The Golden Ratio

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. The Golden Ratio author Mario Livio pages 320 publisher Broadway rating 7/10 reviewer raceBannon ISBN 0767908155 summary Through telling the tale of the Golden Ratio, Livio explains how this simple ratio pops up in all kinds of physical phenomenon and debunks the idea that the ratio is present in many famous man-made structures and art work. Along the way, he provides historical tidbits regarding some of the well-known and not so well-known mathematicians throughout the ages and he tells the story of some of the more famous and not so famous mathematical advances. Finally, he discusses the possibility that mathematics may represent some kind of global truth that exists throughout the cosmos.

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.

Comments

Mathematics not universal?

[s20451]

Didn't read the book.

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?

[Mr. Slippery]

If mathematics are not universal, then the mathematical reasoning that can be conducted to deduce the laws of nature is also not universal.

You're assuming a relationship between mathematics and the "laws of nature" that isn't there. As Einstein put it, As far as the laws of mathematics refer to reality, they are not certain; as far as they are certain, they do not refer to reality."

Mathematics is as socially constructed as any other form of language. It is based on axioms and defintions, not observation of reality. We select those axioms and definitions in a way to be useful to us, just as we select for those lingustic constructs that are useful. But this selection is based on our desire to communicate with others - it is a social construct. Once upon a time if you asked mathematicians what nubmer, when squared, gave negative one, they'd say there was no such number; now, any bright middle school kids know it's i.

"Reality" is also to a large degree socially constructed, since all can ever speak of is our observations, which are socially conditioned. You see what you expect to see or are trained to see. (You don't see the fnords, or Sombody Else's Problem, while the hypothetical planet Vulcan (the one inside the orbit of Mercury, not Mr. Spock's home) was observed several times, as were Blondlot's N-rays.) This is why double-blind protocols are used - though if everyone involved has an expectation, that doesn't help.

What we think of as "reality" is just a model that we mostly share. The electron, for example, is not a component of human experience but a component of a model that unifies and predicts many observations. That is a very good and useful model, but it is entirely conceivable that some extra-terrestrial civilization has (or some future human civilization will have) a model that is just as useful but doesn't contain anything like electrons. (Just like Chinese Medicine has a "patterne-thinking" model of the human being that is radically different than and incompatible with the reductionist model, yet is extremely useful.) What would such an electron-free model look like? I can't tell you, I'm too conditioned by the electron model.

Remember: for any set of observations, there are an infinite number of hypothesis to fit them. There's no end to the curves you can plot through any finite set of data points. We see the points and call them a line, but it ain't necessarily so. The best we can do is eliminate lines that don't go anywhere near the points.

Re:Mathematics not universal?

[photo storm]

You bring out a very subtle fallacy, and one that is tied to philosophical issues regarding mathematics. I bit of history is in order:

The fundamental question is this: is, or isn't, mathematics an extension of logic? A smart man named Frege (read about him here) said, yes, it is. He showed a way to connect formal logic with set theory, which is the basis for mathematics as we know it.

There was only one problem: Russell's Paradox. Bertrand Russell showed that, using Frege's axioms that defined set theory, we have a contradiction - Russell's Paradox. And as any student of logic knows, a contradiction can be used to prove anything at all, which means that mathematics as Frege defined it was not viable.

To make a very long and very interesting story short, Russell (with Alfred Whitehead) attempted to create a foundation for mathematics that would not give rise to Russell's paradox - the Principia Mathematica. And everyone thought the world was cool.

Then, in the 1930s, Kurt Godel came along and smashed a hole in Russell's approach by showing that, given a sufficiently powerful formal system, one will always find unprovable truths and irrefutible falsehoods. So mathematics was, by that line of reasoning, incomplete.

This leaves the door open to a variety of critiques, the most relevant of which is that it is automatically not universal. After all, how could it be - there are things missing! We can't prove everything that is true, and we can't disprove everything that is false!

Godel's argument tells us that we are unable to describe the universal laws of nature using non-universal and incomplete mathematics. That dosen't make mathematics useless - it just places a limit on what we can or cannot do. For instance, we cannot use deductive mathematics to describe the laws of nature in their entirety, because we know that any effort to be complete is doomed to failure - by Godel's theorems.

