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# The Golden Ratio

#### timothy posted more than 10 years ago | from the cradle-of-life dept.

676

raceBannon writes "The book surprised and fascinated me. I thought it was going to be solely about the Golden Ratio. Mario Livio does cover the topic but along the way he throws in some mathematical history and even touches on the idea that math may not be a universal concept spread across the galaxy." Read on for the rest of raceBannon's review.

I have to admit that it is a little spooky to me that this ratio, this irrational number (1.6180339887...), pops up in many varied natural phenomena from how sunflowers grow to the formation of spiral galaxies; not to mention that the Golden Ratio and the Fibonacci Series are related. It makes you want to think that there is a God with a plan.

The Golden Ratio is defined as follows: In a line segment ABC, if the ratio of the length AB to BC is the same as the ratio of AC to AB, then the line has been cut in extreme and mean ratio, or in a Golden Ratio called Phi.

On the flip side, Livio squarely debunks the idea that the Golden Ratio is present in many famous paintings and architecture that has been postulated in previous books. He rightly points out that you can find the Golden Ratio in anything depending on where you decide to place the measuring tape. The idea that the Golden Ratio is such a symbol of universal beauty that it appears by accident in our great man-made buildings and artwork does not carry any weight. I think Livio makes his point.

He also uses the Golden Ratio as a framework to illuminate other historical tidbits about key mathematical figures, guys like Pythagoras and Euclid, who continue to affect the mathematical world to this day. I love this kind of stuff; the historical context of how and why these legends did what they did is very interesting to me. For example, I did not know that Euclid himself did not discover geometry or even make any great new contributions to the field in terms of ways to apply it. What he is famous for is organizing the information into a coherent fashion. He was a teacher of the highest order; so much so that Abraham Lincoln himself used Euclid's texts, unchanged after all those years, to learn the subject back in Lincoln's log cabin days.

The book is not all a philosophical discussion. Livio does use some simple math examples to make his points but it was at a level that I could follow. According to my college professor, I escaped College Calculus by sheer luck. Livio does provide the rigorous math examples in appendices at the end of the book (I did not bother with these).

Finally, Livio takes a shot at the idea that mathematics is a universal concept across the entire universe. To be honest, I have always assumed that it was. After all, I am a Trekkie and this concept goes unstated throughout all four TV series. The idea that mathematics is a human construction and probably holds no water in another civilization that grew up on the other side of the universe makes a lot of sense to me. I have to admit; I need to ponder that one for a while.

I recommend this book. If you like the history of science, your high school algebra class is just a little more than a dark fog in your memory, and you get a charge out of scientific mysteries, you will not be disappointed.

You can purchase The Golden Ratio from bn.com. Slashdot welcomes readers' book reviews -- to see your own review here, read the book review guidelines, then visit the submission page.

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### Book Review: The Golden Shower (-1, Troll)

#### stratjakt (596332) | more than 10 years ago | (#8202962)

This probably isnt first post, but pretty damn close.

### It works, sorta.. (0, Funny)

#### grub (11606) | more than 10 years ago | (#8202970)

Out of every 1000 slashdotters, 1.6180339887... will have had sex with a real woman.

### Re:It works, sorta.. (-1, Troll)

#### Anonymous Coward | more than 10 years ago | (#8203091)

I should have known that the other 998.3819663... out of a thousand would mod me down.. :)

### golden ratio? (-1, Troll)

#### Anonymous Coward | more than 10 years ago | (#8202978)

is this about how much pee you can drink before you vomit?

### I wrote a review.. (-1, Interesting)

#### Tirel (692085) | more than 10 years ago | (#8202980)

for my english class. hope you enjoy it.

"Throughout history, thinkers from mathematicians to theologians have pondered the mysterious relationship between numbers and the nature of reality. In this fascinating book, Mario Livio tells the tale of a number at the heart of that mystery: phi, or 1.6180339887.

This curious mathematical relationship, widely known as the "Golden Ratio," was defined 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 the crystals of some materials, to the shapes of galaxies containing billions of stars. 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 suggested that it is connected to the behavior of the stock market!" The Golden Ratio is a 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 one of the greatest treasures of geometry; such medieval thinkers as mathematician Leonardo Fibonacci of Pisa; and such masters of the modern world as Debussy, Le Corbusier, 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.

### BUT DOES IT RUN LINUX (-1, Offtopic)

#### SIG TR0LL (749566) | more than 10 years ago | (#8203033)

jfioajfoijfoijffia

### From your english class? (4, Insightful)

#### iota (527) | more than 10 years ago | (#8203138)

for my english class. hope you enjoy it.

I presume you got an F. Since is a direct and obvious plagarism of the publisher's description of the book. (see: http://search.barnesandnoble.com/booksearch/isbnIn quiry.asp?isbn=0767908155)
It's obvious, because it doesn't really say anything other than what can be related to the title of the book (which is not unusual for back-of-the-book descriptions)
It's direct, because, well -- I can search google for any sentence in your text and find it.

Lame.

### Actually, from the link listed... (-1, Troll)

#### W32.Klez.A (656478) | more than 10 years ago | (#8202985)

You can purchase The Golden Ratio from bn.com.

A new copy is not available from Barnes & Noble.com at this time.

maybe the editors should check closer next time.

### Re:Actually, from the link listed... (0)

#### RandBlade (749321) | more than 10 years ago | (#8203045)

If you're trying to find a copy of the book, its available at Amazon. Hope that helps.

### Re:Actually, from the link listed... (1, Troll)

#### stratjakt (596332) | more than 10 years ago | (#8203076)

Why the ads anyways? If I wanted the book, I'd be resourceful enough to find it.

Do they get referrer bucks or some other such lame innernet moneymaking scheme?

