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

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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 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.

294 pages, Paperback

First published January 1, 2002

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Mario Livio

47 books223 followers

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Displaying 1 - 30 of 381 reviews
Profile Image for Sabio.
70 reviews12 followers
July 16, 2007
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 simple to the nature of cellular automatons and how they relate to this numeral sequence (sorry, that is another story -- not in this book).

When extirpating the superstitious mind, we must be careful not to throw the baby out with the wash. The author heeds this warning well. The book exposes all the false hopes in this number while still confessing some of its power too.

All to say, humans will make gods out of anything -- even after they think they have given up their gods.

Profile Image for Paul Weiss.
1,250 reviews232 followers
February 18, 2023
A historical exploration of what Livio characterizes as "Golden Numberism"

Personally I would have thought that there was plenty of fodder in our world to put into a thoroughly fascinating full-length book on the irrational number that mathematicians have universally come to call "the Golden Ratio". Mario Livio's THE GOLDEN RATIO is not one of those books.

It is almost exclusively a history book and, at that, it's not even a history of mathematics so much as a dry-as-dust, rather turgid outline of over-wrought historical efforts by numerologists to prove by any means that architects and artists around the world used the golden ratio in their work. A few pages of this kind of information is interesting. An entire book is not.

Paul Weiss
Profile Image for Bradley.
Author 5 books4,099 followers
April 4, 2017
Well, I was expecting something a bit more exciting because of my natural love for Phi, simply because, you know... SPIRALS are EVERYWHERE, Dude.

Still, the author does a palatable job of giving me a fairly decent history of mathematics from the focus of the Golden Ratio, the Golden Triangle, the logarithmic spiral, the Fibonacci sequence... all of which is, of course, the same thing, expressed slightly different with a ton of additional cultural significances.

No surprise here. This is Phi.

However, I did take umbrage against some of the side explanations early on for why ancient or apparently unsophisticated tribes didn't have numbers that counted past four. I mean, sheesh, if we went purely by the mystical importance that the Pythagoreans placed upon the first couple of numbers, we might also believe they couldn't count past five. It's a mistake of the first order, taking a little bit of data and coming to enormous conclusions based on our own prejudices.

That's my problem, I suppose, and he does at least bring up the option that the ancient peoples might have been working on a base four mathematical system, but for me, it was too little, too late. I cultivated a little patience, waiting until we get further along the mathematical histories past the Greeks and into the Hindus and the Arabics where it got a lot more interesting, and then firmly into known territory with the Rennaisance.

Most interesting, but also rather sparse, was the Elliot wave and the modern applications of Phi. I wish we had spent a lot more time on that, honestly.

But as for the rest, giving us a piecemeal exploration of Phi in history, art, and math, this does its job rather well.
Profile Image for Jasmine.
668 reviews47 followers
August 2, 2011
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 were parts of this that felt like bible code. I suppose credit where it's due a lot of this is actually debunking the bible codesque shit, but why is it even the focus of a serious book... perhaps because this is not a serious book.


I mean really the fibonacci sequence describes the birth patterns of rabbits... no it doesn't it describes a word problem about rabbits. if you follow these 5 steps you get an amazing pattern that looks like a Fibonacci sequence by MAGIC. no you get it because you followed 5 steps. I mean I just don't buy any of this as meaningful.
Profile Image for Ms.pegasus.
720 reviews139 followers
January 15, 2015
Livio's book examines the convoluted history and applications of phi. Like π (pi) it is both a constant and an irrational number. It's derivation is deceptively simple. Imagine a line divided into a longer segment (a) and a shorter segment (b); the dividing point is placed so that the longer segment (a) compared to the shorter segment (b) is proportional to the entire length (a+b) compared to the longer segment (a). In other words, a/b=(a+b)/a=phi.

In mathematics, there are many ways to express the same relationship. For example, the above equation can also be written as a/b=(a/a)+(b/a)=phi. Since a/a=1, and b/a is the inverse of a/b, or 1/phi, then phi=1+(1/phi).

