The Golden Ratio: The Divine Beauty of Mathematics

“Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into mean and extreme ratio. The first we may compare to a mass of gold, the second we may call a precious jewel.” Johannes Kepler (p. 15)

While Kepler was a brilliant astronomer and mathematician, his work was also riddled with references to mysticism. The Golden Ratio is a slick, coffee table style book with wonderful illustrations which is very much in the spirit of Kepler. Unfortunately a lot of the math is wrong.

While the author provides the standard definition of \(\varphi\) as the solution to the equation \(\varphi^2-\varphi-1=0\), he then declares that \(\varphi=1.618\) with no indication that this is an approximation. A few sentences later, we learn that \(1.618^2=2.168\). Meisner informs us that like \(\pi\) (which he reports as being equal to \(3.14\)), \(\varphi\) is an irrational number. Given the over-all tone of the book it’s possible that the author obtained the value of \(\varphi\) from a wildly popular novel: the character Robert Langdon in the novel The Da Vinci Code incorrectly defines the golden ratio to be exactly \(1.618\) (Brown 2003, pp. 93–95). Meisner derives the exact value of phi using the quadratic formula on page 51, leaving us to wonder why he didn’t start there as most of the many books on phi do.

There are other troubling errors. On page 28 we are informed that the 3-4-5 right triangle is the only triangle in which the lengths of the sides are in arithmetic progression — clearly any triangle of with legs in a 3,4,5 ratio (6, 8, 10, etc.) will also work. If the sides of a right triangle are \(\sqrt\varphi,1,\varphi\), we obtain a right-triangle known as Kepler’s triangle. Interesting — but we are then informed that this is the only right triangle in which the sides form a three-term geometric sequence.

That being said, there is lots of good math as well. Several of Euclid’s theorems about the mean and extreme ratio (now known as the golden ratio) are wonderfully illustrated along with many of the wonderful mathematical properties of phi. Among these: \(\varphi^{2n}+\varphi^{-2n}\) is an integer for all natural numbers \(n\). My worry is that many casual readers might not be able to sort out the true from the false. Indeed the Race Point Publishing web page displayed several reviews, none of which made any mention of the issues I discussed above. As a result, I can’t recommend this book for its mathematics. For the mathematical joy that is phi, \(\varphi\), \(\pi\), \(e\) & \(i\) by David Perkins is the gold standard in my view.

Gary B. Meisner is the creator of the website www.goldennumber.net and PhiMatrix Golden Ratio Design and Analysis software. His biography at goodreads offers this brief biography: “This (his interest in \(\varphi\)) was inspired by his lifelong interests in mathematics and science, which translated into a career in finance and technology. After earning his CPA, and BS and MBA degrees from two top business schools, he spent most of his career in operational CFO/CIO roles. He is now an independent consultant, and continues his research, writing and software development”.

The software creates a rectangle of variable side lengths whose longer side is split in the Golden Ratio. The user can then overlay this on various images to seek out the Golden Ratio in nature and art. Unfortunately, no physical object can have exact dimensions equal to any real number — hence all instances of \(\varphi\) found in art or nature are approximations. Like Kepler, who found motivation for exploring the orbits of the planets in a very approximate fit between the then-known orbital ratios and the ratios of nested Platonic Solids (using a carefully chosen order in which to nest them) phi-seekers find nearly golden ratios by carefully choosing where they draw their rectangles. While in some cases (the turning ratio of leaves on plants, for example) there are sound mathematical reasons for phi’s appearance we can only speculate as to the intent of the artist whose work appears to utilize phi. Given time and the PhiMatrix software, one can find near-phi ratios in almost any painting, drawing, or building. The meaning, if any, of such discoveries is not clear to me.

The author puts his software to good use finding an incredible number of near-instances of phi. In several cases the choice of where to start and end the golden rectangle seems contrived. For example, on page 129 a vertical golden rectangle is imposed on the U.N. Building in New York. The building’s flat expanse of windows is broken up by three unevenly spaced horizontal rows of windows with contrasting colors. There is a fourth (wider) strip at the top. By placing the foot of the rectangle in the middle of the first set of contrasting windows and its top at the very top of the building the middle strip of contrasting windows divides the entire rectangle in a ratio which is approximately golden. Besides using the middle of one strip and the top of the other to create his rectangle, he ignores the bottom five or six stories. In addition, the top strip (by my rough measurement) is twice as wide as the three intermediate strips — using the middle of the top strip would clearly not create a near-golden rectangle.

In Chapter VI:” A Golden Universe?” the author finds several near-phi ratios in the heavens. For example, if we set the radius of the Earth to be 1, then the combined radii of the Earth and its Moon is approximately \(\sqrt\varphi\). Here’s another: the orbital period of Venus is approximately \(0.6152\) of an Earth year, which is close to \(1/\varphi\). Again, short of some physical reason why these near-phi ratios might be meaningful, they would appear to have very little significance.

The illustrations are wonderful, but there is little to no discussion of the seemingly arbitrary nature of the placement of the golden rectangles. While the errors in each measurement are provided there is no analysis of how the placement was made. For those in search of phi in nature and in art The (Fabulous) Fibonacci Numbers by Posamentier and Lehmann as well as The Golden Ratio by Mario Livio explore much of the same material. In both cases the mathematics is all correct and the presence of phi in art and nature is given a more nuanced treatment.

Richard J. Wilders is Marie and Bernice Gantzert Professor of the Liberal Arts and Sciences at North Central College in Naperville, IL. He is a huge fan of the Fibonacci sequence and the golden ratio. In fact a Fibonacci-based mural adorns the main entrance hall of the Wentz Science Center where he plies his trade as a teacher of mathematics and its history.