### Q: What's the best part about 12 noon at slashdot? (-1, Offtopic)

#### Anonymous Coward | more than 10 years ago | (#8202992)

A: It's 1700 GMT. That means all the Brits have finally given up pretending to work and gone home for the day. According to statistics, post quality soars over 30% on average.

### I loved this book, until... (-1, Offtopic)

#### Anonymous Coward | more than 10 years ago | (#8202994)

...the full-color foldout of the Goatse guy.

What the hell does a tortured bunghole have to do with radio?

### Re:I loved this book, until... (0)

#### Anonymous Coward | more than 10 years ago | (#8203043)

What do you mean foldout? It struck me as more like a stretchout, or maybe a reamout? Inquiring minds want to know.

### The Da Vinci Code (2, Informative)

#### fee^ (94129) | more than 10 years ago | (#8202999)

On the fictional side of this type of thing, those of you into this kinda stuff (like me) should read Dan Brown's 'The Da Vinci Code'. I've read that and 'Angels and Demons'. Both fantastic reads. More Info Here [danbrown.com]

### Re:The Da Vinci Code (1, Informative)

#### Kenja (541830) | more than 10 years ago | (#8203061)

The movie "Pi" (as in the symbol) also touches on the subject via the Kabala and Jewish numerology.

### Re:The Da Vinci Code (1)

#### DR SoB (749180) | more than 10 years ago | (#8203294)

They are talking about Phi not Pi, Phi is an H of a lot cooler then Pi.. - Robert Langdon.

### Re:The Da Vinci Code (1)

#### Thud457 (234763) | more than 10 years ago | (#8203329)

I thought the movie "Pi" was a warning to look for "Intel inside".

### Re:The Da Vinci Code (2, Insightful)

#### Anonymous Coward | more than 10 years ago | (#8203081)

Fictional is absolutely correct, sir. Entertaining, perhaps even compelling, but in the end purely a creation of someone else's imagination. Those who accuse "The Passion of the Christ" of being anti-Semitic should also be willing to point their guns at Dan Brown for being anti-Catholic.

### Re:The Da Vinci Code (3, Insightful)

#### wfolta (603698) | more than 10 years ago | (#8203186)

The Da Vinci Code is gripping fiction, but it's not in the same class as The Golden Ratio.

The Golden Ratio is carefully and deeply researched. The Da Vinci Codes is allegedly based on research, but the "research" behind it is recycling tired old conspiracy theories.

From his statements online and in his forward, methinks Dan Brown is trying to have it both ways: claim it's based on fact but use the plausible deniability of it being a fictional work. It is a gripping read, don't misunderstand me. But you have to remind yourself that it's totally fictional.

### Re:The Da Vinci Code (1)

#### DR SoB (749180) | more than 10 years ago | (#8203321)

You should read the book before passing such judgement. I think the point the original poster was trying to make was the fact that Dan Brown discussing the number Phi in great detail in his book. Phi was on of Da Vinci's favourite numbers.. And it really is a great book, and the point of the "research" and the "tired old conspiracy theories" is the fact that he uses his research as "FACT" to prove/disprove some of those conspiracy theories, which is definitely not "tired" nor "old".

### Re: Da Shitty Code (2, Funny)

#### Anonymous Coward | more than 10 years ago | (#8203191)

Dan Brown is the greatest writer of all time.

Provided we ignore EVERY OTHER WRITER EVER.

### How far is HURD behind release now? (-1, Offtopic)

#### Anonymous Coward | more than 10 years ago | (#8203005)

Does anyone know? 10 years? twelve? If they want to replace Linux for political reasons, they should make something that works.

### The Galaxy?? (4, Funny)

#### Gorimek (61128) | more than 10 years ago | (#8203007)

The concept of math isn't even spread very far on this planet.

### obligatory (0)

#### Anonymous Coward | more than 10 years ago | (#8203203)

I don't have a concept of math, you insensitive clod!

#### Anonymous Coward | more than 10 years ago | (#8203008)

I just heard some sad news on talk radio - Horror/Sci Fi writer Stephen King was found dead in his Maine home this morning. There weren't any more details. I'm sure everyone in the Slashdot community will miss him - even if you didn't enjoy his work, there's no denying his contributions to popular culture. Truly an American icon.

#### Anonymous Coward | more than 10 years ago | (#8203131)

OMG!!! Holy shit! How come I didn't hear about this?! This isn't on Yahoo! or Slashdot or anyplace! Certainly not on CBS or ABC, and definitely not on the New York Times, LA Times, or any other of those Commie-terrorist-loving lefty papers!!! I haven't watched FOX News but I'm sure at least they reported it! I can't believe the control the liberal media has over this country!!!!

-AC (hint:doesn't stand for Anonymous Coward)

#### mblase (200735) | more than 10 years ago | (#8203244)

http://news.google.com/news?hl=en&edition=us&q=ste phen+king&btnG=Search+News -- really? Who reported it?

### aw man (-1, Offtopic)

#### Savatte (111615) | more than 10 years ago | (#8203024)

I thought this book was going to be about getting rich quickly by investing in precious metals.

### The Golden Ratio (4, Funny)

#### Anonymous Coward | more than 10 years ago | (#8203047)

He who has the gold, makes the ratio.

### Mathematics not universal? (5, Insightful)

#### s20451 (410424) | more than 10 years ago | (#8203055)

If mathematics are not universal, then the mathematical reasoning that can be conducted to deduce the laws of nature is also not universal. Hence, if a different civilization has different mathematics, they have different physical laws as well.

This is basically a postmodern viewpoint, that reality is socially constructed. This viewpoint has been largely derided by the scientific community, and has failed to replace science because it hasn't really offered a compelling alternative. The only way I can see it being true is if other civilizations don't "exist" in the universe as humans do.

Do a google search for Alan Sokal for a scientist's viewpoint of postmodern scientific criticism.