Another way to look at the equation is to multiply both sides of the equation by phi. Then phi squared=(1 x phi) + 1. If the format of a binomial equation is adopted, then phi squared-phi-1=0. Written with coefficients of one, the equation reads: (1 x phi squared)-(1 x phi)-(1 x 1)=0. Solving the equation using the formula (-b plus or minus (√b squared - 4ac)/2a, where a=1, b=-1 and c=-1) results in the positive solution of phi=(1+√5)/2 or phi=1.6180339887.....

Livio addresses the key question about phi immediately. “What is it that makes this number, or geometric proportion, so exciting as to deserve all this attention?” (p.7) Versatility is one answer. Imagine the point dividing segments a and b is instead a hinge. Bend b up so that a line connecting the beginning of segment a and the end of segment b for an isoceles triangle. The triangle forms the first leg of a five-pointed star or pentagram. Connecting the points of the star result in a pentagon. Connect the vertices of the pentagon and the result is a smaller pentagram built around a smaller pentagon. Now fold b so that it is perpendicular to a. A rectangle can be formed. Using b as the dimension of a side, cut away a square, and a smaller proportional rectangle remains. The process can be repeated again and again. The recursive properties suggested by the formula for phi is replicated geometrically by the “golden triangle” and the “golden rectangle.”

Livio's book communicates his own sense of excitement to even the dullest student. He explains the rule for generating the Fibonacci Sequence (in case you haven't read the Dan Brown book!). The sequence is as follows: 1,1,2,3,5,8,13,21,34,55,89,144,233.... Now create from these numbers the sequence: 1/1,2/1,3/2,5/3,8/5,13/8,21/13,34/21,55/34,89/55,144/89,233/144.... The decimal equivalents for that series is: 1.0, 2.0, 2.5, 1.6666, 1.625000, 1.615385, 1.619048, 1.617647, 1.618182, 1.617978, 1.618050.... In other words, each number in the series approaches closer and closer to the value of phi.

Livio's book offers much more than a seemingly endless series of numerical games. He calls attention to Fibonacci's pedagogical genius. “In many cases, Fibonacci gave more than one version of the problem, and he demonstrated an astonishing versatility in the choice of several methods of solution. In addition, his algebra was often rhetorical, explaining in words the desire solution rather than solving explicit equations as we we would today.” (p.94) I have always maintained that there is more than one way to explain mathematics, and the method that appeals to me personally is visual. Two of the videos I found most helpful in understanding the math behind this book were: https://www.youtube.com/watch?v=1X-UA... and https://www.khanacademy.org/math/geom....

The copious historical detail that Livio introduces can at first seem like a bewildering distraction. However, for me it highlighted the vast gulf between arithmetic as a fingers and toes counting exercise, and the study of abstract relationships which so enamored the ancient Greeks. I gained a new appreciation for their achievement from reading this book.

Through his scrupulous measurements and application of historical knowledge, Livio dispels many of the myths that surround phi, the so-called “golden ratio.” His examination turns to art, music, and poetry, and he notes how much of this mythology grew out of beliefs about religion and metaphysics that were intertwined with progress in mathematics.

Livio's book is meant to appeal to a wide audience. I was attracted to this book after reading THE EIGHT by Katherine Neville, as well as the DA VINCI CODE by Dan Brown. I gained much more than I expected. Despite the complexity of some of the concepts, anyone who has made it through high school math will comprehend his explanations. This is not because you will need that high school knowledge. Rather, when he launches into a theorem or proposition, you will remember that you once knew the references he is citing! Nevertheless, I have to admit that I found the chapter on periodic tilings and the significance of Penrose tiles difficult to understand. A clearer explanation, to me, was the link https://www.youtube.com/watch?v=QTrM-....

Returning to the question of why phi is important, Livio reveals that the recursive pattern appears in nature: The arrangement of leaves around a branch (phytotaxis), and the arrangement of the scales on a pinecone, for example.

The effort it took to understand this book was rewarded by the satisfaction of communicating with so many historical minds on this one abstract concept.
Profile Image for Woodge.
460 reviews30 followers
March 15, 2011
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: Yeah, you TV Guide-reading eejit! Get yer head out of your ass! This stuff is interesting!
YOU: You are frikkin high.
WOODGE: Okay, yeah, fine, go back to your latest episode of THE APPRENTICE but I'm talking here so Shut It.

