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The Golden Section

Hans Walser
Mathematical Association of America
Publication Date: 
Number of Pages: 
[Reviewed by
Underwood Dudley
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In the first sentence of his Foreword, the author says, “The Golden Section has turned up, since antiquity, in many aspects of geometry, architecture, music, art, and even philosophy.” Before getting around to what is in his book I want to comment on this. It might be misleading.

As most readers of MAA Reviews know, the golden section is that which results when a line segment is divided into two pieces such that the ratio of the whole to the larger is the same as the ratio of the larger to the smaller. That is, if the segment has length \(1\) (which it may as well) and the larger piece has length \(x\), then \(1/x=x/(1-x)\). This gives \(\frac{-1+\sqrt{5}}{2} = .618033988\). The reciprocal of this, \(1.618033988\), has been denoted by a variety of symbols, but \(\phi\) I think is the most common and the one I will use. Professor Walser prefers \(\tau\), which is also fine, and lets \(\rho\) denote \(\phi-1\). The number \(\phi\) has many properties, including \(\phi^2=\phi+1\), which is the same as \(\phi=1+1/\phi\), leading to the unique and picturesque continued fraction \[ \phi=\frac{1}{1+\frac{1}{1+\frac{1}{1+\dots}}}\]

Many readers of MAA Reviews may assume, since Professor Walser says that it has turned up in so many places since antiquity, that the golden section has been part of human consciousness for a long time. This is not so. The earliest use of “golden section” was in a German book published in 1835, a consequence, as one commentator put it, of “German romanticism”. It did not appear in an English mathematical work until 1898. In 1509, Luca Pacioli called it the “divine proportion” but before that it was referred to, when it was referred to at all, as Euclid did, as “division in extreme and mean ratio.” Pacioli’s reasons for the proportion’s divinity had nothing to do with geometry, architecture, music, or art. His reasons were all mystical and must be understood mystically. Roger Herz-Fischler gives them in his valuable A Mathematical History of Division in Extreme and Mean Ratio. “The first is that it is one only and not more.” “The second attribute is that of the Holy Trinity.” The third was that \(.618\dots\) is irrational. The other reasons were equally mystical, that is to say irrational and not to be understood by using reason. Herz-Fischler says (page 150 of the Dover edition), “I insist upon the fact that Pacioli does not recommend the use of [the golden section] in determining the proportions of works of art and architecture.” The belief that \(\phi\) has, through the ages, been purposely built into buildings, paintings, or sonatas, or that it is somehow part of the world or of our brains is one that has sprung up fairly recently.

For example, the idea that a rectangle with dimensions\(\phi\) and \(1\) (or, equivalently, \(1\) and \(\phi - 1\)) is the one that is aesthetically most pleasing seems to have gotten a start in the 1860s, though too many authors repeat this as if it were part of the wisdom of the ages. (In what follows, I am drawing on another invaluable resource, George Markovsky, “Misconceptions about the golden ratio,” College Mathematics Journal 23 (1992), 2–19.) Back then, one Gustav Fechner presented subjects with ten rectangles and asked them which they thought was the nicest. The rectangles varied from a square to one whose sides had the ratio of 2 to 5 — that is, with aspect ratios from 1 to .4. The three rectangles in the middle, those with aspect ratios .57, .62, and .67, were chosen by 76% of the subjects.

Well, of course. Squares are dull, long flat rectangles look as if they had been stepped on, and tall skinny rectangles make us nervous-they look as if they may fall over any minute-so naturally something in the middle gets picked. But the golden section has nothing to do with it. Further studies have shown that \(\phi\)-rectangles are in fact not the prettiest. There is now (at least there was when this was written) a Web poll on the subject, which may be found at, that has rectangles with width 54 and heights 65, 68, 72, 77, 81, 87, 96, 108. The sixth is close to a golden rectangle, but it has been picked as the most pleasing rectangle by only 12% of the 1501 respondents to the poll. The 54-by-72 rectangle is the clear winner, with 30% of the votes. (The percentages of respondents choosing each of the eight rectangles are, respectively, 9, 3, 30, 16, 19, 12, 5, 8.) If you measure books, a handy source of rectangles, you will find that almost none has dimensions that come close to those of a golden rectangle. A golden-rectangle book looks too tall and skinny. Professor Walser’s book measures \(9\) inches by \(6\) inches, not \(6\phi\) inches by \(6\) inches.

The facts are similar for the other bits of nonsense that are attached to \(\phi\). Someone once said, perhaps on mystical grounds, that the height of the human body was divided at the navel in the ratio of \(1\) to \(\phi\), but it is not so, even for the most aesthetically pleasing of humans. People have gone to the trouble of measuring navel heights, one not being able to resist mentioning a “ticklish subject,” and found that the ratio is, on the average, \(1\) to something larger than \(\phi\) The ancient Egyptians did not use \(\phi\) in designing their pyramids. How could they, since they had no notion of irrational numbers? The Parthenon does not have \(\phi\) built into it, as many authors have asserted — the evidence seems rather better for \(9/4 = 2.25\). It must be the human thirst for marvels and wonders, combined with the human ability to see patterns where there are none, that accounts for such assertions.