### Re:Mathematics not universal? (-1, Offtopic)

#### Bongzilla (458471) | more than 10 years ago | (#8203170)

this is just to agree with s20451's point of view

### Re:Mathematics not universal? (1)

#### Trelane, the Squire (608266) | more than 10 years ago | (#8203197)

I always thought that if there were aliens, they would probably be so vastly different from us that they would indeed have different notions completely. I guess that could include even to the point of having different systems than our mathematics.

It's also an assumption to say that aliens would be about our size in body mass. Different size changes the way they would 'look' at the universe.

### Re:Mathematics not universal? (3, Insightful)

#### s20451 (410424) | more than 10 years ago | (#8203303)

Yes, but what would we have in common with them? We both exist in the universe, and can make observations about the universe. Thus, assuming they have a certain amount of technological advancement, we could communicate with them and ask, "How does your planet move around your star?". It would take a whlie for us to communicate our respective definitions of "ellipse" and "gravitation", but surely we would agree that the path is elliptical due to gravitation, even if they had three purple heads and were fifty feet tall.

Then again, if they have no concept of "time" or "movement", then I would argue that they don't exist in the universe in the same way we do.

### Re:Mathematics not universal? (4, Funny)

#### kfg (145172) | more than 10 years ago | (#8203220)

I offer this argument to those who state "You create your own reality."

I kick them in the shin.

Then say, "Why did you do that?"

KFG

### Re:Mathematics not universal? (4, Insightful)

#### Raindance (680694) | more than 10 years ago | (#8203232)

Well, I wouldn't say that "if a different civilization has different mathematics, they have different physical laws as well."

Rather, I'd set mathematics and logic equal (there's a respected tradition that does, see Bertrand Russell, Principia Mathematica). Then, to say that mathematics isn't the same across the universe, one would say that logic isn't the same across the universe.

Now, "Logic isn't universal" is a damn meaningless statement. It'd translate into "Logic cannot describe [timespace-area/context] X." Which is, of course, a logical assertion about X.

I think either the reviewer's portrayal of the argument or the argument itself is bogus.

RD

### Something I learned from Martin Gardner... (5, Informative)

#### kzinti (9651) | more than 10 years ago | (#8203059)

Something I like about the golden ratio is that it is the number that is exactly 1.0 greater than its reciprocal. This makes it easy to remember the exact value: just solve

x = 1 + 1/x

You'll get a quadratic with the solutions (1 +/- sqrt(5))/2, or 1.618... and -0.618...

### Re:Something I learned from Martin Gardner... (4, Interesting)

#### product byproduct (628318) | more than 10 years ago | (#8203213)

A way more fun way to solve x = 1+1/x:

On a calculator:

2) press [1/x] [+] [1] [=]
3) GOTO 2

In other words this converges to the golden ratio! It takes a while, so normally you do this when you're bored.

### math is not universal? (3, Insightful)

#### liquidpele (663430) | more than 10 years ago | (#8203060)

How could it not be?
Math is just a way of describing objects, forces, and interactions... how could you describe them differently??

### Re:math is not universal? (5, Insightful)

#### greatmazinger (747174) | more than 10 years ago | (#8203124)

Math is just a way of describing objects, forces, and interactions..

Ummm, no. That's not math. That's physics. Math is more abstract and one can do math without associating any of the concepts with "reality". One you use math to model reality, it becomes science and engineering.

### Re:math is not universal? (0)

#### Anonymous Coward | more than 10 years ago | (#8203287)

show me math that has no connection with reality.. ask someone who can't see, hear, smell, or touch how mathematics works

### Re:math is not universal? (0)

#### Anonymous Coward | more than 10 years ago | (#8203188)

Say you visit a planet where the dominant species, the one responsible for things like math and science, experiences everything double due to their funny optical and other sensory apparatus. How would you describe the concept of "one" to such an entity?

### Re:math is not universal? (0)

#### Anonymous Coward | more than 10 years ago | (#8203205)

Why should it be? The math we used was invented by us. It's a high level concept.

### Different aliens (1)

#### Trelane, the Squire (608266) | more than 10 years ago | (#8203250)

If the aliens were different (even far different) than we, they would indeed describe them differently. They would SEE ('see?') them differently.

### Re:math is not universal? (1)

#### robbway (200983) | more than 10 years ago | (#8203256)

We should be able to apply our mathematics to everything. However, if our thought process is drastically different, they ("aliens" if you will) may not recognize it as a corollary to their ideas. They may not even have a use for it.

For instance, math doesn't really need to add. The concept can be completely explained with the concepts of negative numbers and subtraction.

To get a real handle on the concept of different mathematics models, take the extremely difficult class of Abstract Algebra. It's called Algebra because it wraps itself around the ideas of open sets, closed sets, and operations as a generic concept.

### Movie (4, Informative)

#### savagedome (742194) | more than 10 years ago | (#8203072)

The movie PI [imdb.com] is also a very compelling watch for those who are interested.

### Re:Movie (4, Funny)

#### ath0mic (519762) | more than 10 years ago | (#8203181)

...which I think also holds the record for the "longest movie title in history."

:)

### Re:Movie (0)

#### Anonymous Coward | more than 10 years ago | (#8203336)

Amusingly, Pi the movie is actually more about Phi (1.617) than Pi (3.14). There is a lot more discussion about spirals (including lots of examples, such as cream in coffee, hurricanes, etc.) than about the properties ascribed to Pi.

As pointed out above, Phi is neat because it solves x=1+1/x, and just about any self-repeating shape (5-sided star, golden rectangle) has it. You can compute any of the Fibbonacci numbers with it directly (F(n)=((phi^n - phiC^n)/sqrt(5)) where phi is 1.617... and phiC is 1/phi, IIRC), and it has uses in generating realistic imagery in computer graphics.