Anyway, Euclid put it thusly: 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. The resulting ratio is phi, an irrational number equaling approximately 1.6180339887... it goes on forever without repeating a pattern. Phi is said to be the most irrational of irrational numbers. (The most famous irrational number is pi, the ratio of a circle's circumference to it's diameter.) Irrational numbers are called that because they can't be expressed as a ratio of any two rational numbers.

YOU: Dude, I'm falling asleep here.
WOODGE: Oh, don't be such a baby, the book is much more interesting than the nitty gritty of numbers 'n' stuff.
YOU: Huh? Did you say something? Donald Trump was saying something profound.
WOODGE: Mm-hm.

Anyway, this book was a breeze to read, even I was surprised. It delves into history, art, astronomy, philosophy, poetry, and is full of good quotes and fun historical facts. It also debunks a number of myths associated with the Golden Ratio. Much of this erroneous stuff can be found in other books treated as facts but Mario Livio, a theoretical astrophysicist by trade, gets behind the mumbo jumbo and gives you the straight dope. Maybe you've heard that the Golden Ratio is all over the Parthenon or was a big factor in building the pyramids or was the basis for many of Piet Mondrian's paintings? But that's just wrongedy-wrong-wrong! But some of the diverse places that the Golden Ratio actually does appear includes: the petal arrangements of roses, pentagrams, Platonic solids, the shape of distant galaxies, nautilus shells, and accounting fraud.

Much more than just blathering on about a freakin' number, this book gets into history and touches on Euclid, Pythagoras, Alexander the Great, Galileo, Johannes Kepler (and the fact that his mom was arrested for being a witch -- Burn her! Burn her! She's a witch!) and art history, and whether or not God was a mathematician.

And of course there's Fibonacci and his series of numbers which have a very close relationship to phi. The Fibonacci series begins 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89... and each successive number is the sum of the preceding two numbers. If you take one of the numbers and divide it by it's previous number you get closer and closer to phi as you go further in the series. Fibonacci numbers are found everywhere; from the number of petals in a flower, to the number of spirals on a pineapple; to phyllotaxis (Greek for "leaf arrangement"); to the family tree structure of bees, et cetera.

I thought it all was pretty cool to tell you the truth.

WOODGE: So doesn't that sound pretty cool?
WOODGE: Hello? Anybody?
Profile Image for Kara Babcock.
1,952 reviews1,295 followers
December 28, 2020
This is one of the oldest (perhaps the oldest?) physical books I own and have yet to read. Goodreads suggests I’ve had it for nearly a decade. Oops. The truth is, I was never excited to read this. I love reading math books! But I am not particularly enamoured of books that explore one or two “special numbers,” and phi is perhaps my least favourite special number. The blurb from Dan Brown on the cover didn’t help. See, phi has been egregiously sexed up and romanticized by people, turned into a mystical number that recurs exactly throughout art and nature, and ascribed aesthetic properties it doesn’t deserve. I was nervous this book would repeat these claims. Well, I owe Mario Livio an apology. Not only does he critically challenge those claims and debunk a lot of the hogwash surrounding the golden ratio, but he also takes it upon himself to tell a broader and more complete story than focusing solely on this number. The Golden Ratio: The Story of Phi, the World's Most Astonishing Number is a good story of the intersection of mathematics and society, and it provided one key insight that, as a math and English teacher, I find very valuable.

You would be forgiven for, having begun the book, thinking that Livio has entirely forgotten about phi for the first couple of chapters. Rather, he explores the history of numbers and counting in general, eventually ended up in ancient Babylon and Greece and making some connections with geometry. This creates a much richer backdrop for Livio’s later exposition of the golden ratio, and it also broadens the reader’s awareness for how various cultures developed and practised mathematics at different points in history. For example, Livio discusses the Rhind/Ahmes papyrus, which famously provides insight into Egyptian mathematics about 3500 years ago. He emphasizes the papyrus’ purpose as a teaching/reference tool—it specifically explains how to do practical calculations. Fast-forward a couple of millennia, and Fibonacci was doing the same thing—writing tutorials, essentially, for accountants.

