first on the subject written for the layperson. Phi is the golden ratio of antiquity (1.6180339887), a never-ending number so lauded for its harmonious qualities that in the 16th century it was dubbed the divine proportion. It is related to phenomena as diverse as the petal arrangements of roses, the breeding patterns of rabbits and the shape of our galaxy. Phi is also claimed to have been crucial in the design of the Great Pyramids, the composition of the Mona Lisa and the construction of Stradivarius violins. Livio (The Accelerating Universe) carefully investigates these and other claims and does not hesitate to debunk myths perpetuated by overzealous enthusiasts he calls "Golden Numberists." This is an engaging history of mathematics as well, addressing such perennial questions as the geometric basis of aesthetic pleasure and the nature of mathematical objects. Useful diagrams and handsome illustrations of works under discussion are amply provided. Livio is gifted with an accessible, entertaining style: one typical chapter bounds within five pages from an extended discourse on prime numbers to a clever Oscar Wilde quote about beauty to an amusing anecdote about Samuel Beckett and finally to an eminently clear explanation of G"del's incompleteness theorem. With a guide to the history of ideas as impassioned as Livio, even the math-phobic can experience the shock and pleasure of scientific discovery. This thoroughly enjoyable work vividly demonstrates to the general reader that, as Galileo put it, the universe is, indeed, written in the language of mathematics.
Copyright 2002 Reed Business Information, Inc.

From Booklist

Numbers aficionados will delight in astrophysicist Livio's history of an irrational number whose fame is second only to that of pi. It's called the golden ratio and was discovered by Euclid more than 2,000 years ago. It seems that for any line divided into two unequal segments, the resultant lengths of the two segments and the original line can be formed into a ratio that equals phi, or 1.618 . . . This curiosity of plane and solid geometry might have remained just an oddity had the ratio not cropped up in unusual places, from the structure of crystals to botany to the shape of spiral galaxies. This unending surprise drives Livio's narrative, which he spices with profiles of people obsessed by this ubiquitous number. Some have tried to prove that the ratio was the design principle for the Parthenon; Kepler was crazy about phi; and there's a whole mathematical community devoted to Fibonacci numbers, whose permutations produce phi again and again. Livio's encyclopedic selection of subjects, supported by dozens of illustrations, will snare anyone with a recreational interest in mathematics. Gilbert Taylor
Copyright © American Library Association. All rights reserved