### Pi the movie (4, Interesting)

#### cryptochrome (303529) | more than 10 years ago | (#8203356)

I literally watched that movie 2 nights ago. Spooooooky....

Not bad (aside from one glaringly obviousl mathematical error). The thing that I mulled over the most was the proposition that a large integer could be a number of fundamental significance. In the movie it was 216 digits long. I had always figured all the really fundamental numbers were irrational. After thinking about it and looking up on the internet it seems there are actually only 6: pi, e, i, 1, 0, and phi (and arguably, -1). And the first five can be directly related with the equation:

e^(pi*i) + 1 = 0

phi is not directly related to the others in such a manner (In the movie the god number is somehow tied to both pi and phi). Although pi and phi both happen to be ratios that are also irrational. But to get back to my original point, the suggestion that any number of a truly fundamental significance besides 0 and 1 would be not only rational but an integer seems improbable.

### The Da Vinci Code (0, Offtopic)

#### Mork29 (682855) | more than 10 years ago | (#8203074)

I'd suggest The Da Vinci Code [amazon.com] to people who are all sorts of interested in this sort of thing. Da Vinci played a small part in all this fun Phi stuff, and evidence of it can be found in his paintings. Besides, this is just a great book that everybody should read! They point out many places where one can find the "Golden Ratio" within this fine book.

### Re:The Da Vinci Code (1)

#### mblase (200735) | more than 10 years ago | (#8203209)

Da Vinci played a small part in all this fun Phi stuff, and evidence of it can be found in his paintings.

He was far from the first, and I don't just mean Aristotle. The ancient Egyptians believed that this "sacred ratio" was important enough to embed in their art and constructions. Many Egyptian temples employ rectangluar archways designed according to the Golden Ratio (phi).

At the Great Pyramid of Giza, the ratio of the length of one side of the base to the perpendicular height of the pyramid is about 2/phi, making the slant height of the pyramid side proportionately equal to phi. The result is that each side of the pyramid is a Golden Triangle.

### Sounds like an interesting book (1, Funny)

#### Anonymous Coward | more than 10 years ago | (#8203077)

With a title like that I was expecting 320 pages of this:

1.6180339887498948482045868343656381177203091798 05 76286213544862270526046281890244970720720418939113 74847540880753868917521266338622235369317931800607 66726354433389086595939582905638322661319928290267 88067520876689250171169620703222104321626954862629 63136144381497587012203408058879544547492461856953 64864449241044320771344947049565846788509874339442 21254487706647809158846074998871240076521705751797 88341662562494075890697040002812104276217711177780 53153171410117046665991466979873176135600670874807 10131795236894275219484353056783002287856997829778 34784587822891109762500302696156170025046433824377 64861028383126833037242926752631165339247316711121 15881863851331620384005222165791286675294654906811 31715993432359734949850904094762132229810172610705 96116456299098162905552085247903524060201727997471 75342777592778625619432082750513121815628551222480 93947123414517022373580577278616008688382952304592 64787801788992199027077690389532196819861514378031 49974110692608867429622675756052317277752035361393 ...

### Why wouldn't math be known across the universe? (5, Insightful)

#### ObviousGuy (578567) | more than 10 years ago | (#8203078)

What reasons would there be for an alien to not understand or accept that one plus one equals two. Any being capable of human-equivalent level of thought would be able to count objects. Whether they did in this in base-2 or base-3 or base-10 or base-12, it doesn't matter because all these bases can be reconciled to each other.

Could there be some areas of mathematics that humans have discovered that has not been discovered by an alien race? Sure. Prior to Newton there was no calculus and so Kepler had to discover the period of planetary orbits using geometry and algebra. But this does not mean that Kepler would not have used calculus if it had been available to him, only that such a concept had not yet been thought of.

But counting and simple addition and subtraction are mathematical operations that are mastered even by animals. It is fairly condescending to assume that aliens could not even fathom those levels of mathematics.

### Re:Why wouldn't math be known across the universe? (4, Interesting)

#### Sique (173459) | more than 10 years ago | (#8203233)

With just adding and subtracting you get something called 'constructive Mathematics'. It's a subset of Mathematics, and it's missing some important axioms necessary for quite usual things like calculus (nonexistance of a supreme for any limited set of numbers) and algebra (no selection axiom, thus you can't prove that infinite dimensional vector fields have a base).

### Re:Why wouldn't math be known across the universe? (3, Insightful)

#### Apostata (390629) | more than 10 years ago | (#8203241)

quote: "It is fairly condescending to assume that aliens could not even fathom those levels of mathematics."

And it's fairly narrow-minded to assume that another life form in the galaxy has a frame of perception that's even remotely similar to ours. This is more than just saying "what if they see in infra-red!", but rather to say that we feel mathematics is the de facto language of the universe because it (as with terrestrial life on Earth) doesn't yet have a competitor.

### Re:Why wouldn't math be known across the universe? (1)

#### Suicyco (88284) | more than 10 years ago | (#8203263)

Seeing how Archimedes did his thing with volumes [thewalters.org] simply through deduction and observation, I find it hard to believe that others in the universe wouldn't have the same basic powers of observation and logic. Archimedes almost discovered integral calculus through what amounts to puzzling out how to calculate volumes of strange objects. Its a leap of logic and happened in the absence of advanced mathematics as we know them. I suppose its possible that our way of doing math is very different, however at the lowest level I can't see how it would matter, ie. on/off binary logic. Charge, absence of charge. That happens all over the universe doesn't it?