See, I appreciate this, because most approaches to discussing the golden ratio focus on the idea that its use in architecture, art, etc., invokes certain ingrained aesthetic ideals in us. These approaches further seek to ground the golden ratio in the idea that its proponents and adherents throughout history have sought it out as a result of being fascinated with mathematical beauty. Livio, on the other hand, reminds us that a great deal of mathematics was (and remains) practical. It’s true that the Pythagoreans were a semi-mystical cult that believed their discoveries reflected the beauty of nature—but the problems they solved were motivated by questions of geometry and arithmetic that were relevant to life in Greece at the time. This has remained true throughout history: our development of mathematical approaches is driven by our needs as a society. The adoption of Hindu-Arabic numerals, for example, didn’t happen because they are “more beautiful” than Roman numerals—the accounts liked them better for arithmetic!

It might seem strange for a mathematician, especially one who loves pure math, to be arguing against the idea that beauty should be a foundational concept of mathematics. And I’m not, not really. But I agree with Livio that viewing mathematics in the past through a lens of beauty/aesthetics is ultimately an ahistorical reading that confuses more than it illuminates. Understanding the emphasis on practical applications for math helps us understand its place in our society.

And this is where The Golden Ratio really got me. Several chapters examine whether well-known artists used the golden ratio in their work. Livio discusses the works themselves, as well as numerous scholarly intrepretations both for and against the idea that the golden ratio played a part. I appreciate his extensive use of references and the way he engages with the topic as objectively as possible. Most importantly, Livio suggests that our desire to spot the golden ratio in this artwork undermines and devalues the artists’ general mathematical brilliance. If the aesthetic quality of a work of art were simply the matter of using the right shape of rectangle everywhere, what does that say about art and artists? Why wouldn’t we have made a computer program that can generate “the perfect work of art” by now? No, Livio concludes, the brilliance of these works of art is independent of their use, or lack of use, of the golden ratio. It comes from a far deeper grounding in mathematics than we care to credit—from the use of perspective to plane geometry, math is everywhere in art. He points out how some artists, like Durer, studied mathematics purposefully to improve and influence their artistic output.

I teach math. I also teach English. People treat me like a unicorn because of this, but I really don’t see them as all that different. Neither did Charles Dodgson, who wrote Alice in Wonderland. Livio cites numerous other poems and literary works that use math, as subject matter or metrical inspiration or both. He reminds us that this siloing of STEM is a recent and very artificial phenomenon, that throughout the majority of history, STEAM indeed was the rule of the day. The idea that if you have an artistic sensibility you must somehow be allergic to mathematics is ahistorical and untrue, for as Livio points out here, many of the most celebrated and famous artists studied, understood, and used math in their work.

In this way, The Golden Ratio provides a far more valuable story than simply “the world’s most astonishing number” (which phi is not). Livio’s tangents into philosophy, history, art, and music remind the layreader that mathematics is not this alien construct that only super-intelligent people can appreciate or do. It is fundamental to our lives, to our praxis, and to our pleasure—not for any innate beauty it possesses, but for the way its practice can help us create what we consider beautiful. The golden ratio does not play as big a role in this process as some want you to believe. Rather, as is usually the case, the truth is far more wonderful and broader in scope than the simple idea that one number can rule them all.

Originally posted on Kara.Reviews, where you can easily browse all my reviews and subscribe to my newsletter.

Creative Commons BY-NC License
Profile Image for RKanimalkingdom.
496 reviews65 followers
October 1, 2018
Am I nerd for voluntarily reading this? Yep.
Do I care? Nope.

I'm not the brainest person on the planet. Actually, I'm not very geeky at all. But I love to learn. I like to spend time analyzing and picking things apart ; dissecting the material and discussing it with someone. However, not everything enchanted me and this book straight up annoyed me.
Essentially, I went into this book expecting to be, I don't know, told about Phi. The discovery, the relevance, the applications and help it provided to the world? Wouldn't you, based on this title? But what I got instead was pages and pages of conspiracy theories of phi and the "debunking of it".

Review Continued Here

Profile Image for EJ Natale.
25 reviews1 follower
July 20, 2011
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.

The author goes into lengthy sidebars about art and astronomy that do not play into the material. The author prattels on about Kepler and some of his work and how it related to phi, and then that all of his work was bunk. The author writes at length about how some artist might have based their paintings on the golden ratio, but then writes that the dimensions are off. This book is a waste. It is hoaky.

