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

Jul 15, 2007

Sabio rated it

Shelves: math_physics

When people leave organized religion, they often look for mystical awe elsewhere. Math is a not infrequent haven of new agers who gave up on the tried and true faiths of their parents. Real math takes lots of work, whereas quick mystical attachment takes very little effort.

This book shows how many people have read far too much into Phi (1.6180339887 ...) [The Golden Ratio]. The author shows how, Phi is prevalent in nature, but it is not magically so. Phi's prevalence is due simp...more

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Apr 01, 2009

Tim rated it

Mathematical constants make engaging characters in the popular imagination. At least the rash of books for general audiences in this vein published in the last two decades suggests this. Astro-physicist Livio's leading character is a somewhat less well-known constant - those special numbers discovered or created by mathematicians over the centuries. Phi - the so-called Golden Ratio - has been known since Euclid. Geometrically, given a line AB cut by point C, where AC > CB, then locate C on...moreMathematical constants make engaging characters in the popular imagination. At least the rash of books for general audiences in this vein published in the last two decades suggests this. Astro-physicist Livio's leading character is a somewhat less well-known constant - those special numbers discovered or created by mathematicians over the centuries. Phi - the so-called Golden Ratio - has been known since Euclid. Geometrically, given a line AB cut by point C, where AC > CB, then locate C on the line so that AC/BC = AB/AC. This equality yields Phi, an irrational number beginning with 1.618. Livio calls Phi the 'most irrational' of irrational numbers since, written as a continued fraction, it consists entirely of ones and thus converges slower than any other irrational number written as a continued fraction. The character of Phi turns up in many places - for example in construction of the pentagon and pentagram, in Fibonnaci numbers, in mathematical descriptions of the arrangement of buds in a sunflower or leaves on a plant stem, in the shape of galactic spirals. Particularly important in the history mathematics and in Livio's account is the Fibonnaci sequence where the ratio of successive numbers increasingly approximates Phi. In its many appearances Livio conveys the magic of Phi. But Livio is a debunker too. Phi's magic, he argues, lies not in some natural aesthetic known to ancient peoples, artists, musicians, or others. Most such claims for the use of Phi - in the construction of the pyramids, in painting, in music, in architecture, or elsewhere - he shows to be 'number juggling', claims founded on ambiguities in measurement and interpretation without strong psychological foundation. Yet the romance of the Golden Ratio does raise for Livio foundational questions about mathematics. As he asks in his closing chapter, "Is God a Mathematician?" Do mathematicians discover constants like Phi and the rest of mathematics all of which enjoy some objective natural or Platonic existence? Or are mathematical constants and the rest of the edifice of mathematics somehow just the creations of brains such as ours? His own conclusion is that "Our mathematics is the symbolic counterpart of the universe we perceive" where our perceptual apparatus (collectively as well as individually) is a kind of evolving computer program, a rule-based system which, in interactions with nature - ourselves and the world - generates mathematics, culture, the arts and the uncounted endeavors to which the human presence on this planet gives rise.(less)

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Mar 15, 2011

Woodge rated it

Shelves: mathematics, non-fiction

Here I go all math geeky again. I picked up this slim book (about 250 pages) a couple years ago and then I started thinking about it and felt compelled to read it. (Voices in my head. You know.) The golden ratio, or phi (pronounced "fee"), was first discovered by Euclid (remember him from geometry class?). Somewhere around 300 B.C. Euclid--

YOU: Whoa-whoa-whoa, wait a minute, Woodge... you actually read another book about math. For fun?! Are you for real?
WOODGE...more

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Apr 16, 2011

Scarlett Sims rated it

Shelves: history, leisure, maths, philosophical, real, scholarly, science

I bought this book several years ago. I'll admit it, I bought it because of the Da Vinci Code. Dan Brown even has a quote on the cover! Which, after I read the book, I thought was kinda weird, because the author spends a lot of time debunking ideas about the appearance of the Golden Ratio in art and architecture. I liked the skeptical approach he took when looking at claims of it being used in the pyramids or the works of Da Vinci.

Really, anyone could read this book, ie non mathema...more

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Dec 09, 2008

Kyle Wright rated it

· review of another edition

Having expected a book filled to the brim with Phi related information, I feel let down by the end result. Livio's book covers a broad history of mathematics and geometry in general, with instances of phi thrown in where context allows. While I did learn a lot of interesting information about the origins of math, I felt that Livio left little space for the phi-related aspects, which was the primary purpose of my reading this book.