Artists did not use \(\phi\) when composing, even though the writings of those whose heads have been turned by \(\phi\) are full of pictures of pictures, from Michelangelo to Seurat and beyond, with golden rectangles in them. I would undertake to find a golden rectangle in any picture, even a Jackson Pollock: pictures have so many points of significance in them that would not be hard to find four that come close to forming a rectangle similar to a \(1\)-by-\(\phi\) one. Indeed, on the cover of Professor Walser’s book we see a reproduction of da Vinci’s Mona Lisa with three golden-looking rectangles superimposed on it. Two of them may have been inserted for decoration only, but the one framing the subject’s face looks as if it was there to illustrate Leonardo’s use of \(\frac{-1+\sqrt{5}}{2}\). But it doesn’t quite fit. Its right-hand side almost goes to the start of the hair but not all the way; its lower edge is a bit below the chin. Did Leonardo lay out a golden rectangle before starting to paint? I suppose we cannot know for sure, but I think that the face came first, not any geometry.

Some writers have claimed that Mozart used \(\phi\) to divide his piano sonatas into parts. John Putz, writing in Mathemtics Magazine (“The golden section and the piano sonatas of Mozart,” 68 (1995), 275–281), convincingly showed that \(\phi\) is not in the sonatas by giving a mathematical explanation of how misguided people could conclude that it was. But he could not resist writing, “Mozart may have known of the golden section and used it.” He took that back, sort of, in his next sentence, but ended with, “Perhaps the golden section does, indeed, represent the most pleasing proportion.” Thus does nonsense rise up, though feebly, even in publications of the MAA.

Though \(\phi\) is not part of the physical world, there is no doubt that it turns up in many different places. The reason for that is easy to see. It is an instance of Richard Guy’s Law of Small Numbers, that there are not enough small integers available for the many tasks assigned to them. The origin of \(\phi\) is \(x^2=x+1\), a quadratic equation with very small integer coefficients. Quadratic equations occur often in mathematics, so it is no surprise that among them the one for \(\phi\) would occur more than once. If \(\phi\) were a root of \(25x^2=26x+24\), then I would be surprised to encounter it in more than one place, but as it is I am no more astonished than I am that there are three dimensions, three ships of Christopher Columbus, three subjects in the trivium, three degrees of burns, and three bags of wool (one for my master, one for my dame, and one for the little boy who lives down the lane). Three comes up a lot, and so does \(x^2=x+1\).

Professor Walser, to his credit, indulges in no \(\phi\)-nonsense at all. His book is devoted to simple mathematics, mostly geometry, that in one way or another involves \(\phi\). It is clearly written and contains material that will be new to most readers of MAA Reviews. For example, let us construct a fractal by starting with a line of length \(1\) and adjoining to it two other lines of length \(s\), emanating from the end of the line at 120-degree angles, making a \(y\)-shape. The proceed by self-similarity, constructing two more smaller \(y\)s with branch length \(s^2\), then four more, and so on. If \(s\) is too small the branches won’t touch, if s is too big they will overlap, but if \(s\) is just right there will be in the limit no space between them. Just right is \(s = 1/\phi\). There are many more examples with a wide variety of fractals.

For another example, take an ellipse and construct a circle whose diameter is the line segment joining the foci of the ellipse. For which ellipse will the area of the circle be the same as the area of the ellipse? For the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2}=1\) with \(\frac{a}{b}=\phi\). The reason for this, of course, is that the geometry gives \( \left(\frac{a}{b}\right)^2=\frac{a}{b}+1\).

The regular pentagon is full of \(\phi\)s, and the Fibonacci sequence is also \(\phi\)-ful. Though \(\phi\) is not everywhere, it keeps popping up. Inscribe a regular icosahedron in a unit cube and you will find that its side length is \(1/\phi\). The book is filled with delightful material like that, clearly explained and plentifully illustrated, all at a level well within the grasp of any reader of MAA Reviews. Here is one final example, selected to show that \(\phi\) does not occur only in geometry. Let us play a coin-tossing game where the first player to toss a head wins. The game is biased in favor of the player who tosses first, of course, so let give the second player two tosses after each toss of the first player. Even taking turns as A, B, B, A, B, B, A, … leaves the advantage with A, so let us weight the coin so that the chance of tossing a tail is greater than \(1/2\). The probability of a tail that makes the game fair is, as you may have guessed by now, \(1/\phi\).

Get this book, have fun with \(\phi\), and marvel at its ubiquity in mathematics. The number needs no \(\phi\)-foolishness about rectangles and navels to make it, and this book about it, interesting

Woody Dudley ( teaches at DePauw University. His Numerology (an MAA publication) contains some material on \(\phi\).

The table of contents is not available.