I find it interesting it is pretty much built upon the mythos of the golden ration that Dan Brown built with the "Da Vinci Code," but the author goes out of his way to state that he will not cover the mona lisa.

Save your time and money. Do not read this steam pile of self indulgent crap.
Profile Image for Jill.
425 reviews223 followers
August 9, 2014
The Fibonacci sequence (and its consequent relationship to the Golden Ratio) is one of my favourite things. No, really. So I went into this book already interested and somewhat informed. Not sure if that would make a difference, though, because Livio's treatment of this topic is really solid. For one thing, it's clearly written -- which always helps for the artsy reader -- and while formulae and proofs litter the pages, concrete examples and pictures show up frequently. The structure of the book is excellent -- each section moves smoothly and, often, chronologically into the next, while still bring divided by subject. Pretty inspired (though the final chapter seems a bit abrupt, especially at the end).

Livio also sees himself as a bit of a detective, using geometry and timeline analysis to check whether work commonly attributed to Golden Ratio influence is actually of that ilk -- and generally, it isn't. It's kinda fun to see a dude who loves this number so much still be critical of its cultish status -- DON'T FUCK AROUND WITH PHI, BITCHES; ain't no real mathematician got time for that.

Really informative, surprisingly narrative, and a nice, but critical, love letter.
Profile Image for Mikael Lind.
175 reviews55 followers
August 8, 2013
A fascinating historical expose about how a single number, phi, has (or is believed to have) influenced human creation within such different fields as music, art, architecture and, of course, mathematics of different kind.

The book's strength is that you don't have to be a mathematical minded person to be able to understand it. I could follow the mathematical formulas roughly by the mathematical knowledge I gained more than fifteen years ago, but even though I was persistent enough to try to follow Livio's formulas, this is not crucial to the story of the book.

I read this book from start to finish in one week, and read nothing but this book - something unusual for me since I usually read 3-4 books at the same time. It's a well-written and sometimes mind-boggling popular science book that makes you interested in the world of mathematics. I highly recommend it!
Profile Image for Gendou.
597 reviews271 followers
July 28, 2013
Phi has some surprising mathematical properties, which are eventually discussed here and there throughout the book.

Mostly, this book is a history of mathematics. From the etymology of numbers, to the Pythagorean brotherhood, and the discovery of incommensurability, and finally, to modern day mathematics.

The book dispels myths of Phi's use in famous works of art, construction of the pyramids, etc.

I find Livio to be a trustworthy author, who prefers demystification over hyperbole, which I respect. But sometimes the drawn out history lesson left my interest to wander. Happily, he would always tie it back to them mathematics, which kept me interested.
Profile Image for Tim.
93 reviews
April 1, 2009
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 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.
Profile Image for Ob-jonny.
224 reviews3 followers
May 7, 2014
This is a great book about number theory in general and is much more than just the discussion of phi, the golden ratio. It is truly amazing to see how often this number and ratio are found in nature. The widths of the spirals of pinecones and various flowers display the ratio as due patterns in the breeding of rabbits. But some of the numerical properties of the number are equally fascinating. For instance 1/phi is equal to 1 + phi. There is also a section on prime numbers which is just as interesting as the discussion about phi and it provides some of the variety to the book I mentioned. For instance, there are some very large prime numbers which are only composed of the number 1. There is an interesting discussion of the longest prime number found. Another thought-provoking discussion is the tendency for numbers starting with the lower digits appearing in nature and statistics. For instance, collections of numbers tend to have 1 as a first digit much more often than 9 and the transition is continuous between 1 and 9. It has something to do with how the range of numbers starting with 1 is much wider than for larger numbers when looking at them on a logarithmic scale. This book will really get you thinking and is especially suited for people interested in number theory.
Profile Image for Christopher Litsinger.
735 reviews7 followers
September 2, 2010
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 very catchy, but it will give you a far more accurate idea of what you're in for.
Profile Image for Cassandra Kay Silva.
704 reviews277 followers
June 28, 2011
I was so excited to get this book. I have a minor obsession with the golden section/ratio. I have always somewhere deep in my heart hoped that string theory would turn out to have strings vibrating at ratios or frequencies somehow related to the golden section. Unfortunantly I already knew everything in this book. Nuts~! I was hoping to get some new information. I don't think that is a fair reason to say that the book was not great. It was still really fun.
Profile Image for Raed.
271 reviews68 followers
March 13, 2021
we imagine ourselves walking in a long hallway - a hallway so long that we cannot see the end of it, neither where we came from, and behind us, we hear the sound of a ball which bounces. When we turn around, there is no doubt, there is a bouncing ball. it gets closer and closer. At one point, inevitably, it overtakes us and continues its course, still leaping. And it moves away, until it gradually disappears from our sight. The question is not: is the ball bounces ? Because it is a fact; it bounces. But rather: Why does it bounce? How did she start to bounce back? Did anyone kick it? Or is it a special ball that likes to wiggle on its own?
Does this space have a particular physics that forces the ball to bounce indefinitely?