When Livio does manage to address phi directly, he d...moreHaving expected a book filled to the brim with Phi related information, I feel let down by the end result. Livio's book covers a broad history of mathematics and geometry in general, with instances of phi thrown in where context allows. While I did learn a lot of interesting information about the origins of math, I felt that Livio left little space for the phi-related aspects, which was the primary purpose of my reading this book.

When Livio does manage to address phi directly, he does so by debunking preexisting phi-myth. There are numerous instances of claims of phi related architecture, art, writing, and natural phenomenon, and Livio shows that many of these are not true, or at the very least, conjecture with no solid evidence. I wish that instead of spending so much time telling what phi is not and where it isn't found, Livio would have spent more time on instances where phi occurs. If I were to write a book about "glurbl," but spend the whole time telling you that cars aren't glurbl, neither are bowling balls, and despite popular belief, the Guggenheim museum is not a glurbl, by the end you would probably have no better idea of what glurbl is. That is what I felt about most of this book.

As the end of the book drew near, Livio managed to eke out a few examples of phi and better explain it's place in mathematics. Overall, however, I thought the book was better as a general overview of the history of mathematics and geometry, with the occasional contextual highlight on phi, than it was a comprehensive tome solely about phi. I was also disappointed that Livio proposed many theories, but he never really took a stand on the issues, instead leaving me with a vague impression that either could be right, and that he didn't care much which it was.

While not nearly the impressive book on phi that I had hoped for, Livio managed to present an interesting book on the general history of mathematics and geometry. I suppose that if you are interested in phi, then this is a good first read, as it will help you approach the other books with a more critical mind. Just don't expect this to be a book full of amazing phi occurrences and phenomenon.(less)

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Aug 01, 2011

Jasmine rated it

Shelves: romanian

so I stayed up past bedtime tonight to finish this book, not because I love this book, but because I would give anything to not be reading it anymore and now I'm not.

I'm not a platonist. I don't look at concepts made up by humans and say those describe things humans see so they must have a magical relationship to truth. I actually weirdly assume when people make things up those things should be related to what is true so it is a given they will relate to true things.

there w...more

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Aug 30, 2009

Tiffany rated it

I found the seed of the book interesting, but after long strings of repetitions of examples, I got a little bored. I think the author has hit on a voice that works for people with some mathematical background and interest, but I think he's overshot for people who didn't enjoy learning math (on the other hand, would they have picked up the book in the first place?). I confess that I skipped the proofs at the back. But I did enjoy the beginning of the book - brief bits of the history of mathematic...more

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Jul 29, 2011

Jason Mccool added it

As a bit of a geek, I thought this sounded interesting. Although it did have interesting parts, it seemed like the real meat of the book could have been condensed considerably. It is good in providing a summary of nearly everything that's ever been said or written about the golden ratio, but it's probably a little exhaustive for the average reader, or even the average geek. A lot of it reads more like a thesis than book - maybe not surprising since it's nearly 300 pages dedicated to one number. ...more

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Jan 11, 2011

Lucas Brendel rated it

Recommends it for: Math enthusiasts, anyone who is interested in the wonders of math in the world

This book explains how the mathematical value of phi(fee) exists in the world and how it occurs naturally in many areas of the world, and the many people in the world who were fixated with the number, and how its amazing qualities gave it the title of the Golden Ratio. The good part is you do not need to be a math master to be able to rad this. All equations and use of math are simple enough that anyone who has taken high school math can read this book and comprehend the topic. This book is g...more

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Aug 04, 2010

Michael rated it

Finally finished this. I had to let go of it about two thirds of the way through because of the tiny font size. Nevertheless, I enjoyed reading about PHI in part because of the all the references to it's relevance throughout history. There are also plenty of suggestions for further reading for those that are really taken by specific topics along the way. Although I really enjoy the subject of mathematics, I don't consider myself a mathematician yet I found the Livio's book easy enough to follow ...more

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Aug 12, 2010

Tom rated it

An interestingly written historical account and perspective of the most irrational or irrational numbers.â Livio debunks many of the attempts to find Phi in architecture, art, music, poetry; yet, he also provides sound rationale for finding Phi in many places. His descriptions of number theory would certainly pull me in; however, when he went into the philosophical, he pushed me away. Overall, I enjoyed the discussion of Phi, its history, how it has intrigued people for centuries, and how it r...more

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Dec 19, 2011

Migdalia rated it

Shelves: science-math, history-science

Mario Livio knows how to write in a way to make complex ideas comprehensible to the layman. He also makes it an entertaining read.

