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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.
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.
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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
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...
The movie PI is also a very compelling watch for those who are interested.
Free XBox, PS2
More here.
REM Old programmers don't die. They just GOSUB without RETURN.
There is a really great page that explains the relation between the Golden ratio and the Fibonacci numbers here
The fibonacci number is the series 1,1,2,3,5,8... where every number is the sum of the two numbers before it. What does this have to do with the golden ratio? Everything! Just check it out, you'll be amazed.
Different civilizations use different math systems, but they're all related by basic concepts. Addition is addition, no matter how you write it down. Any two math systems can be translated back and forth.
-B
The flaw in that line of thinking, which many on /. are making, is assuming that what we percieve singularly is similarly percieved by another species. Let's take a thought walk for a moment... First, we percieve an object, say a book, at a single position in time/space. While we can percieve the entirety of it in space, our perceptions cannot percieve simultaneously the entirety of it's temporal measurement. Therefore, we see on book in the now.
Now another lifeform comes along, one which can percieve the entirety of the book in time/space. They percieve not only a different book than we are capable of, but further, they may percieve each temporal book as a seperate item, just as we percieve spacially translated objects as seperate. So where we see a single book, they see an infinite number of books. We can only assume that their method of counting would differ from ours, or that we would be unable to correlate ours to theirs because we can not percieve the many, only the one.
Assuming that another specias percieves the universe the way we do is the height of hubris, and the largest flaw in alien contact scenarios. Our mathematical beauties when percieved on a larger scale may be no more than a mere curiosity, instead of the vaunted unchanging laws.
Just a thought.
You can have it fast, accurate, or pretty. Pick any 2.
Two other interesting books: Zero: The Biography of a Dangerous Idea by Charles Seife.
Trigonometric Delights by Eli Maor.
Both books cover the a lot of historical ground in mathematics.
--- Ban humanity.
a la "I refute it thus?" ... except Samuel Johnson's method was a lot more friendly than yours!
Fun problem:
A. Take a number. Add 1.
B. Take a number, square it.
For what number are the answers from (A) and (B) equal?
By now, you know the answer from the context of the question.
The book is a pretty good read, though it drags in a few places (the draggy places are still readable).
One other book I learned more from is called
An Imaginary Tale by Paul J. Nahin , which is the story of the imaginary number (square root of minus 1). It is written in clear language and is intended to shed light on the topic, rather than mysticize the "imaginary" phrase in its name.
If this is anything similar to the book "A history of Pi" by Petr Beckmann then I'm all over it. Petr Beckmann was able to put down some ideas and link them together in that book that I was very captivated by. It was technical enough to hold it's own, but he explained it well enough for most people to understand. At least explain it to the people who had enough interest and brain capacity to be interested in the subject and read the book.
a * (sqrt(5) + b)
and:
a * (2.23606797749978969640 + b)
are exactly equivalent computationally using double-precision floating-point arithmetic on a 32-bit processor. In fact, the second statement will execute more quickly because there's no need to perform an expensive sqrt() operation.
The reason that these statements are equivalent is that floating-point calculations can't deal with irrational numbers. Heck, they can't even deal with most rational numbers. A lot of people don't realize this, but the even simple fraction 1/3 can't be represented exactly as a floating point number. All the function call sqrt(5) does is calculate the floating-point number that's closest in value to the square root of 5. If you can provide that to the program as a constant, then you save an unnecessary computation step.
Obviously for readability you'd define a named constant (e.g., SQRT5) rather than just using the number in place explicitly, but my point is that you're still better off defining the value of that constant as 2.23606797749978969640 rather than as sqrt(5).
No. I would agree he is a solopsist. I would contend there is a difference between one who believes there is an intangible ideal underlying perceived reality and one who believes reality is all in his own mind.
For instance, as a physicist I may admit to there being number under everything and idealized states that do not exist in reality. This does not, however, mean that I admit to the nonexistence of matter or objective reality at all.
As a Buddhist I might ask "Does a dog have Buddha nature?"
KFG
If your model and my model differ, then we disagree about reality, but we are probably not correct (either of us).
I agree with you completely, except that the very point of postmodernist approaches is that there is no such thing as correct or incorrect; there is observation and perception, which are sometimes shared and sometimes different. Science does NOT define reality; it simply provides a set of observations that are repeatable by anyone who follows the correct procedures and uses the right tools. These reproducible "facts" are of course of extreme use to our society, and I don't think postmodernists would claim that science is anything less than a boon to our society and our understanding of our universe. However, they would argue that there is no such thing as "correct" and "incorrect" because our understandings of such are merely anchored within the model we hold of the world around us.
So DJerman, I agree completely with you, and I think you agree completely with postmodernist critique as it is understood in the academy. The "Individual creates reality" kinds of statements are made by people who don't understand the inherent critique made by postmodernism.
Remember, postmodernism isn't really a philosophy per-se because even its strongest proponents (Michel Foucault should immediately come to mind) realized that it was not very useful for creating new interpretations of human nature. Rather, it is a tool for criticism and critique, a way of addressing the shortcomings of modernist notions of progress and the human condition. It was often said during Foucault's life that his own works of history were not really postmodernist, because in order to make a strong historical argument he had to betray many of the tenets of the postmodernist "we can't really know anything" dilemma.
Of course, I've been immersed in postmodernism for so many years that I'm not even sure I exist anymore. I don't call myself a postmodernist - I think the term has no meaning since postmodernism is not really an ideology (although it is taken to be one by people who have a little knowledge of it and think it sounds good); however, I think the postmodernist critique is an important one and very relevant for addressing many of the shortcomings of our perception of our condition.
And that's how we turn short observations into 400-page works in academia...
I did not design this game/I did not name the stakes/I just happen to like apples/And I am not afraid of snakes-AniD
> math may not be a universal concept spread
> across the galaxy...
For those who aren't reading between the lines...
Mathematics is one of the very strong arguments
for a Creator. Math didn't "evolve". The idea
that mathematics in not universal is an attempt
to argue against the idea (or implication) of
a Creator. The basic idea being that math
"evolved" in ways peculiar to the local
environment much as biological organisms (read
Humans) have evolved in ways peculiar to the
local enviroment.