Book Review

The

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 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. based on phi, even after he spends a lot of time dis- First, it seems to suggest that somehow the cube cussing phi. A favorite gambit of Livio is to ask is related to phi. Fortunately, when one reads chap- rhetorical questions such as the one on p. 178: ter 4, one learns that “The Golden Ratio, φ, plays “Short of intellectual curiosity, for what reason a crucial role in the dimensions and would so many artists even consider employing the properties of some Platonic solids.” As one might Golden Ratio in their works?” The placement of this expect, the cube is not one of these solids. Another question is interesting because it follows a long sec- oddity is that while phi is present in the various tion generally showing that artists have not been proportions of the dodecahedron, it is interesting using phi in their work in any significant way. Of to note that the dodecahedron in the painting is dis- course, part of the answer to the question is that torted by the perspective that Dali used. Thus the people keep writing books and papers extolling proportions that we see in the painting itself are the aesthetic virtues of phi. With so much being not those of the dodecahedron. Livio makes no at- written about phi by “experts”, many artists feel tempt to actually measure any of the dimensions strong pressure to at least look at phi. or to relate what we see to phi. He then proceeds At times the book appeals to mysticism. It talks to ask: “Why did Dali choose to exhibit the Golden about the “mystical” properties of integers and re- Ratio so prominently in this painting?” This is as- peats a lot of nonsense about 666, the number of tonishing because he did not give any evidence the beast; there is even a ridiculous formula relat- that phi is present in any significant way or that ing 666 and phi. In particular, we are expected to Dali had any interest in displaying phi in his paint- be amazed (p. 23) that sin 666◦ + cos(6 × 6 × 6)◦ is ings. Since Dali wrote about his paintings, one a “good approximation” of the negative of phi. would expect that he would have mentioned his use Doing some “research” of this type, I was amazed of phi if that was of importance to him. to find that tan 666◦ + tan 666◦ (about -2.75276) is Another way of expanding what it means to “sort of” a good approximation of −e. In his dis- “use phi” is to take all applications of cussion of pyramidology, Martin Gardner shows numbers as applications of phi. Of course, one can how in the absence of any rules one can torture express the Fibonacci numbers in terms of powers numbers to come up with just about any result one of phi, but Livio, like most authors writing to as- wants. tonish people, neglects to mention that repre- Livio describes the rectangle construction that senting the Fibonacci numbers in terms of phi phi enthusiasts are so fond of (pp. 85–86). The would make it much harder to “use” them in many fact that one gets a spiral of rectangles is consid- applications. In particular, one can use the Fi- ered amazing. Of course, one can do the same bonacci numbers happily without ever knowing thing with any rectangle by dividing it into two about phi. Most properties of the Fibonacci num- pieces: a smaller rectangle similar to the original bers are best derived from the recurrence relation rectangle, and another rectangle that always has Fn = Fn−1 + Fn−2 , rather than by using phi. The fact some fixed proportion. One can then create a spi- that Fibonacci numbers can be written in terms of ral of smaller rectangles that converges to a point. phi is a special case of the much more general re- Why the spiral derived from phi should be called sults available as part of the theory of linear re- “the Eye of God” is not explained. It is also not men- currence equations with constant coefficients. √tioned that the rectangle having dimensions 2 by On p. 86 Livio notes that if one takes any two 2 is even more amazing, since if one divides the positive integers and forms a series in which each long side in half one gets two rectangles similar to new term is the sum of the preceding two terms, the original rectangle instead of just one rectangle then eventually the ratio of a term to the preced- and a square, as one does with phi. ing term converges to phi. He holds this out as an In addition to its tendency to exaggerate the amazing fact but does not mention that in general “uses’’ of phi, the book contains outright errors. For if one picks any linear recurrence to generate terms example, on p. 19 we are told that “we could even in such a sequence, one will find that consecutive argue theoretically that the fact that 13 is a prime terms converge to some ratio that depends only on number, divisible only by 1 and itself, gives it an