That being said, this is by no means an easy read. Look, you're dealing with Phi here - a mathematically complex number whose beauty is in its simplicity [I know that sounds contradictory.] IF you can understand the ideas behind it and that spring from it. That's MATHEMATICS with a capital word. Going in, you need to have some basic understanding of it, or y...more

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Jul 20, 2011

EJ Natale rated it

Terrible book. Poorly written. Vague. No direction.

This book is more numerology. The author creates loose and thin parallels to Phi, then refutes them. This happens repeatedly throughout the book.

The great pyramids might be built based on a ratio similar to phi. Oh, no, maybe not.
Oh, these painting might contain phi built into some of the geometry. Oh, wait, nope. They don't. The artist didn't even know what phi is.
The content makes no sense.

...more

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Sep 01, 2010

Christopher Litsinger rated it

· review of another edition

Shelves: read-in-2010, terrible, non-fiction

My review for this book will consist of the suggestion of a new title: "In which the author describes in great detail several ways in which the Golden Ratio was documented to be used in art and architecture and then proves those ways to be false with very little detail, and then rambles on for a bit about some other number theory and whether or not God is a mathematician, but generally leaves you somewhat less impressed with Phi than you were to begin with"
I'll admit it's not ver...more

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Jun 30, 2010

Upom rated it

Shelves: mathematics, golden-ratio, philosophy, nonfiction

The Golden Ratio turned out to be somewhat interesting, but ultimately tedious book about phi. The book does a decent job of covering what exactly phi is, its fantastic mathematical properties, as well as some of its appearances in nature. That being said, much of the books is a critique of where it DOESN'T appear. Although debunking claims of appearing in the golden pyramids and art is important in scholarship, it certainly takes away from the magic of the number, as well as makes for boring re...more

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Sep 21, 2011

Jeff rated it

Shelves: mathematics

Mario Livio makes mathematics very accessible by writing about compelling, complex ideas in very simple language. As an astrophysicist he brings a strong practical application of mathematics to his exposition.

In The Golden Ratio, he uses the history and mathematics related to the number phi as a jumping off place to explore a variety of ideas, but specifically how mathematics effectively and surprisingly models a number of disparate physical phenomena. He also effectively argues th...more

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Feb 24, 2009

Cameron rated it

Shelves: documentary-biography

I know it is very strange that I was reading a book about Math but this was really cool. This number and ratio are talked about in the book "The Da Vinci Code" and I thought it was interesting in that book. when I saw a friend of mine reading this book I thought I would give it a try. It is amazing to think about how this number comes up so often in so many different things such as math, geometry, architecture, art, Music and even poetry (though I did not understand that part). There a...more

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Aug 15, 2008

Jamie rated it

The full title of this book is The Golden Ratio: The Story of PHI, The World’s Most Astonishing Number. Now, I know you may not believe me when I say this, but despite this being a BOOK about a NUMBER, it's really not as exciting as you might expect.

Still, I enjoy stepping outside my own areas of expertise or experience and looking into new areas like mathematics. I think the problem is that I’ve been spoiled by stuff like Bill Bryson’s excellent A Brief History of Nearly Everything,...moreThe full title of this book is The Golden Ratio: The Story of PHI, The World’s Most Astonishing Number. Now, I know you may not believe me when I say this, but despite this being a BOOK about a NUMBER, it's really not as exciting as you might expect.

Still, I enjoy stepping outside my own areas of expertise or experience and looking into new areas like mathematics. I think the problem is that I’ve been spoiled by stuff like Bill Bryson’s excellent A Brief History of Nearly Everything, which manages to take material that should by all rights be boring and make it engaging by focusing on the quirky personalities and the bigger picture of how it relates to our everyday lives now. Unfortunately that wasn’t really the case with The Golden ratio.

Briefly, a golden ratio (a.k.a., "the divine ratio" or "the mean and extreme ratio") is achieved when a line is divided into two segments such that the ratio of the smaller segment to the larger is the same as the ratio of the larger segment to the entire line.