### What about aliens at the molecular level? (1)

#### Trelane, the Squire (608266) | more than 10 years ago | (#8203288)

or the quantum level? What if the aliens lived in different dimensions? Also, to say the laws of this backwater place in the immense universe are the same everywhere is a little arrogant

### Re:Why wouldn't math be known across the universe? (5, Interesting)

#### arbour42 (731167) | more than 10 years ago | (#8203367)

Prior to Newton there was no calculus

In a fascinating book, a Hindu scholar and monk, Sri Tirthaji, discovered in the Hindu Veda scriptures the basis for our math system. There he found shortcuts for most all our math work - easy ways to do difficult long divisions in a matter of seconds, quadratic formulas, PI to over 32 digits, the Pythagorean theorem (much before the Greeks), derivatives, calculus.

Our math is actually from the Vedas, and the Arabs got it from them, and then spread it through the Western world. The Vedas are at least several thousand years old.

The book is called Vedic Mathematics or Sixteen Simple Mathematical Formulae from the Vedas [amazon.com] and can be found at amazon or used book stores.

It's one of the major works of genius of science. The first time i read it, it was shocking how advanced it was, and simple! Any division such as 1.748362 / 59487 can be done long handed (pencil and paper) in a minute.

Our math system, how it was discovered or invented, who knows and how far back, is absolutely brilliant.

### Re:Why wouldn't math be known across the universe? (1)

#### mslinux (570958) | more than 10 years ago | (#8203372)

Could there be some areas of mathematics that humans have discovered...

Math is not discovered, it's made up. One plus one is two because men made rules that make it that way. Also, you can use mathematical models to prove that math is not consistent... Read up on Godel's Theorems. The only thing that makes math work are the rules that we all must accept (these are called axioms). If we do not accept the rules, then math doesn't add up... pardon the pun.

### Thumbs up (1, Offtopic)

#### haxor.dk (463614) | more than 10 years ago | (#8203092)

I own the book, bought it a year ago myself. A good read.

If you're looking for something a bit along the same lines, but sprinkled with history, religion and conspiracy, I can recommend "the Da Vinci Code" by Dan Brown.

### Numbers are numbers (4, Insightful)

#### Wind_Walker (83965) | more than 10 years ago | (#8203121)

Regardless of what radix used (10-based, hex, octal, etc) "6" of something is still "6". Call it whatever you want, some alien name. There are still 6 items there.

By the same nature, prime numbers are always prime. There exist a certain number of things (5, 7, 11, etc) and cannot be evenly divided. Period. We call them prime numbers, and we use our base-10 radix. Aliens could call them Borgolsmocks in their base-182, but the fact still remains that a Borgolsmock cannot be divided evenly.

And I firmly believe that no intelligence would survive for long without a knowledge of mathematics. Counting the days for crop rotation, the ability to evenly divide food among the tribe, and communication of the number of animals in a herd... mathematics will be generated in the evolution of any intelligent species.

And it is truly universal.

### Re:Numbers are numbers (1)

#### Radix37 (670836) | more than 10 years ago | (#8203262)

Regardless of what radix used (10-based, hex, octal, etc) "6" of something is still "6".

What if you used radix 37?

### Re:Numbers are numbers (1, Funny)

#### Anonymous Coward | more than 10 years ago | (#8203337)

Aliens could call them Borgolsmocks in their base-182, but the fact still remains that a Borgolsmock cannot be divided evenly

Please define the following in absolute, universal terms:

cannot:
be:
divide:
evenly:

### A god with a plan? (5, Interesting)

#### mindstrm (20013) | more than 10 years ago | (#8203129)

Why does this make you think there is a supreme being, with a plan? Just beause things work out?

The balance and beauty of nature and all that?

OF COURSE there is a pattern, and things work out. Look at evolution.

You take a puddle in the middle of nowhere.. it has an ecosystem in it with a perfectly balanced population (too many, it dries up, too few, they reproduce...). Would these little creatures say "Oh wow! Look how there is JUST enough water for each of us! There must be a GOD!".... silly, right?

Nature seems balanced in the world, becuase that world produced nature... they are intertwined, infinitely.

Irrational numbers only seem strange because of the way we choose to look at things... the fact that it doesn't reduce to some fraction in our counting system doesn't *mean* anything holy or significant....

The fibonacci series and the golden ratio are related? Sure are.
(The ratio of successive numbers in the fib. series approaches the golden ratio as you go upwards)

But it's not so weird, is it? A sunflower.. the way it grows, it builds on itself.. in a spiral... naturally following this series.

Is it some grand creator that made it that way, or is it just the plain, obvious way for something to grow?

What would be evidence of a creator would be if things did NOT follow what was natural and obvious. If these things did NOT follow the golden ratio and other straight math.. if we could find no explanation for why things had a weird ratio, or weird behavior.. no explanation from the current or possible past enviroment to explain how something evolved.... come to me with that, then we can talk about a creator.

Until then, i'ts just nature.

### Re:A god with a plan? (2, Funny)

#### Geckoman (44653) | more than 10 years ago | (#8203281)

What would be evidence of a creator would be if things did NOT follow what was natural and obvious. If these things did NOT follow the golden ratio and other straight math.. if we could find no explanation for why things had a weird ratio, or weird behavior.. no explanation from the current or possible past enviroment to explain how something evolved.... come to me with that, then we can talk about a creator.

If God had intentionally inserted all these frequently recurring constants and ratios everywhere, then they, like the Babelfish, would be proof of God's existence. That would defy faith, and He would disappear in a poof of logic.

Then, unfortunately, I'm afraid we'd all get hit by a bus....

### Re:A god with a plan? (2, Interesting)

#### danmitchell (691749) | more than 10 years ago | (#8203382)

Since I'm taking a class on general relativity this semester, I'll weigh in with a quote of Albert Einstein.

What I'm really interested in is wheter God could have made the world in a different way; that is, whether the necessity of logical simplicity leaves any freedom at all.

In other words, maybe nature is what it is because God created it that way, or maybe it is what it is because it has to be.

