Livio: The Golden Ratio
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Mario Livio, The Golden Ratio : The Story of Phi, The World's Most Astonishing Number. New York : Broadway Books, 2002. viii + 294 pages, with appendices, "further reading", and index; illustrated.
This is another book whose title worried me some before I started reading. The "Golden Ratio", usually known by its short name, the Greek letter "phi", has for some time been the object of obsessive mysticism. Weren't ancient buildings like the Acropolis or the pyramids all built adhering to the dictates of the Golden Ratio? Isn't it well known that all those Medieval and Renaissance paintings used Golden Ratios everywhere? Isn't all of nature infused with the Golden Ratio, and doesn't that say something about how we were visited by aliens in ancient times?
Did the author really think that phi was "the world's most astonishing number"? It's debatable; phi is a relatively interesting number, one of the few identifiable irrational, transcendental numbers that does have a way of showing up in nature. Astonishing? Perhaps, but no matter. Its close relationship to the Fibonacci sequence (the sequence of ratios of successive Fibonacci numbers converges to phi) is probably enough for mystical status but also probably points to the reason why phi is so popular with nature. The author discusses many interesting and illuminating examples.
...the Golden Ratio is the most irrational of all irrational numbers in the following sense. Recall that the Golden Ratio is equal to the continued fraction composed entirely of 1s. That continued fraction converges more slowly than any other continued fraction. In other words, the Golden Ratio is farther away from being expressible as a fraction than any other irrational number. [pp. 113—114]
Livio also spends a fair amount of attention examining claims that the Golden Ratio
- infuses the design and construction of the pyramids;
- was ubiquitously used by the Greeks in public buildings;
- was the basis of composition in the paintings of numerous celebrated Renaissance painters; and
- is known to be the "most attractive" proportion that people perceive.
Numerous authors have claimed that the Golden Rectangle is the most aesthetically pleasing of all rectangles. The more modern interest in this question was largely initiated by a series of rather crankish publications by the German researcher Adolph Zeising, which started in 1854 with Neue Lehre von den Proportionen des menschlichen Körpers (The latest theory of proportions in the human body) and culminated in the publication (after Zeising's death) of a massive book, Der Goldne Schnitt (The golden section), in 1884. In these works, Zeising combined his own interpretation of Pythagorean and Vitruvian ideas to argue that "the partition of the human body, the structure of many animals which are characterized by well-developed building, the fundamental types of many forms of plants,...the harmonics of the most satisfying musical accords, and the proportionality of the most beautiful works in architecture and sculpture" are all based on the Golden Ratio. To him, therefore, the Golden Ratio offered the key to the understanding of all proportions in "the most refined forms of nature and art." [pp. 178—179]
The history of such claims is interesting anthropology and the author adopts an admirably analytical and objective stance in analyzing them. Virtually all claims turn out to have no substance to them beyond keen imaginations and wishful longing. Thus, this volume provides a nice antidote to rampant number mysticism.
All the attempts to disclose the (real or false) Golden Ratio in various works of art, pieces of music, or poetry rely on the assumption that a canon for ideal beauty exists and can be turned to practical account. History has shown, however, that the artists who have produced works of lasting value are precisely those who have broken away from such academic precepts. In spite of he Golden Ratio's importance for many areas of mathematics, the sciences, and natural phenomena, we should, in my humble opinion, give up its application as a fixed standard for aesthetics, either in the human form or as a touchstone for the fine arts. [p. 200]
I did have one very small peeve: I would like to have seen more paragraphs that collected thoughts into shorter, more coherent units; I thought the author's paragraphs rambled somewaht. But I got past it easily enough and enjoyed the book.
The author gets high marks from me for scienticity in this book, which I found very readable without sacrificing precision in its analysis. It's a very good example of rational, analytical thinking in action.
-- Notes by JNS