If you assume a unit length (like "1 foot" for one of the line segments) and do some math, you end up with the irrational number "Phi," which approximates to 1.61803… Phi is like the underappreciated cousin of another irrational number, Pi, which you may remember from basic geometry as 3.14159…yadayadayada. Irrational numbers like Phi and Pi have the quality that they can’t be expressed neatly and exactly as the ratio of two whole numbers. You can calculate them out to an infinite number of decimal places and they’ll never end or repeat.

So with that little math lesson out of the way, this book goes into how ancient philosophers, mathematicians, artists, musicians, architects, and even Mother Nature made use of Phi to make aesthetically pleasing or interesting stuff. Or that they didn’t. One of the author’s more annoying habits is to spend a whole chapter conjecturing about how, say, the ancient Egyptians used Phi to design the Great Pyramids of Giza, only to reveal in the last paragraph that no, they didn’t after all. Psych!

Still, Phi does really show up in some interesting places, like the shapes of spiral galaxies or nautilus shells or mathematical oddities like Fibonacci numbers. It’s an intoxicatingly cool idea to think that Phi is a kind of building block that ties the universe together in some mysterious way if we could only fully comprehend it. Like if you calculated it out to enough decimal places, you’d find a hidden message like this from the Almighty Maker himself.

But some interesting anecdotes and mathematical gymnastics aside, the author rarely succeeds in making things that interesting for any period of time. Fascinating as the underlying ideas are, this seems like a topic that could be covered sufficiently in a magazine article, journal monograph, or encyclopedia entry more appropriately than a whole book. (less)

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Nov 05, 2011

Jeffrey rated it

This book gives you the history of Phi from an historical / meaning perspective. Livio certainly speaks to the impractical mythical use of Phi and how it rationally can be dis-proved of merit. But at the same time, he speaks to the issue that Phi certainly has its practical applications. And one reason to pursue Phi and all it means is the mere fact that "Mathematics as we know it captures only a tiny part of the vast space of all possible simple sets of rules that might describe the workin...more

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Nov 14, 2008

melydia rated it

If you divide a line so that the ratio of the smaller to the larger is equal to the ratio of the larger to the whole, you have the golden ratio, phi. There has been an abundance of literature on the presence of phi in a number of unexpected locations, and this book addresses many of these appearances intelligently. It is organized more or less historically, starting with the Pythagoreans' obsession with phi (due to its presence in the pentagon and other neat little number tricks) and continuing ...moreIf you divide a line so that the ratio of the smaller to the larger is equal to the ratio of the larger to the whole, you have the golden ratio, phi. There has been an abundance of literature on the presence of phi in a number of unexpected locations, and this book addresses many of these appearances intelligently. It is organized more or less historically, starting with the Pythagoreans' obsession with phi (due to its presence in the pentagon and other neat little number tricks) and continuing through the present. The author avoids doctoring numbers to fit phi into famous works of art and architecture, and indeed debunks several such cases. While some of the direct appearances of phi are pretty nifty (such as leaf growth patterns on plant stems), much of the book covers subjects that are only related to phi by a few generations, usually through the pentagon or the Fibonacci numbers. I do not fault the author for this; tangents are to be expected in books about such a narrow subject as a single number.

The final chapter, "Is God a Mathematician," includes leading theories in response to that question (yes, no, and sort of) and Livio's personal opinion. I understand the desire to address such a topic, since mathematics is pretty amazing and phi is no small example of this, but this chapter seemed sort of forced, like the author was at a loss on how to wrap up the book. The explanation of the dual nature of light was sort of random, and the rather unsubtle promotion of Stephen Wolfram's then-unpublished book (which was not well received by the math community) was sort of irritating. I imagine that Livio's desire was to instill a lingering thirst for knowledge in his reader, to encourage further study, but it felt more like an advertisement for a newfangled religion that will change the way you look at the world. Despite the final few pages, I found this book to be informative and quite readable, which is always high praise for a book about math. Perhaps if Livio had left out his personal opinion I would have finished it feeling more satisfied. (less)

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Jan 03, 2008

jed rated it

Recommends it for: those who buy into the golden ratio hype

I normally really dig these, half math, half art, half real life books. But by the second half of this book Mario Livio seems more focused on debunking golden ratio theories rather than adding his own. No doubt, that's a thankless task. But nevertheless, it made for a rather downer of a book.