### I rememeber this from... (5, Interesting)

#### gpinzone (531794) | more than 10 years ago | (#8203133)

Donald in Mathmagic Land. It was a great little video Disney produced back in 1959 with Donald Duck. The narrator goes off the topic at times, but the overall animated descriptions of the golden ratio and its related golden values were awesome. Unfortunately, this Disney short is not available on VHS or DVD currently. Look to eBay to find a long lost copy of it.

### Furniture design (5, Interesting)

#### hulap0pr (562242) | more than 10 years ago | (#8203134)

The golden ratio concept is a big part of furniture design. Case pieces (boxes, bureaus, etc...) appear more balanced and pleasing to the eye when the golden ratio is followed. Go home and measure your highboy...

### Re:Furniture design (1)

#### cj79 (718594) | more than 10 years ago | (#8203247)

Case pieces (boxes, bureaus, etc...) appear more balanced and pleasing to the eye when the golden ratio is followed.

Same can be said for centerfolds.

### This topic... (1)

#### ArmenTanzarian (210418) | more than 10 years ago | (#8203152)

Is covered many many many times in many other books (did I mention the word many).

I just finished the somewhat overrated, but entertaining Da Vinci Code [google.com] which mentions this in addition to several other interesting. The presentation is that of fiction, which adds entertainment, but detracts from the believability.

Also the movie Pi [google.com], which I probably need not mention here, speaks of this to some length.

Final question being, does this book really add to my knowledge of the subject? I think I've heard all of the examples of where this ratio can be found in nature, is this guy just beating a dead horse? The review doesn't really imply that there's anything new here.

### muscle memory (1)

#### Matey-O (518004) | more than 10 years ago | (#8203156)

How hard was it to write this without mispelling ratio as ration?

### Re:muscle memory (0)

#### Anonymous Coward | more than 10 years ago | (#8203237)

Funny... I make that same mistake because of muscle memory. Same with type->typo.

### Re:muscle memory (0)

#### Anonymous Coward | more than 10 years ago | (#8203242)

Thanks to Metal Gear Solid, I can rarely spell ratio without the n. Now there's a mathematical ration for you.

### universal math? (4, Interesting)

#### mblase (200735) | more than 10 years ago | (#8203159)

The idea that mathematics is a human construction and probably holds no water in another civilization that grew up on the other side of the universe makes a lot of sense to me.

Yes and no. Mathematics is just a way of modelling things abstractly. Even things like counting from one to ten is a model for concrete objects, and provides a way of, say, making sure the number of cows you have today is the same as the number of cows you had yesterday. At the higher level, mathematics lets you model shapes, motion, acceleration, and gravitational collapse of entire stars.

The most common types of mathematics we use include decimal arithmetic (trading with money), algebra (solving for unknown quantities), and geometry (simplifying the world into abstract shapes). Hundreds of other branches of mathematics exist to model different things in different ways, and none of them are "right" -- they all are optimized for particular problem-solving.

With that in mind, I find it inconceivable that advanced civilizations on other planets would not have some kind of mathematics, and at least share the natural numbers with us (not necessarily base ten, though). If all you're doing is raising food for your family, then even arithmetic may be more than you need to bother with. But anything that involves abstract problem-solving, measurement, and/or exchange of goods for trade is going to need some kind of math. The models they use may bear no resemblance to the ones we use, but that doesn't mean it's not math.

### How does one dispute math as a universal concept? (2, Insightful)

#### hellfire (86129) | more than 10 years ago | (#8203162)

I'd be fascinated to hear more about this. I want to get the book but I'm impatient and want to discuss it now! :)

I would think that math in some was is universal, in the sense that every sentient creature has to figure out a method of counting. Some creatures count in base 10, others base six, maybe base 12. Other creates could figure out a counting base we haven't thought of yet. However, if they have a method of counting and measuring, I'm sure we'd have a method of translating their mathmatical models to our own, without too much trouble.

Perhaps the definition of math here is different than mine? Thoughts?

### Opinion, Mr. Spock? (2, Funny)

#### ackthpt (218170) | more than 10 years ago | (#8203163)

"Captain, I believe there is a 1.6180339887 percent probability that any security officers beamed down to the planet will survive."

### who trusts this review... (-1, Troll)

#### Anonymous Coward | more than 10 years ago | (#8203165)

...since it is written by someone using a johnny quest character for his alias!?

### Definition FYI (3, Interesting)

#### Prince Vegeta SSJ4 (718736) | more than 10 years ago | (#8203166)

Golden Ratio:

The number 1.618..., which is half the sum of one plus the square root of five (1+SQRT(5))/2. This number was known in ancient times, and has many interesting properties in many fields. In Fibonacci series, the higher one goes in the series, the closer the ratio between a number and it's predecessor comes to the Golden Ratio.

From "The Technical Analysis of Stocks, Options & Futures" - William F. Eng

Geez, I never thought my Trading and /. would come together. Then again it is delving into the Uber Math Geek world - lol

### Awful (1)

#### dubbayu_d_40 (622643) | more than 10 years ago | (#8203177)

This is more of a historical book than anything else. Since it is called the golden ratio, I expected more about "the golden ratio" that what it offered. I wish I had read the Amazon customer reviews before I wasted my money on this junk.

### First-contact scenarios? (5, Insightful)

#### bravehamster (44836) | more than 10 years ago | (#8203183)

Finally, Livio takes a shot at the idea that mathematics is a universal concept across the entire universe. To be honest, I have always assumed that it was. ... The idea that mathematics is a human construction and probably holds no water in another civilization that grew up on the other side of the universe makes a lot of sense to me.