the story begin by 1 1 2 3 5 8 13 21 34 55 ....(the fibonacci sequence) that the nature have always tendency to use it ,we know that complexity is never an evidence for god existence ,we can use simple instructions in a simple programing language and the result after 1000 iteration will be so beautiful and so complicated ,the question here is can we create a program with a great complexity and wait for a very complicated results, the answer is simple , "we can't" (we don't reach this level yet),the secret of life is in the instant t=0, in the Big Bang ,it is clearly that all constant of the universe are pre fixed, and it so clear that the beginning is so epic because complexity is too high..
scientist never ask about t=0 he ask about t=1 , why nature love mathematics, is it that an evidence for the existence of a mathematician ,

from very complicated beginnen to higher chaos to spiral galaxies to earth to humans with their paper looking for mathematical concept : the ball still bounces , the ball bounces with a very beautiful math patterns,and every day we understand more and more , and everyday we feel the beauty of bounces, the beauty of life , the beauty of the creation,

Profile Image for Ajeje Brazov.
724 reviews
October 17, 2017
"La sezione aurea o rapporto aureo o numero aureo o costante di Fidia o proporzione divina, nell'ambito delle arti figurative e della matematica, indica il rapporto fra due lunghezze disuguali, delle quali la maggiore è medio proporzionale tra la minore e la somma delle due." (da Wikipedia)

Il numero risultante da quel rapporto è un numero irrazionale pari a 1,618...
Un numero, un rapporto che ritroviamo o che molti studiosi hanno trovato in molte opere dell'uomo e opere della natura.
Il libro ripercorre la storia di questi studi che partono, presumibilmente, dall'epoca dei Babilonesi fino ai giorni nostri, e ancora oggi il tutto è velato dal mistero, come dice Einstein: "Quella del mistero è la più straordinaria esperienza che ci sia dato vivere. E' l'emozione fondamentale situata al centro della vera arte e della vera scienza."
Le lettura è stata molto interessante, con spunti riflessivi e didattici molteplici. Si è andato da capitoli pieni di nomi di studiosi, quali: fisici, matematici, astronomi, poeti, pittori, con un'infinità di date, che mi hanno distolto la mente dal fulcro del libro, a capitoli, invece, pieni di spiegazioni, esempi, studi che mi hanno sbalordito ed entusiasmato ad approfondire. Tra questi studi, che mi hanno interessato molto, ci sono sicuramente "i frattali":
(un video sui frattali, altro che LSD :-PP)
Poi c'è "un giochino", ma viene da uno studio accurato fatto da Le Corbusier, redatto nella sua opera: "Modulor", dove rapportando l'altezza di un uomo o donna con l'altezza dal proprio ombelico a terra, darebbe un numero molto vicino al rapporto aureo, cioè circa 1,618... (provateci, a me è venuto 1,67 :-D)

"Vedere il mondo in un granello di sabbia
E un paradiso in un fiore selvaggio,
Tenere nel palmo della mano l’infinito
E l’eternità in un’ora." (William Blake)
Profile Image for Rosemary Paul.
8 reviews
January 23, 2013
It is a great book. I finished it in 2 days. Very stimulating in that I love books that try to give a relationship between numbers and the nature of reality.
PHI 1.6180, not to be confused with PI 1.14159, is considered the Golden Ratio. Discovered by Euclid over two thousand years ago.
The book 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.
271 reviews
August 27, 2009
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.
Profile Image for Lady-R.
227 reviews114 followers
October 4, 2021
Una muy documentada dismitificación del uso consciente de la proporción áurea en el mundo del arte y una curiosa relación de cómo las matemáticas están presentes en la naturaleza (o al revés).