It started out well, I was retelling nearly every line of the book to anybody around me that would listen. I must have been quite a bore!

But then, suddenly, I wanted to put the bo...more

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Nov 10, 2011

Brian Sison rated it

Shelves: history, math, non-fiction

This is a full history of most math topics, with a focus on Phi, the Golden Ratio. The author starts with a trip through the development of numbers, from counting numbers, to integers, to rationals, to reals, all the way to complex numbers. He then goes into a study of the development of the golden ratio, from geometric contructions of pentagons and pentagrams. He also devotes plenty of team to the Fibonacci sequence and its close relation to Phi.

It's a solid math book, but does h...more

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Aug 26, 2009

White rated it

This book was given to me by my daughter and son-in-law. Excellent read! I was familiar with the golden ration already from a few of the science reads I have done but this book zeros in on it 100 %. You go from the last page mesmerized by the beauty of all existence with an insurmountable awe with the connection of everything. Not even 4 years of studying philosophy can expose you to the awe of this one book.

This book is a mathematical utopia.

A must read.

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Jun 14, 2009

Mrs_M rated it

This book was given to me by my daughter and son-in-law. Excellent read! I was familiar with the golden ration already from a few of the science reads I have done but this book zeros in on it 100 %. You go from the last page mesmerized by the beauty of all existence with an insurmountable awe with the connection of everything. Not even 4 years of studying philosophy can expose you to the awe of this one book.

This book is a mathematical utopia.

A must read.

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Jul 20, 2008

Kevin Hartzog rated it

As a biologist, I do have some interest in the natural phenomenon of the golden ratio as it appears in Botany and in gastropod shell formation. However, Livio's presentation was rather tedious and laborious. Livio spent much of the book disproving the use of the golden ratio in art and architecture. That, coupled Livio encompassing such a wide spectrum which also spanned music, biology, chemistry, physics and astronomy resulted in short expositions that never completely satisfied nor explained h...more

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May 15, 2008

Teodora rated it

Recommends it for: geeks, wonderers, adventurers, naturalists, stoners, people with patience

Recommended to Teodora by: ahhhh, math and I go some time back

Between 1 and 2, these pretty whole numbers, lies a number so fascinating that you might be overwhelmed with the beauty of quantifying beauty's perception.
Enter Phi= 1.6180339887....
This humber can explain the difference between the architecture of the Guggenheim as opposed to that of any classical courthouse (picture columns and squares).
The latter are commensurable numbers unlike Phi, which defines rose petal growth, mollusk shell growth, The proportions in Kate Moss's face,...more

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Apr 12, 2011

David Dranchak rated it

· review of another edition

Mario Livio tells the story of "1.6180339887," i.e., "phi." Phi or the "Golden Ratio" has fascinated both scientist and artists for centuries. Livio discusses the significance of the Golden Ratio and investigates "phi-fixated individuals."

Mario Livio is an internationally known astrophysicist, a bestselling author, and a popular lecturer.

Learn More about Mario Livio by visiting his website:

http://www.mariolivio.com

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Oct 29, 2009

James B. rated it

The number phi (approximately 1.618) is a ratio of successive numbers in the Fibonacci Sequence, first described by Euclid geometrically as the means-extreme ratio, that appears in many surprising places in nature and mathematics. This ratio shows part of the beauty of the mathematical design of the natural world. Why God chose to use this ratio is a mystery that needs to be further explored.

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Aug 20, 2008

Gregory rated it

You don't need a background in math to enjoy this book. Phi is a mathematical constant. Given a line of finite length, if you divide the line into two segments, such that the ratio of the whole line to the larger segment is the same as the ratio of the larger segment to the smaller segment, that ratio is Phi. It was dubbed the Golden Ratio owing to its many fascinating properties. The ratio is an irrational number, 1.61803...; that it cannot be expressed as a fraction was taken in ancient times ...more

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Jun 08, 2011

Doug rated it

Shelves: nonfiction

The author debunks a lot of the instances where Phi supposedly appears in art, architecture, music, and poetry but it still is amazing the Phi is so pervasive in nature. Such an interesting question whether Phi (and all numbers and mathematics in general) exists as a result of human thinking or as something that exists in nature that humans discovered, it blows my mind.

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