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5.0 von 5 Sternen Remarkable Catalyst for Thought 17. März 2003
Format:Gebundene Ausgabe
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, they 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.
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5.0 von 5 Sternen "The World's Most Astonishing Number" 7. April 2003
Von Timothy Haugh - Veröffentlicht auf
Format:Gebundene Ausgabe|Von Amazon bestätigter Kauf
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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4.0 von 5 Sternen Interesting book, interesting number 9. Februar 2003
Von mrliteral - Veröffentlicht auf
Format:Gebundene Ausgabe|Von Amazon bestätigter Kauf
In mathematics, there are a few irrational numbers that stand out from the infinitely crowded field. First in the bunch would probably be the square root of two, which was kind of the "first" irrational number. Then there is the everpresent pi, and then the less well-known but completely important "e". If there was a fourth place prize, however, it would probably go to the golden number, phi, or roughly 1.618.
In this book, Livio gives a brief history of mathematics and phi's place in it. Intimately related to the Fibonacci numbers, a sequence of numbers in which any given number is the sum of the previous two (after the first couple); these numbers (1,1,2,3,5,8,13...) have shown up in some unlikely places such as sunflowers and nautilus shells. Livio shows us the significance of phi in both the mathematical and physical world.
Livio also makes a good case that phi may be the most overrated of all numbers. Although it has a wonderfully golden name, it actually doesn't live up to its reputation; Livio shows that phi's presence in art and architecture is more fictional than real and that there is nothing about phi that automatically confers aesthetic beauty. A good portion of the book is dedicated to debunking these golden myths.
Overall, this is a good book. Livio's writing is appealing to both mathematician and non-mathematician alike. He does have a tendency to meander from his topic, which can be distracting (even if entertaining), although he eventually does get back on track. For those who like reading about math and the significance of certain numbers (I have also read books on pi, e, i, 0 and infinity), this is a worthwhile read.
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5.0 von 5 Sternen All About 1.618033988749894848204586834365638117720309179... 14. Januar 2003
Von R. Hardy - Veröffentlicht auf
Format:Gebundene Ausgabe
Of all the irrational numbers, the best known is pi, which shows up all over the place. However, if you read _The Golden Ratio: The Story of Phi, The World's Most Astonishing Number_ (Broadway Books) by Mario Livio, you will gain an appreciation for the ubiquity of another irrational with all sorts of amazing properties. You can try this one on your calculator: Phi equals 1.6180339887... (As an irrational, its string of numbers goes infinitely beyond the decimal point, and you can be sure computers have calculated it to millions of places). Take the inverse of that number; that is, divide it into one. You will get 0.6180339887...; in other words, the inverse looks just like phi itself, but with a zero instead of one left of the decimal. Or try this: start with a 1, followed by a 1. The next number will be the two previous ones added together, which is 2; the next number, in turn, is again the two previous ones added together, which is 3. The series goes 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... This is the famous Fibonacci sequence, and is investigated widely within this book as it is intimately connected to phi. Take any number in the series and divide it by the number before it, and you will get a number close to phi; the higher the number in the series, the closer the result comes. (13 / 8 = 1.615 ; 55 / 34 = 1.6176....).
These sorts of number tricks abound in Livio's book, and the mathematics is not daunting. It is also a history of phi, which turns out to be a representative slice of the history of mathematics. Euclid knew the number, but Leonardo Fibonacci in the twelfth century developed the series with its ratio. It shows up in breeding rabbits; spirals in pine cones, sunflowers, galaxies, and hurricanes; tilings and fractals; and many more surprising places. Livio has enormous fun giving and explaining all these examples. Showing up as it does all over the place, perhaps phi is just being seen because that is what is being looked for. Livio, whose day job is being Head of the Science Division at the Hubble Space Telescope Science Institute, is refreshingly dismissive of attempts to try to see a Golden Ratio in everything, which people have tried to do for centuries. It isn't in the pyramids, nor in the Parthenon, nor in Leonardo's paintings.
Without forcing the issue, however, it is easy to see that the Golden Ratio, logarithmic spirals, and Fibonacci numbers are all over the place; there is even a _Fibonacci Quarterly_ mathematical journal. This leads to larger final issues, which Einstein expressed as the question, "How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?" Do mathematical concepts have a universal and timeless existence "out there" and are just waiting for us to discover them? Or is mathematics a human invention that resides only within the human brain? It can't be surprising that this classic conundrum is not definitively solved here. Livio's ideas about it, however, well expressed and tied to this remarkable numerical constant, are well worth thinking about.
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5.0 von 5 Sternen another mysterious and fascinating irrational number like pi and e 21. Januar 2008
Von Michael R. Chernick - Veröffentlicht auf
Several years ago I prepared a review for amazon on this book. Since that time there have been many others to contribute. There are those like me who found it fascinating and gave it five stars, others that gave it a 4 or a 3 because they quibbled with the author over some mathematical issues and finally agroup that really hated it and found it boring and gave it only 1 or 2 stars. Some of those in the third group claim to be mathematicians but thought the book had too detailed. I don't see how a true mathematician could not love this book. Here is what I wrote that I still believe.

The book is 253 pages and 10 appendices about a number called the golden ratio. I give it 5 stars. It is a book for mathematicians and non-mathematicians alike. The first question I asked was how can an entire book be devoted to one number. Well Beckman wrote a book about the number pi and certainly that was interesting. There is a lot to say about the geometry of pi and many mathematical and statistical properties it has. Some including the Buffon needle problem are related by Livio in this book. He contrasts pi to the golden ratio (phi) which also has geometric and mystical properties. The quantity pi is a transcendental number meaning it is not the solution of any algebraic equation. On the other hand phi is algebraic as it is the solution to a quadratic equation.
Other strange properties of phi are:
1. If you subtract 1 from it you get its reciprocal
2. Add 1 to it and you get its square

To see the marvelous algebraic and geometric properties of phi you need only scan through the 10 appendices. Scan through the book and the pictures show you the many artistic properties related to phi.

Although algebraic phi is an irrational number. By applying the quadratic formula to its solution (see Appendix 5 in the book) you will see that its solution involves the square root of 5. Pythagoras and his followers in ancient Greece were said to have discovered irrational numbers (a natural consequence when you study right triangles) and hid this knowledge from the populous.

Phi is defined by Euclid as the "extreme and mean ratio". As Livio quotes Euclid " A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser". This leads to an equality of proportions that yields phi=1.6180339887 rounded to ten decimal places.

Livio also discusses the relationship between the ratio and our concept of beauty (i.e. the quality of the perfect face). It is also interesting that in his new book on the impossibility of solving the 5th degree polynomial by radicals Livio relates the Galois theory of groups to concepts of symmetry. There he also attributes our perception of besuty to symmetry.

If you have the time read the book thoroughly. Write a review that adds to what has been said if you like. Or skim through the pages and appreciate the artist properties of phi along with its algebraic and geometric properties. Read about fractals and myths. Enjoy this wonderful book!
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5.0 von 5 Sternen Magic Kingdom of the Golden Ratio 17. Januar 2003
Von "mimeka" - Veröffentlicht auf
Format:Gebundene Ausgabe
"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.....
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