346 NOTICES OF THE AMS VOLUME 52, NUMBER 3 advantage over 10, because most fractions would to find the golden ratio somewhere can be irreducible in such a system.” Given that divis- alter two numbers by ±1% and alter their ibility properties are independent of the base, this ratio by roughly ±2%. statement makes no sense. I was surprised to find the following discussion On p. 116 the book says that “Jacques Bernoulli’s in Livio’s book (p. 47) without attribution: association with the Golden Ratio comes through another famous curve.” The curve referred to is the The second point that is often ignored logarithmic spiral. However, the definition of the by the too-passionate Golden Ratio afi- logarithmic spiral does not depend on phi, as can cionados is that any measurements of be seen from its equation r = aecθ in polar coor- lengths involve errors or inaccuracies. It dinates, where a>0 and c>0. This curve spirals is important to realize that any inaccu- infinitely often in both directions if −∞ <θ<∞. racy in length measurements leads to a If one permits c =0then one gets a circle. One can yet larger inaccuracy in the calculated certainly use phi as a parameter, but clearly one can ratio. For example, imagine that two also use any other positive number as a parame- lengths, of 10 inches, each, are mea- ter. It is seriously misleading to claim that the sured with a precision of 1 percent. This properties of the logarithmic spiral somehow de- means that the result of the measure- pend on phi. Even though the book has ten math- ment of each length could be anywhere ematical appendices that contain formulas, between 9.9 and 10.1 inches. The ratio nowhere in the book does the formula for the log- of these measured lengths could be as arithmic spiral appear. Of course, the formula for bad as 9.9/10.1=0.98, which repre- the logarithmic spiral would reveal that the curve sents a 2 percent inaccuracy—double has no special dependence on phi. The claims about that of the individual measurements. the logarithmic spiral being related to phi are re- Therefore, an overzealous Golden Num- peated at several places in the book. berist could change two measurements Chapter 8 has some interesting material about by only 1 percent, thereby affecting the tilings and quasi-crystals. It is a shame that this ma- obtained ratio by 2 percent. terial is not developed with more technical details. Even though Livio is aware of my paper and Chapter 9, the final chapter, contains a long dis- quotes it in various places, it is not even men- cussion about the unreasonable effectiveness of tioned in the notes for the chapter where the pre- mathematics. The breezy way in which discussions ceding paragraph appears. This chapter also dis- of phi, the Fibonacci numbers, god, relativity, and cusses the Great Pyramid and seems to follow the string theory all roll into one another serves to outlines of the discussion in my paper, again with- glorify the role of phi. out any attribution. For example, compare p. 6 of The book also suffers from sloppy scholarship. my article with p. 56 of Livio’s book. As in my In several places it follows my paper closely with- paper he includes the link to Martin Gardner’s dis- out giving any attribution. For example, a key point cussion of pyramidology, which is the crank dis- that I addressed in my paper (p. 5) was to develop cipline of predicting the future by playing around some way of determining whether a measurement with various measurements from the Great Pyra- can actually be an indicator of the presence of phi. mid. In my paper I pointed out that some of the In my paper I proposed that people use a ±2% “facts” that Martin Gardner used in his classic range around phi to at least treat the claim of the book, Fads and Fallacies in the Name of Science, to presence of phi as being worthy of consideration. debunk the pyramidologists are actually based on I gave the rationale for this as follows: their work. Again, no citations are given to my work. To his credit Livio concludes that the Golden Another point overlooked by many Ratio was most likely not consciously incorporated golden ratio enthusiasts is the fact that in the design of the Great Pyramid. measurements of real objects can only I think that Livio lost a great opportunity. If he be approximations. Surfaces of real ob- had focused on the mathematics of phi and spent jects are not perfectly flat. Furthermore, less time on trying to astonish people with dubi- it is necessary to specify the precision of ous claims, he would have done the mathematical any measurements and to realize that in- community a great service. Given his ability to accuracies in measurements lead to write, he would have also produced a much more greater inaccuracies in ratios. For interesting book. example, a ±1% variation in the mea- surement of two lengths can lead to a roughly ±2% variation (0.99/1.01 ≈ 0.98 to 1.01/0.99 ≈ 1.02) in the ratio that is computed. Thus someone eager

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