Review of the golden ratio

Euclid defined what has come to be known as the Golden rule as “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser. ” In the beginning of the book Mario Livio quotes Einstein saying, “The fairest thing we can experience is the mysterious. It is the fundamental emotion, which stands at the cradle of true art and science. He who knows it not and can no longer wonder is as good as dead, a snuffed out candle. ” For Mario, the Golden ration evokes absolute wonder so he tries to share this with in this book.

He illustrates that it has inspired thinkers from all disciplines like no other number in the history of mathematics. On the contrary one wonders if one number even an extra-ordinary one as Mario claims Phi to be could sustain a whole book. But his in depth knowledge of the arts is sufficient and he moves across history and art to make alive the material he requires. He illustrates that what makes a surprise party effective is the whole gratification that we all feel when suddenly exposed to unexpected appearances.

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He uses this analogy to express that Mathematics and the Golden ration have this same effect. Majority of readers can conjure up dim memories from geometry class of the irrational number pi. A Theoretical astrophysicist Livio gives pi’s overlooked relative phi it’s deserved attention with this account, the first on the subject written for the layperson. Phi is the golden ratio of antiquity (1. 6180339887), an infinite number so famous for its humongous qualities that in the 16th century it was declared the divine proportion.

It is compared to phenomena as detached as the petal arrangements of roses, the breeding patterns of animals like rabbits and our galaxy’s shape. The number is also claimed to be foundational in the in the design of the Great Pyramids, the composition of the Mona Lisa and the construction of Stradivarius violins. Mario Livio (The Accelerating Universe) carefully analyses these concepts and does not hesitate to discredit myths perpetuated by overzealous enthusiasts he chooses to call the “Golden Numberists. ”

This is an engaging recount of mathematics because is addresses such crucial questions as the geometric basis of aesthetic pleasure and the obscure nature of mathematical objects. The book also includes useful diagrams and illustrations of various works under discussion. Livio is gifted with an accessible, entertaining style and he easily uses five pages for one chapter about an extended discourse on prime numbers then shifts to a clever Oscar Wilde quote about beauty to a humorous anecdote about Samuel Beckett then settles with a clear explanation of G”del’s incompleteness theorem.

With an in-depth guide to the history of ideas as illustrated by Livio, even the math-phobic can experience the unexpected shock and gratification of scientific discovery. Throughout history, thinkers ranging from mathematicians to theologians have analyzed the mysterious correlation between numbers and the nature of reality. Euclid discovered the Golden Ratio, more than two thousand years ago because of its imperative role in the construction of the pentagram. Magical properties have been attributed to this phenomenal number. From that time the ratio has been identified in the most astonishing variety of places.

Psychological studies have even investigated if the Golden Ratio is the most aesthetically pleasing proportion extant. It has even been said that the creators of the Pyramids and the Parthenon employed put it to use in creating these masterpieces. Among the works of art, it is said to feature in Leonardo da Vinci’s Mona Lisa to Salvador Dali’s The Sacrament of the Last Supper. Believe it or not but even poets and composers are believed to have used it in their works. Lately it has been discovered to have a connection in the behavior of the stock market.

The Golden Ratio also tells the human accounts of numerous phi-fixated people, including the followers of Pythagoras who believed that this proportion revealed the hand of God. An astronomer Johannes Kepler, said phi is the greatest treasure of geometry. A Renaissance thinker/mathematician, Leonardo Fibonacci of Pisa; and masters of the modern world as Goethe, Cezanne, Bartok, and physicist Roger Penrose all attest to the power of this number. Wherever his quest for the true meaning of phi takes him, Mario Livio exposes the world as a place where order, beauty, and eternal mystery will always and absolutely coexist.

As the calculations in the book demonstrate, the precise value of the Golden Ratio (the ratio of AC to CB in Figure 2) is the never-ending, never-repeating number 1. 6180339887 . . . , and such never-ending numbers have intrigued humans since time in memorial. One story sates that when the Greek mathematician Hippasus of Metapontum discovered, in the fifth century b. c. , that the Golden Ratio is a number that not a whole number (like 1, 2, 3, . . . ) or even a ratio of two whole numbers (like fractions 1/2, 2/3, 3/4, . . . which are known presently as rational numbers), this absolutely shocked the other followers of the famous mathematician Pythagoras (they were called the Pythagoreans). Their worldview (which is greatly described in the second chapter of the book ) was based on total admiration for the arithmos. (This is the intrinsic properties of whole numbers or their ratios–and their presumed role in the cosmos. ) The revelation that there exist figures, like the Golden Ratio, that go on and on forever without displaying even the slightest repetition or pattern caused an absolute philosophical crisis.

Legend even states that, overwhelmed with this humongous realization, the Pythagoreans sacrificed a hundred oxen in horror although this is highly unlikely. This is because they were strict vegetarians. What should be noted however is that many of these stories are entirely based on poorly documented historical material due to the date factor. The exact date of the discovery of numbers that are neither whole nor fractions is not known with absolute certainty. On the contrary, some researchers place the discovery in the fifth century b. c. which is consistent with the dating of the stories just described.

What is however clear is that the Pythagoreans believed that the existence of such numbers was so horrific that it must evidence some sort of cosmic error, one that should be suppressed and kept secret from the human race. “The fact that the Golden Ratio cannot be expressed as a fraction (as a rational number) means simply that the ratio of the two lengths AC and CB in Figure 2 cannot be expressed as a fraction. ” This implies that however hard one may try or search, they cannot find some common measure that is contained, let’s say, 31 times in AC and 19 times in CB.

Such two lengths that have no common measure are called incommensurable. The discovery that the Golden Ratio as an irrational number was in turn the discovery of incommensurability. Still on the Pythagorean Life, the philosopher and historian Iamblichus, who was descended from a noble Syrian family, describes the violent reaction to this discovery. This thoroughly engaging piece of work expertly demonstrates to the reader that, as Galileo put it, the universe is without doubt, written in the language of mathematics.