From what I understand, the vast majority of realistic first-contact scenarios postulated involve using mathematics as a common ground to bridge the language barrier. 1 + 1 equals 2 in every language on earth (except New Age holistic 1 + 1 = 3 crap). It makes sense and it works everywhere. It would be awfully damn hard to build a spaceship without mathetmatics, let alone trying to calculate launch trajectories or transfer orbits. Unless they had such an intuitive grasp of higher level mathematics that they don't even consider it worth talking about, I don't see how any species that had no concept of math could ever rise above the level of pointy sticks and sharpened rocks. And even then you'd probably want to keep track of how many rocks you had to make sure Lurg over there didn't *borrow* a few.

### Right you are! (0, Offtopic)

#### davek99999 (749198) | more than 10 years ago | (#8203196)

"After all, I am a Trekkie and this concept goes unstated throughout all four TV series." This is exactly how I feel about the Star Trek universe. All four TV series. You said it, my friend.

phi is gay! :P

### Intelligent marketting strategy for stupid ideas (2, Funny)

#### dtfinch (661405) | more than 10 years ago | (#8203231)

He probably just caught on to the idea that if you say something outrageous enough in your books, like math is wrong, people will buy them.

### Phi (5, Funny)

#### Rupert (28001) | more than 10 years ago | (#8203240)

I hate it when people use extreme amounts of decimal precision when talking about irrational numbers. Really, is 1.6180339887 (or 1.6180339887498948482045868343656) much more informative than 1.618? If you're going to do calculations with it, use the exact value:

1/2 * (sqrt(5) + 1)

and sort out the irrational bits at the end, rather than introduce rounding errors at the beginning.

That's just a rationalisation, of course. My real reason for complaining about decimals is that it feels wrong. 1.6180339887 does not look like a profound number. It's like the number is a beautiful woman, and the decimal representation is the pornographic pictures she posed for when she was young and needed the money. Yes, it looks like her, and it may even be useful. But the real thing is *so* much better.

### The real question is... (1)

#### Eightlines (536572) | more than 10 years ago | (#8203252)

What are the dimensions of this book? Is the size of the book pleasing to the eye? I always enjoy these things when someone takes a bit of a step back and relates the physical format of the book to the subject at hand.

### Groklaw (-1, Offtopic)

#### digitalchinky (650880) | more than 10 years ago | (#8203260)

Groklaw, is it down, or just under higher than normal load given the SCO thing today?

### hmm.. (1)

#### andy55 (743992) | more than 10 years ago | (#8203261)

Finally, Livio takes a shot at the idea that mathematics is a universal concept across the entire universe.

This seems like a tall, tall order. I've been into math/geometry/visual related software [soundspectrum.com] for years now and am now transitioning into making my living off it. However, the fact that there are still many fundamental mysteries in mathematics always raises doubt on the things like our origins, God, and the universe. Pi is the best example of that. It's no puzzle to me why countless minds have tried to be the hero (or the mathematician version of one), to unlock pi's mystery, but no one has yet to really break through. The film Pi is an excellent and enjoyable film, and considers the magnitude (as well as the price) of unlocking pi's mystery.

I'd like to day I'm open minded, but whew. Perhaps such things are more considerable when you start to consider all the various matter/energy theories floating around out there. There's still gigantic mysteries still out there for cosmology and physics (dark matter, open universe, dark energy, unification of gravity into the standard model), so I suppose we should never be too hasty to close the door on counterintuitive or far-fetched theories. I'd love to hear anyone who can paraphrase the thrust of this person's arguments, etc.

### math and humanity (3, Insightful)

#### jstoner (85407) | more than 10 years ago | (#8203265)

It's an interesting question: how far could a civilization get without math? IANA historian, but it seems to me the more sophisticated a (human) civilization, the better its mathematics. The Aztecs did develop a fair amount of math completely independently of Eurasian civilizations.

Could a race become spacefaring without math? Could they develop the radio communications we could use to detect them? I suppose they could if the circumstances of their environment or adaptation (Low-gravity, bio-radio communications) allowed it.

But how would you arrive at the necessary conclusions without an abstracted intellectual framework like math? Maybe progress would just be slower.

Hmmm... makes you wonder what we're still missing.

### If you like math history (1)

#### hey! (33014) | more than 10 years ago | (#8203268)

Try Journey Through Genius [barnesandnoble.com] by Will Dunham.

It covers a sampling of many of the great theorems and proofs of mathematics in a form that anyone with high school math can follow, as well as giving interesting insights into the personalities of the mathematicians (where this can be known). Most of them were, um eccentric. It is nice to know that Euler at least was well adjusted, if you couldn't exactly call him normal.

Euclid is represented twice here: once for his proof of the Pythagorean theorm and once for his proof of the infinitude of primes.

### Don't confuse Syntax for Content (4, Insightful)

#### Master Switch (15115) | more than 10 years ago | (#8203276)

While the syntax we use for mathmatics is culturally defined, the content beneath them is not. We humans discover, not invent mathmatical constructs. As much as we would like to think we create, we do not. We iterate and find the best fit solutions.

### Bullshit (0)

#### Orion442 (739483) | more than 10 years ago | (#8203278)

If a creature can't distinguish between "more than" or "less than" of an object, it cannot be labeled "intelligent" to begin with.

### Debunking constants (4, Interesting)

#### hcg50a (690062) | more than 10 years ago | (#8203301)

Livio squarely debunks the idea that the Golden Ratio is present in many famous paintings and architecture that has been postulated in previous books.

This doesn't sound exactly right.

I think it may be the case that writers have attributed the use of phi in art when there was no such intentional use by the artist.

But the very nature of phi makes it unlikely not to appear in certain contexts.

Same with pi.

The thing I love about math is that it has utterly nothing to do with reality or the universe or anything at all.

Typically, however, physicists make assumptions that match, more or less closely, to what is happening in the real world, so the conclusions from such assumptions match, more or less closely, to what is actually happening in the real world.

