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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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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