Aunque, en su mayor parte, el libro se centra en la relación de la proporción áurea y la serie Fibonacci, también es interesante la introducción que hace a la historia de los números.
Profile Image for Shaw.
32 reviews11 followers
March 23, 2016
This was a wonderful book, full of fun thought provoking ideas on the history of knowledge and learning. Two thumbs up, five stars, highly recommended.
Profile Image for Mirko.
136 reviews6 followers
January 27, 2019
Very good book about mathemathics and the golden ratio (1.618), a number which suprising presence and application is found in art, physics, statistics and many other areas.
Profile Image for David.
134 reviews20 followers
May 21, 2013
Reading this book was a stimulating experience. Exploring the mathematical principles and theories studied by various past cultures and how they employed them isn't a new idea but this author is uniquely good at it. Other authors in this genre, such as Amir Aczel can sometimes be guilty of spending too much time on sculpting the biography of a math genre and leaving its concepts severely under-explained. Livio however, created what I felt to be an adequate mix between math teaching an math biography. The result for me was to become greatly interested in a genre of study I formerly felt to be too clouded in obscurity for anyone but the balding, quirky, pocket-protector and horn-rimmed glasses wearing geeks of the world to make sense of.

I appreciated the sensibility Livio showed in this work, specifically in his loyalty to the scientific method. This genre tends to produce the occasional crank who with a theory that sounds beautiful but fails to stand up to unbiased and rigorous testing. It's admirable when a someone can instead treat their theory with reservations and honestly subject it to a barrage of tests, attempting to disprove it, before presenting it to the world. And while Livio does bring up scores of wild theories proposed throughout history on the relation of the golden ratio, pi, and the Fibonacci sequence to any piece of soul-stirring art, music, or architecture, he faithfully presents the holes and counter-points, reserving his enthusiasm for the truly provable and profound. This book is not meant to disprove the connection between irrational numbers and our sense of beauty, harmony and proportion, but rather to help sift through the weak theories and teach the principles surrounding those that have proven themselves and stood the test of time.

The book surveys various people and cultures who employed mathematics in their building, artistic, and musical endeavors in a chronological order, trying to determine how deep their knowledge was and how deliberate their use of higher mathematics. It sorts through the claims made by different authors and researches that individuals from the pyramid builders to the early philosophers to the renaissance painters to the modern artists/musicians perhaps knew of the irrational numbers and incorporated them into their works. Necessary to understanding each group or famous person to which the question of expert mathematical foreknowledge is posed, is a brief study of their biographical information and contributions to history. So this book functions as a nice primer to studying some of these people, especially Euclid, Plato, and Pacioli. As well as studying various people it also spends a couple chapters on the presence of irrational numbers (or our ability to recognize them) in nature. Needless to say, it goes into great detail on each topic. Surprisingly, the level of detail never becomes so intricate that the book loses rhythm & readability, and the author doesn't hide behind the voices of the many people and works he cites throughout this work.

I really enjoyed this book and I suppose the purpose of many works of a scientific or scholarly nature - such as this one - are to transfer an understanding or at least stimulate an interest in an idea, and to that end I am quite satisfied with it.
Profile Image for Kyle Wright.
84 reviews7 followers
December 9, 2008
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 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.
Profile Image for Eleclyah.
340 reviews40 followers
January 20, 2013