But there is no reason why some utterly alien intelligence can't make a set of assumptions that match their reality, which would be utterly alien to us, yet still valid, and still recognizable by mathematicians, if not physicists.

This is the giant flaw at the end of the book Contact, by Carl Sagan. Ellie discovers a message in the constant pi, placed there by an intelligence. If this were a constant of physics, that would imply the existence of some incredibly advanced intelligence that engineered the universe to contain a constant with precisely that value. This is somewhat plausible, and I believe it was Sagan's intent.

But he picked pi, which actually has nothing at all to do with this or any other universe.

What kind of incredibly advanced intelligence can possibly engineer that? I can only think of One.

### The Golden Ratio in statistics (1)

#### careysb (566113) | more than 10 years ago | (#8203316)

1.6180339887% of all statistics are made up.

### The answer is 42! (1)

#### MalaclypseTheYounger (726934) | more than 10 years ago | (#8203317)

Obviously the people that are truly running the show and know the answer to the meaning of life have 13 fingers, and use a base 13 numbering/math system.

6 X 9 = 42 in base 13.

### The Golden Ratio (0, Redundant)

#### Anonymous Coward | more than 10 years ago | (#8203346)

The golden mean (proportio divina or sectio aurea), also called golden ratio, golden section, golden number or divine proportion, usually denoted by the Greek letter phi, is the number phi = (1 + sqrt 5)/2 = approx. 1.618033 ... the unique positive real number with phi^2 = phi + 1 and the equally interesting property phi-1 = 1/phi.

Two quantities are said to be in the Golden ratio, if "the whole is to the larger as the larger is to the smaller", i.e. if (a+b)/a = a/b. Equivalently, they are in the golden ratio if the ratio of the larger one to the smaller one equals the ratio of the smaller one to their difference: a/b = b/(a-b).

After simple algebraic manipulations (multiply the first equation with a/b or the second equation with (a-b)/b), both of these equations are seen to be equivalent to (a/b)^2 = a/b + 1 and hence a/b = phi.

The fact that a length is divided into two parts of lengths a and b which stand in the golden ratio is also (in older texts) expressed as "the length is cut in extreme and mean ratio".

The ancient Egyptians and ancient Greeks already knew the number and, because they regarded it as an aesthetically pleasing ratio, often used it when building monuments (e.g., the Parthenon). The pentagram so popular among the Pythagoreans also contains the golden mean. It is also sometimes used in modern man-made constructions, such as stairs and buildings, woodwork, and in paper sizes, however it is a myth that the European formats (such as A4, which is actually cut to 4 decimal places of sqrt 2) are cut in the golden mean. Recent studies showed that the Golden ratio plays a role in human perception of beauty, as in body shapes and faces.

A possible reason for its supposed attractiveness is shown by the Golden rectangle, which is a rectangle whose sides a and b stand in the Golden ratio. If from this rectangle we remove a square with sides of length b, then the remaining rectangle is again a Golden rectangle, since its side ratio is b/(a-b) = a/b = phi. By iterating this construction, one can produce a sequence of progressively smaller Golden rectangles; by drawing a quarter circle into each of the discarded squares, one obtains a figure which closely resembles the logarithmic spiral theta = (pi/2 log(phi)) * log r.

Since phi is defined to be the root of a polynomial equation, it is an algebraic number. It can be shown that phi is an irrational number. Because of 1+1/phi = phi, the continued fraction representation of phi is 1+1/(1+1/(1+...)) = [1; 1, 1, 1, ...]

The number phi turns up frequently in geometry, in particular in figures involving pentagonal symmetry. For instance the ratio of a regular pentagon's side and diagonal is equal to phi, and the vertices of a regular icosahedron are located on three orthogonal golden rectangles.

The ratios of justly tuned octave, fifth, and major and minor sixths are ratios of consecutive numbers of the fibonnaci sequence making them the closet low integer ratios to the golden mean. James Tenney reconceived his piece For Ann (rising), which consists of up to twelve computer generated upwardly glissandoing tones, as having each tone start so it is the golden ratio (in between an equal tempered minor and major sixth) below the previous tone, so that the combination tones produced by all consecutive tones are a lower or higher pitch already, or soon to be, produced.

The explicit expression for the Fibonacci sequence involves the golden mean. Also, the limit of ratios of successive terms of the Fibonacci sequence equals the golden mean. From a mathematical point of view, the golden ratio is notable for having the simplest continued fraction expansion, and of thereby being the "most irrational number" worst case of Lagrange's approximation theorem. It is also the fundamental unit of the algebraic number field Q(sqrt 5) and is a Pisot-Vijayaraghavan number.

The golden mean has interesting properties when used as the base of a numeral system.

1. 6180339887 4989484820 4586834365 6381177203 0917980576 2862135448 6227052604 6281890244 9707207204 1893911374 8475408807 5386891752 1266338622 2353693179 3180060766 7263544333 8908659593 9582905638 3226613199 2829026788 0675208766 8925017116 9620703222 1043216269 5486262963 1361443814 9758701220 3408058879 5445474924 6185695364 8644492410 4432077134 ...

### Lincoln's distorted texts (0)

#### Anonymous Coward | more than 10 years ago | (#8203349)

Abraham Lincoln himself used Euclid's texts....

This explains a lot, especially Lincoln's distorted views of North versus South....

### 1.61803399 (2, Informative)

#### karmaflux (148909) | more than 10 years ago | (#8203363)

More here. [vashti.net]

### 1.6?!! I THOUGHT THE RATIO WAS 9 TO 4?! (0)

#### enigmals1 (667526) | more than 10 years ago | (#8203379)

Man where have I been... I guess I have to get this book now. I always thought it was 9:4 or 21/4 to 1. *shrug*
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