Also, there are some specific areas of mathematics that lead to direct examples of non-universal, but nonetheless consistent interpertations of nature. Take, for instance, Euclidean and differential geometry. Euclidean geometry is the geometry of flat planes, whereas differential geometry describes abstract mathematical notions. It was once thought that Euclidean geometry is "sufficient", and that it is the simplest way of representing spacial relationships. However, as it turns out, differential geometry is actually much more simpler when it comes to dealing with, say, the theory of relativity - even though it is not intuitively connected to our perception of the universe.

So in short, we have two different "geometries", each of which can, supposedly, explain spacial representation. Both are valid, but one is much more useful. Neither is universal. And yet, there is no contradiction.

I don't know about anyone else, but I think this stuff is interesting.

Re:Mathematics not universal?

[Listen Up]

Wow, the shear ignorance in this entire article and book write-up is amazing. Not to truly upset anyone, but everyone here on Slashdot also appears to have a high school alegbra or entry level college mathematics background.

To start with, Mathematics is not just as human as poetry. Where do you get that idea? Yes, pure mathematics (which is my passion in life) is essentially pure thought. BUT, nothing in mathematics is just 'made up'. All mathematics is based on fundamental, logical axioms (truths), and if anything were to violate those axioms, or the completely logical conclusions drawn therefrom, it would not be mathematics. You can think of mathematics as a grand puzzle, with each discovered piece and each mathematical truth found spelling out a larger picture. You can create bogus logic, bogus mathematical problems but it does not make it true mathematics.

You are also confusing human representation with mathematics in your other statements. On a fundamental level, a law is a law, mathematically/physically/logically/universally. The universe is not ruled by human imagination (i.e. completely imaginary human created friend(s) as in religions) and therefore the system to understand our universe has to follow the same sets of rules as the universe (even rules involving possible pure chaos, as in some areas of quantum theory). Without mathematics, our universe and all that lies within it could only be understood on a physical observation level. Mathematics is the language of the universe, it is the language of physics.

For a slightly deeper explanation, let me explain that Mathematics does not involve physical representations as you were taught in HS and earlier. For example, the number 1 as opposed to a capital S to complete addition (which is a logical law) means that 1+1=2 is the same as S+S=* because the system is beyond the physical characters used to represent the logic. The logic would not be different in an alien society. The laws of the universe do not change, therefore the same logic would be implemented. Using a 1 or an S would make no difference.

There is sooooo much more, but just reading this story and people's posts makes me sad on a certain level. One of the oldest "truths" in the world...The person who is always least understood is not an artist, it is a mathematician.

Something I learned from Martin Gardner...

[kzinti]

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...

Why wouldn't math be known across the universe?

[ObviousGuy]

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?

[arbour42]

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 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:math is not universal?

[greatmazinger]

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.

A god with a plan?

[mindstrm]

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.

I rememeber this from...

[gpinzone]

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

[hulap0pr]

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...

First-contact scenarios?

[bravehamster]

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.

Phi

[Rupert]

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.

Why do we need cardinality?

[Xoder]

I see a lot of nay-sayers in this thread talking about "How could some alien not understand that one of one thing and two more of that same thing make three?"

You are assuming that everyone has a concept of cardinality. Realistically, people don't have much of one beyond the number six (yes, there are outlyers for whom eight objects in a group is eight objects not one-two-three-four-five-six-seven-eight objects). If a being had no concept of cardinality, that would make many things more difficult, but many others much easier. This organism would not think of a system as the sum of its parts, but rather as a cohesive whole (or rather the cohesive whole). It is likely that they would be philosophical geniuses compared to us. There are creatures of this type toward the end of Calculating God by Robert J. Sawyer (See your favorite bookseller and/or your local library), and their possible existance is not implausable.

Fibonacci

[Lewie]

This book was a great light read, the math is not difficult and some of the classic paintings and such were really cool to see.

The most interesting part of the book for me was the correlation between Fibonacci and the Golden ratio. As I read it, as you ascend the Fibonacci sequence the ratio between the current number and the one before it converges on the golden ratio. F20 divided by F19 is as near the golden ratio to as many decimal places as any of us have use for, probably.

An interesting "party trick" was also mentioned that I remember vividly. Take any two numbers and add them, then take the new number and the larger of the first two and add them, then take the new sum and the old sum and add, ala Fibonacci. Continue for twenty or so iterations and the 20th number divided by the 19th will be damn close to the golden ratio. This is, I think, because any such construction is a linear multiple of the base Fibonacci set (see prev. paragraph). When you divide, the common multiple falls off and you still get Phi. I thought that was pretty cool. :)