"La sezione aurea" ovvero "Opere famose che non hanno nulla a che spartire con ɸ"
Non è mia abitudine lasciare da parte un libro, ma quando è troppo, è troppo.
La storia di ɸ, un numero irrazionale che vale 1,618..., e del suo sorprendente ricorrere negli ambiti più svariati, è un argomento che ben si sarebbe prestato alla stesura di un saggio valido.
Quello di Mario Livio mi ricorda vagamente un trattato esoterico un po' fuori luogo, in cui gli appassionati di ɸ (categoria di cui lui, per fortuna, non fa parte, altrimenti avrei preso il libro e l'avrei buttato fuori dalla finestra) vengono descritti come una specie di setta di fanatici che fa carte false pur di veder comparire più o meno ovunque il rapporto aureo.
La maggior parte del testo serve a disquisire sul perché e sul percome ɸ non ha niente a che vedere con le piramidi di Giza, con il Partenone, nonché con un buon numero di opere famose appartenenti alle arti figurative; da questo numero è esclusa la Monna Lisa di Leonardo da Vinci, su cui l'autore non si sofferma, in quanto sul famigerato quadro è stato detto e scritto talmente tanto da non consentirgli formulare un'ipotesi che non sia ambigua (!!!) riguardo alla presenza o meno del rapporto aureo nell'opera.
Dopo aver passato più di metà libro discorrendo del più e del meno, anche di argomenti che con ɸ hanno davvero poco a che spartire, finalmente Livio si decide a parlare delle opere in cui il rapporto aureo è stato utilizzato consapevolmente.
Cita un piccolo numero di artisti per lo più ignoti o semi ignoti, senza portare come esempio alcuna opera, e quando uno sta per perdere le speranze eccolo che tira fuori dal cilindro (era ora!) un nome altisonante: Le Corbusier!

Ma l'aver arrancato per 250 pagine prima di ritrovarmi a questo punto è stato troppo.
Il tema è trattato piuttosto superficialmente, e come se non bastasse ci sono diversi errori di traduzione: "nominatore" al posto di numeratore, ad esempio, o frasi totalmente sconnesse.
Magari ero io che da questo libro mi aspettavo troppo, non so. Comunque, secondo me non vale proprio la pena leggerlo. Nemmeno se siete appassionati dell'argomento!
Profile Image for Mark.
270 reviews8 followers
May 28, 2012
Mario Livio's The Golden Ratio nicely balanced the last book I read about the world's most astonishing number. Actually, it far surpassed it! While Livio debunks the opinion of others that phi is conspicuous in the ancient pyramids and other monumental ancient works of art, his lively discussion of other places where we are surprised to find phi is enlightening and entertaining. Whether you are curious to know more about the golden number, the golden ratio, the golden triangle, rectangle, rhombohedra, golden sequences, or golden trees I heartily recommend this book!

If it's been a few years since you finished school and like me you "returned" most of the higher math you learned to the teacher, you may have to concentrate to follow some of the math proofs.

On a personal level, I am tempted to use Benford's Law in our local community center to engage a few of the local leaders. I hope I can find time in the next few weeks! Also, this book also leaves me wanting to read Livio's latest book "Is God a Mathematician?" and a few recent works by others on chaos theory and complexity systems.
Profile Image for Teodora.
14 reviews2 followers
May 15, 2008
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, and many other beautiful structures found in nature is incommensurable.

"They say that the first human to disclose the nature of commensurability and incommensurability to those unworthy to share in the theory was so hated that not only was he banned from [the Pythogoreans'] common association and way of life, but even his tomb was built, as if [their] former colleague was departed from life among humankind.'

Profile Image for Gabrielam13.
140 reviews23 followers
December 17, 2016
Mi-a plăcut foarte mult această carte pentru că am învățat foarte multe lucruri pe care nu le știam, cum ar fi legătura dintre șirul lui Fibonacci și Phi, cât și multe alte proprietăți fascinante ale lui Phi. Însă dincolo de noutățile matematice însoțite de explicații foarte simple și ușor de înțeles, mi-a plăcut că Mario Livio a făcut o prezentare exhaustivă a conceptului, prezentându-l nu numai din punct de vedere istoric și analizând evoluția lui în matematică, ci și influența pe care a avut-o în artă, în domeniul picturii, a muzicii, iar ulterior chiar și în fizică și astronomie. Cartea se încheie cu un mic capitol filosofic care abordează tema naturii matematicii: invenție sau descoperire, lăsând astfel deschis apetitul cititorilor pentru a continua investigația și dincolo de această carte.

Dacă matematica ar fi predată într-un mod la fel de interesant și în școală, sunt sigură că ar exista mai mult interes și apreciere pentru ea !
Profile Image for Jimmy Ele.
234 reviews90 followers
August 17, 2016
I thoroughly enjoyed this book. The appendixes in the back in particular were especially helpful when it came to the Mathematical proofs. The origin of Phi all the way through Fibonacci and beyond was well documented and eye opening. I would recommend this book for anyone who has an interest in Mathematics and/or the history of ideas.
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