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Book Review: the Golden Ratio, Volume 52, Number 3

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Book Review: the Golden Ratio, Volume 52, Number 3 Book Review The Golden Ratio Reviewed by George Markowsky The Golden Ratio were being made about Mario Livio the golden ratio. Broadway Books, 2003 The results of my re- Paperback, 304 pages, $14.95 search were published in ISBN 0-7679-0816-3 “Misconceptions about √ the Golden Ratio” (The The number (1 + 5)/2=1.618 … is widely College Mathematics known as the golden ratio, φ and phi. Phi appears Journal, Vol. 23, No. 1, in many different equations and formulas and has Jan. 1992, 2–19). This many interesting properties. Many people have paper debunks many of heard marvelous tales about phi and how it per- the more prominent meates art and nature. My first exposure to phi was claims about phi and in a comic book entitled Donald in Mathmagic documents their perva- Land, which later became an animated cartoon sive presence in the seen by millions of people. As I grew up I kept see- mathematical literature. ing the same “facts” repeated in many different For example, the name places, including popular books on mathematics, “golden ratio” is a nineteenth-century creation and is not an ancient name for phi. Furthermore, it various mathematics textbooks, newspapers, and does not appear that phi was used to design either even in scholarly papers. It seemed as if every- the Great Pyramid or the Parthenon. For example, body knew these basic “facts” about phi. the Parthenon is 228 feet and 1/8 inch long, 101 Around 1990 I decided to give a talk to the Uni- feet and 3.75 inches wide, and 45 feet and 1 inch versity of Maine Classics Club and thought that the high. Taking the obvious ratios of length/width golden ratio would be a fascinating topic for this and width/height yields the number 2.25, which is audience. During the preparation of the talk I col- quite far from phi, which is 1.618…. The number lected all of the usual stories about the golden 2.25 = 9/4 is the ratio of two squares, and further ratio being used to design the Great Pyramid and study indicated that the gate to the Acropolis was the Parthenon, as well as about its aesthetic prop- built using this same ratio. erties and its use by painters. I found the references It also does not appear that Leonardo da Vinci to be quite vague, and in the process of trying to used phi, nor is phi present in the proportions of make my talk more precise, I actually began to the United Nations building in New York. look up measurements of buildings. Much to my Furthermore, in a large number of informal audi- surprise, the results did not support the claims that ence participation events, I have found that peo- George Markowsky is professor of computer science at the ple do not pick golden rectangles more frequently University of Maine. His email address is markov@ than others (in fact, they are often picked less fre- maine.edu. quently than others), so that the statements about 344 NOTICES OF THE AMS VOLUME 52, NUMBER 3 the aesthetic superiority of phi do not stand up to analyzes them in the same manner. Later in the empirical tests. These claims and others are de- chapter he reproduces a diagram from my paper molished in my paper in some detail. (he attributes the diagram to me, but does not give Since publishing my paper, I have tried to get a reference) that shows forty-eight rectangles of dif- people, in particular mathematicians, to tell the ferent proportions that I have used a number of truth about phi. Phi has many interesting mathe- times to ask people which rectangle they find most matical properties that deserve to be brought to the pleasing. public’s attention. It is, however, a disservice to On p. 183 of his book, Livio states: mathematics to mix the interesting properties of phi with dubious claims about its importance in art, You can test yourself (or your friends) architecture, human anatomy, and aesthetics. on the question of which rectangle you Mario Livio’s book, The Golden Ratio, is a broad prefer best. Figure 84 shows a collection survey of the properties of phi. The book is some of forty-eight rectangles, all having the 270 pages long, counting ten appendices, and same height, but with their widths rang- bounces along describing various mathematical ing from 0.4 to 2.5 times their height. properties of phi, while at the same time trying to University of Maine mathematician astonish the reader. It is the constant desire to as- George Markowsky used this collection tonish the reader that gets Livio into trouble and in his own informal experiments. that is undoubtedly the source of the subtitle: The Interestingly, Livio does not reference my paper Story of Phi, the World’s Most Astonishing Number. and does not quote my conclusion: When I first heard about this book, I was hopeful that it would finally put many of the bogus stories In the experiments I have conducted so about phi to rest, but unfortunately this book does far, the most commonly selected rec- not quite do so. tangle is one with a ratio of 1.83. For example, in his discussion of the Parthenon, Also, Livio does not point out that there are ac- Livio quotes from my paper and gives proper at- tually two golden rectangles in the diagram—one tribution. However, I believe that he waffles on the is oriented with the long dimension horizontal and issue of whether phi was used in the design of the the other with the long dimension vertical. Parthenon. On p. 74 he states: Livio could have performed a valuable service to So, was the Golden Ratio used in the the mathematical community had he written an ac- Parthenon’s design? It is difficult to say curate book about phi that treated it in a balanced for sure. While most of the mathemat- manner and that consistently and thoroughly de- ical theorems concerning the Golden bunked the various misconceptions about phi that Ratio (or “extreme and mean ratio”) ap- continue to circulate. Throughout this book, Livio pear to have been formulated after the struggles with the problem of wanting to “amaze” Parthenon had been constructed, con- the public without going too far and losing re- siderable knowledge existed among the spectability, but unfortunately he does not suc- Pythagoreans prior to that. ceed in solving it. I take strong issue with his conclusion that it is He deserves credit for surveying a wide range difficult to say for sure whether phi was used in of sources about phi, but in my opinion he is very the construction of the Parthenon. It seems to me inconsistent in how he uses them. In some cases that to even entertain the notion there has to be he does an effective job of debunking nonsense, but some reason to believe that it was true. It is clear in others his debunking is halfhearted. In some that the Greeks were not as enamored of phi as peo- cases he omits data that would be harmful to es- ple became once it received the name golden ratio tablishing phi as the “most astonishing” number. in the nineteenth century. Calling phi division into Unfortunately, he also seems interested in spawn- mean and extreme ratio does not generate great ex- ing some new myths. citement on the part of artists and architects. I For example, on p. 9 he discusses Salvador Dali’s have found no credible evidence that phi was ever “Sacrament of the Last Supper”. The first “fact” that used by Greek artists and architects for any pro- we are presented with is that the canvas measures ject at all. approximately 105.5 inches by 65.75 inches, which In Chapter 7 of his book Livio discusses the “are in a Golden Ratio to each other.” The ratio possible presence of phi in various paintings and 105.5/65.75 is approximately 1.605, which is close its role in aesthetics. Again, he closely parallels my to, but not equal to phi. If it was important for the paper but does not cite the paper either in the text painting to have phi as the ratio of its width to or in the notes to the text. In his discussion of height, why not use a canvas of size approximately Leonardo da Vinci, Livio reproduces exactly the 106 inches by 66 inches, which has a ratio of 1.606, painting and drawing discussed in my paper and which is even closer to phi? We are next told that: MARCH 2005 NOTICES OF THE AMS 345 Perhaps more important, part of a huge the recurrence relation and not on the starting dodecahedron (…) is seen floating above points. In particular, pick any two positive integers the table and engulfing it. … As we shall and use the formula Fn = Fn−1 +2Fn−2. One will see in Chapter 4, regular solids (like the find that eventually the ratio of consecutive terms cube) that can be precisely enclosed by will approach 2. In this regard, phi is no more a sphere with all their corners resting amazing than just about any other number. on the sphere, and the dodecahedron in Livio devotes a fair amount of space to dis- particular, are intimately related to the cussing Luca Pacioli and his work on the “Divine Golden Ratio. Proportion”. Livio notes that Pacioli ends up rec- ommending a system of proportions for art not This paragraph is odd for a number of reasons.
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